Schrödinger Equation – Fundamentals Aspects

#### **Chapter 1**

## Schrödinger Wave Equation for Simple Harmonic Oscillator

*Noor-ul-ain, Sadaf Fatima, Mushtaq Ahmad, Muhammad Rizwan Khan and Muhammad Aslam*

#### **Abstract**

In physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke's law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.

**Keywords:** harmonic motion, frequency, pendulum, displacement, amplitude

#### **1. Introduction**

A system that uses simple harmonic motion (SHM) is known as a harmonic oscillator.

A physical system called a harmonic oscillator experiences a restoring force proportionate to the displacement when it is moved away from its mean position.

A wave equation that describes the behavior of quantum particles is the Schrödinger equation. A harmonic oscillator's energy levels can be demonstrated by the Schrödinger equation to be quantized, which means that they can only take on specific discrete values. The Schrödinger equation has the effect of restricting the possible energies that an oscillator that is harmonic can have [1, 2].

A physical system known as harmonic oscillator oscillates at a frequency proportional to the displacement from its equilibrium position and is governed by a restoring force *Fr*. The *Fr* is proportional to the displacement from its mean position. This means that the system tends to return to its equilibrium position when disturbed from it, and the rate at which it oscillates is determined by the strength of the restoring force and the mass of the system. An equation of simple harmonic motion which is sinusoidal function of time with constant amplitude and frequency can be used to describe the

motion of harmonic oscillator [1, 3]. The two examples of harmonic oscillator are mass connected to the spring and a simple pendulum. Harmonic oscillators are important in physics and engineering because they provide a useful model for many physical systems and can be used to analyze and predict the behavior of those systems [3, 4].

#### **2. Classical behavior of simple harmonic oscillator**

The simple example of linear harmonic oscillator is a mass attached to a wall by means of a spring as illustrated in the following **Figure 1**.

**Figure 1.** *Shows the experimental device for the study of the spring-mass system [1].*

#### **2.1 Expression for potential energy of simple harmonic oscillator**

Hooke's law states that the force required to stretch or compress a spring is proportionate to the distance extended or compressed from its original length. Mathematically, this relation can be expressed as:

$$F \propto \infty$$

$$F \propto -\infty$$

$$F\_r = -k\infty \tag{1}$$

Where, *Fr* is the force applied to the spring, *x* is the displacement of the spring from the original length, and *k* is a constant which is known as spring constant and represents the stiffness of spring [5].

Hooke's law applies to all elastic materials, not just springs. It is an important concept in physics and engineering because it helps to understand and predict the behavior of systems that involve elastic materials, such as springs, rubber bands, and other materials. Hooke's law is also the basis for the design of many mechanical systems, such as shock absorbers, suspension systems, and other devices that rely on the properties of elastic materials [6, 7].

When an object is displaced from its equilibrium position, a restoring force acts on it to push or pull it back toward that position. The *Fr* is directly proportional to the displacement from the equilibrium position and also acts in opposite direction [5]. This force is present in many physical systems, such as springs, pendulums, and massspring systems, and it plays a vital role in the behavior of these systems [3, 4].

*Schrödinger Wave Equation for Simple Harmonic Oscillator DOI: http://dx.doi.org/10.5772/intechopen.112381*

$$F = -\frac{dv}{d\mathbf{x}}\tag{2}$$

∵Force *F* can be expressed as negative derivative of potential energy V. The work done in stretching spring to distance dx

$$W = F \times \text{distance}$$

$$P.E = F \times d\mathbf{x}$$

$$-dv = F \times d\mathbf{x}$$

$$dv = -F \times d\mathbf{x}$$

$$dv = -F \times d\mathbf{x}$$

From Eq. (1)

$$F = -k\mathbf{x}$$

$$dv = k\mathbf{x} \times d\mathbf{x} \tag{3}$$

$$\text{Integrate Eq. (3) within limits } 0 \to \infty$$

$$\int dv = +\int\_0^x kx dx$$

$$V = k \int\_0^x x dx$$

$$V = k \lim\_{0 \to \infty} \frac{x^2}{2}$$

$$V = k \left(\frac{\varkappa^2}{2} - \frac{0}{2}\right)$$

$$V = k \frac{\varkappa^2}{2}$$

$$V = \frac{1}{2}kx^2\tag{4}$$

Where *x* is the distance from equilibrium position [8, 9].

The plot of potential energy (*V*) of a particle executing simple harmonic motion against displacement from its equilibrium length is a parabola as illustrated in the following **Figure 2**.

#### **2.2 Expression for frequency of linear harmonic oscillator**

The frequency of a harmonic oscillator is the number of complete oscillations or cycles it completes per unit time. The frequency of a harmonic oscillator depends on the physical characteristics of the system, such as its mass and stiffness.

According to second law of motion

**Figure 2.** *The potential energy for a simple harmonic oscillator [6].*

$$\mathbf{F} = \mathbf{m}\mathbf{a} \tag{5}$$

$$\text{Comparing Eqs. (1) and (5)}$$

$$\mathbf{ma} = -\mathbf{k}\mathbf{x}$$

$$m\frac{d^2\mathbf{x}}{dt^2} = -k\mathbf{x} \quad \text{\textquotedblleft } a = \frac{d^2\mathbf{x}}{dt^2}$$

$$\frac{d^2\mathbf{x}}{dt^2} + \frac{k}{m}\mathbf{x} = \mathbf{0} \tag{6}$$

Eq. (6) is a second-order differential equation. The general solution of this Eq. (6)

$$\propto = A \sin at\tag{7}$$

 $\text{We know } \omega = \sqrt{\frac{k}{m}}$ 
$$\omega = A \sin \sqrt{\frac{k}{m}} t \tag{8}$$

We know that

$$
\alpha = 2\pi \theta t$$

$$
\alpha = A \sin 2\pi \theta \tag{9}$$

Comparing Eqs. (8) and (9)

$$A\sin\sqrt{\frac{k}{m}}t = A\sin 2\pi\theta t$$

$$\sin\sqrt{\frac{k}{m}} = \sin 2\pi\theta$$

$$\sin^{-1}\sin\sqrt{\frac{k}{m}} = \sin^{-1}\sin 2\pi\theta$$

$$\therefore \sqrt{\frac{k}{m}} = 2\pi\theta$$

$$\theta = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\tag{10}$$

*Schrödinger Wave Equation for Simple Harmonic Oscillator DOI: http://dx.doi.org/10.5772/intechopen.112381*

Eq. (10) gives the frequency of the simple harmonic oscillator, where *ϑ* the frequency, *k* is the spring constant, and m is the mass of a linear harmonic oscillator. The above equation determines that the frequency of a harmonic oscillator is directly proportional to spring constant's square root and inversely proportional to mass's square root. This means that by increasing the stiffness of the spring or by decreasing the mass of the oscillator, the frequency of an oscillator will increase [8].

Generally, the frequency of harmonic oscillator is an important characteristic that determines its behavior and can be used to analyze and predict its motion. The frequency of a harmonic oscillator can be measured experimentally using various methods, such as by measuring the time period of its oscillations or by analyzing its response to external forces.

$$\text{"} \therefore \theta = \frac{c}{\lambda} \nabla = \frac{1}{\lambda} \theta = c \nabla$$

$$2\pi c \nabla = \sqrt{\frac{k}{m}}$$

$$\nabla = \frac{1}{2\pi c} \sqrt{\frac{k}{m}}$$

⊽ is wave number

For two particles connected to each other through a spring as in diatomic molecule, we use term reduced mass *μ* [10].

$$
\nabla = \frac{1}{2\pi c} \sqrt{\frac{k}{\mu}} \tag{11}
$$

#### **3. Quantum mechanical treatment of simple harmonic oscillator**

The wave function is a mathematical representation of a quantum system's state in quantum mechanics. All of the information about a particle or a group of particles, including their position, momentum, and energy, is contained in the wave function. It is a complex-valued function depends on position and time of particle. It is denoted by symbol Ψ [11].

Probability of finding the particle at a certain position is proportional to absolute square of wave function. It is also used to determine the probability density of finding a particle within a certain volume of space.

In quantum mechanics, wave function is a fundamental concept used to calculate many properties of quantum systems, such as energy levels, transition probabilities, and scattering cross-sections. The wave function is also used to describe the behavior of systems that exhibit wave-like properties, such as electrons, atoms, and molecules [12, 13].

The wave function follows the Schrödinger equation, which is a differential equation that determines how the wave function evolves over time. The Schrödinger equation is used to determine the temporal evolution of quantum systems and to predict particle and system behavior under different conditions [14].

#### **3.1 Representation of wave function**

In quantum mechanics, the wave function can be represented in several ways, depending on the context and the physical system being described. Here are three common representations [15]:

#### *3.1.1 Position representation*

In this position representation, Ψ(x,t) gives the probability amplitude of finding a particle at position x at time t. The position representation is used for systems with definite position, such as single particle in a box or a molecule. In this representation, a wave function is typically denoted as Ψ(x,t) i.e., function of position and time. Its mathematical form can be written as: Ψ(x,t) = A(x,t) \* exp(iφ(x,t)) where A(x,t) is the amplitude of the wave function and φ(x,t) is its phase. The amplitude is a realvalued function that describes the intensity of the wave, while the phase is a realvalued function that describes the position of the wave in space and time [16, 17].

#### *3.1.2 Momentum representation*

In this representation, the wave function is function of momentum rather than the position. The wave function Ψ(p,t) gives the probability amplitude of finding a particle with momentum p at time t. The momentum representation is useful for systems with definite momentum, like a free particle. In this representation, wave function is typically denoted as Ψ(p,t) and is function of momentum and time. Its mathematical form can be written as: Ψ(p,t) = B(p,t) \* exp(iχ(p,t)) where B(p,t) is the amplitude of the wave function in momentum space and χ(p,t) is its phase. This amplitude is realvalued function that determines the intensity of the wave in momentum space, while the phase is a real-valued function that describes the position of wave in momentum space [16–18].

#### *3.1.3 Energy representation*

In this representation, the wave function is a function of energy. The wave function Ψ(E) gives the probability amplitude of finding a system with energy E. The energy representation is useful for systems with definite energy, like a particle in the potential well. In the energy representation, wave function is typically denoted as Ψ(E) and is a function of energy. Mathematically, it can be written as

$$\Psi(\mathbf{E}) = \mathbf{C}(\mathbf{E}) \* \exp(\mathrm{i}\Psi(\mathbf{E})) \tag{12}$$

where C(E) is the amplitude of the wave function in energy space and ψ(E) is its phase. This amplitude is real-valued function that determines the intensity of wave in energy space, while the phase is a real-valued function that describes the position of the wave in energy space.

In each representation, Ψ is a complex-valued function satisfies the Schrödinger equation. It can be normalized, which means that the integral of the absolute square of the wave function over all space or momentum or energy is equal to one, ensuring that the probability of locating a particle in the system is one.

The mathematical form of wave function can be used to calculate various properties of the system, such as probabilities of finding the particle in a certain position, momentum, or energy state [19].

#### **3.2 Boundaries conditions**

For the harmonic oscillator, the two common boundary conditions are described as follows [20].

#### *3.2.1 Normalizability condition*

The wave function must be normalizable, which means that the integral of the absolute square of the wave function over all space must be finite. This assures that probability of locating a particle in the system is one [19].

#### *3.2.2 Continuity condition*

The wave function must be continuous and differentiable at the ends of the range. This ensures that the probability density and its first derivative are continuous and smooth throughout the range of motion.

For the harmonic oscillator, the boundary conditions are typically satisfied by using a particular type of wave function, called the Hermite polynomials. The Hermite polynomials are a set of orthogonal polynomials that satisfy both the normalizability and continuity conditions. They form a complete basis set for the wave function of the harmonic oscillator, allowing the solution to be expressed as a linear combination of these polynomials [6].

#### **3.3 Schrödinger wave equation for harmonic oscillator**

The mathematical form of the wave function in quantum mechanics depends on the physical system being described and the representation being used. However, in general, it is a complex-valued function that satisfies Schrödinger equation [8, 21].

In Quantum mechanics, the one-dimensional time-independent Schrödinger wave equation for harmonic oscillator follows as [22]:

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} + \frac{2m}{\hbar^2} (E - V)\psi = \mathbf{0} \tag{13}$$

But the potential energy of the simple harmonic oscillator is V=<sup>1</sup> <sup>2</sup>*Kx*2, therefore

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} + \frac{2m}{\hbar^2} \left( E - \frac{1}{2} K \mathbf{x}^2 \right) \Psi = \mathbf{0} \tag{14}$$

Or

$$\frac{\partial^2 \psi}{\partial \mathbf{x}^2} + \frac{-mK\mathbf{x}^2}{2}\psi = \frac{-2mE}{2}\psi$$

$$\frac{mK}{\hbar^2} = \alpha^2 \frac{2mE}{\hbar^2} = \varepsilon$$

Them

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} - \alpha^2 \mathbf{x}^2 \,\mathrm{\boldsymbol{\nu}} = -\varepsilon \boldsymbol{\nu} \tag{15}$$

This is Schrödinger's equation for harmonic oscillator [23–25]. Here x2 is the coefficient of *ψ*, so it is difficult to obtain its solution. Hence we will find its asymptotic solution

When x ! <sup>∞</sup> <sup>α</sup><sup>2</sup> x2 > >*ε* So we can write:

$$\frac{\partial^2 \boldsymbol{\Psi}}{\partial \mathbf{x}^2} - \alpha^2 \mathbf{x}^2 \,\boldsymbol{\Psi} = \mathbf{0} \tag{16}$$

Its solution is *<sup>ψ</sup>* <sup>¼</sup> *<sup>e</sup>*�*αx*2*=*<sup>2</sup>

$$\frac{\partial \Psi}{\partial \mathbf{x}} = \pm a \mathbf{x} e^{\pm a \mathbf{x}^2 / 2}$$

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} = \frac{\partial}{\sigma \mathbf{x}} \left( \pm a \mathbf{x} e^{\pm a \mathbf{x}^2 / 2} \right) = a^2 \mathbf{x}^2 e^{\pm a \mathbf{x}^2 / 2} \pm a e^{\pm a \mathbf{x}^2 / 2} = (\pm a) e^{\pm a \mathbf{x}^2 / 2} a^2 \mathbf{x}^2$$

Value of *<sup>α</sup><sup>x</sup>* is larger hence we take *<sup>α</sup>*<sup>2</sup>*<sup>x</sup>* ð Þ <sup>2</sup> � *<sup>α</sup>* <sup>≈</sup>*α*<sup>2</sup>*x*<sup>2</sup>

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} = a^2 \mathbf{x}^2 e^{\pm ax^2/2}$$

Or *<sup>∂</sup>*2*<sup>ψ</sup> <sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup>*x*<sup>2</sup> *<sup>ψ</sup>* or *<sup>∂</sup>*2*<sup>ψ</sup> <sup>∂</sup>x*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup>*x*<sup>2</sup> *<sup>ψ</sup>* <sup>¼</sup> <sup>0</sup> Now we take *<sup>ψ</sup>* <sup>¼</sup> *<sup>e</sup>*�*αx*<sup>2</sup> *=*2

Because it obeys the condition that ∣*ψ*∣ <sup>2</sup> decreases with increasing x General solution:

$$\boldsymbol{\Psi}\_{(\mathbf{x}) \equiv \boldsymbol{\mathcal{J}}\_{(\mathbf{x})}} \boldsymbol{e}^{-\alpha \mathbf{x}^2/2}$$

Differentiating w.r.t *x*

$$\frac{\partial \Psi}{\partial \mathbf{x}} = f\_{\left(\alpha\right)} e^{-\alpha \mathbf{x}^2/2} \left(-a\mathbf{x}\right) + e^{-a\mathbf{x}^2/2} \frac{\partial \mathbf{f}}{\partial \mathbf{x}}.$$

Again differentiating w.r.t *x*

$$\frac{\partial^2 \mathcal{Y}}{\partial \mathbf{x}^2} = f\_{\left(x\right)} \left[ e^{-\alpha x^2/2} (-a) + (-a\mathbf{x}) (-a\mathbf{x}) e^{-\alpha x^2/2} \right]$$

$$+ (-a\mathbf{x}) \, e^{-\alpha x^2/2} \frac{\partial \mathcal{Y}}{\partial \mathbf{x}} + \frac{\partial \mathcal{Y}}{\partial \mathbf{x}} e^{-\alpha x^2/2} \left( -a\mathbf{x} + e^{-\alpha x^2/2} \frac{\partial^2 \mathcal{Y}}{\partial \mathbf{x}^2} \right)$$

$$\frac{\partial^2 \mathcal{Y}}{\partial \mathbf{x}^2} = e^{-\alpha x^2/2} f\_{\left(x\right)} \left( -a + a^2 \mathbf{x}^2 \right) + \frac{\partial \mathcal{Y}}{\partial \mathbf{x}} e^{-\alpha x^2/2} (-2a\mathbf{x}) + e^{-\alpha x^2/2} \frac{\partial^2 \mathcal{Y}}{\partial \mathbf{x}^2}$$

$$\frac{\partial^2 \mathcal{Y}}{\partial \mathbf{x}^2} = e^{-\alpha x^2/2} \left[ \frac{\partial^2 f}{\partial \mathbf{x}^2} - 2a\mathbf{x} \, \frac{\partial \mathcal{Y}}{\partial \mathbf{x}} + \left( a^2 \mathbf{x}^2 - a \right) \mathbf{f} \right]$$

*Schrödinger Wave Equation for Simple Harmonic Oscillator DOI: http://dx.doi.org/10.5772/intechopen.112381*

Substituting values of *ψ* and *<sup>∂</sup>*2*<sup>ψ</sup> <sup>∂</sup>x*<sup>2</sup> in Eq. (15)

$$e^{-\alpha^2 \hat{f}} \left[ \frac{\partial^2 f}{\partial \mathbf{x}^2} - 2\alpha \mathbf{x} \, \frac{\partial f}{\partial \mathbf{x}} + \left( \alpha^2 \mathbf{x}^2 - a \right) \mathbf{f} \right] - \alpha^2 \mathbf{x}^2 f e^{-\alpha \mathbf{x}^2 / 2} = -\epsilon f e^{-\alpha \mathbf{x}^2 / 2}$$

$$\text{Or } \frac{\partial^2 f}{\partial \mathbf{x}^2} - 2\alpha \mathbf{x} \, \frac{\partial f}{\partial \mathbf{x}} + (\epsilon - a) f = \mathbf{0} \tag{17}$$

Now substituting y = ffiffiffi *<sup>α</sup>* <sup>p</sup> x and *<sup>f</sup>*ð Þ *<sup>x</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>y</sup>* converting into standard Hermite polynomial equation

y = ffiffiffi *α* p *x* then *dy dx* <sup>¼</sup> ffiffiffi *α* p

$$\frac{\partial \mathbf{f}}{\partial \mathbf{x}} = \frac{\partial \mathbf{f}}{\partial \mathbf{y}} \cdot \frac{\partial \mathbf{y}}{\partial \mathbf{x}} = \sqrt{\alpha} \frac{\partial \mathbf{f}}{\partial \mathbf{y}}$$

$$\frac{\partial^2 \mathbf{f}}{\partial \mathbf{x}^2} = \frac{\partial}{\partial \mathbf{x}} \left(\frac{\partial \mathbf{f}}{\partial \mathbf{x}}\right) = \frac{\partial}{\partial \mathbf{x}} \left(\sqrt{\alpha} \frac{\partial \mathbf{f}}{\partial \mathbf{y}}\right) = \frac{\partial}{\partial \mathbf{y}} \left(\sqrt{\alpha} \frac{\partial \mathbf{f}}{\partial \mathbf{y}}\right) \frac{\partial \mathbf{y}}{\partial \mathbf{x}} = \alpha \frac{\partial^2 \mathbf{f}}{\partial \mathbf{y}^2}$$

Substituting values of *<sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>x</sup>* and *<sup>∂</sup>*2*<sup>f</sup> <sup>∂</sup>x*<sup>2</sup> in Eq. (17), we get

$$a\frac{\partial^2 f}{\partial \mathbf{y}^2} - 2a\frac{\mathcal{Y}}{\sqrt{a}}\sqrt{a}\frac{\partial f}{\partial \mathbf{y}} + (\varepsilon - a)f = \mathbf{0}$$

$$a\frac{\partial^2 f}{\partial \mathbf{y}^2} - 2a\mathbf{y}\ \frac{\partial f}{\partial \mathbf{y}} + (\varepsilon - a)f = \mathbf{0}$$

$$\frac{\partial^2 f}{\partial \mathbf{y}^2} - 2\mathbf{y}\ \frac{\partial f}{\partial \mathbf{y}} + \left(\frac{\varepsilon}{a} - \mathbf{1}\right)f = \mathbf{0}$$

Now f(x) = H(y)

$$\frac{\partial^2 H}{\partial y^2} - 2y \, \frac{\partial H}{\partial y} + \left(\frac{\varepsilon}{a} - 1\right) H = 0 \tag{18}$$

This is standard Hermite differential equation [22]. It can be expressed as

$$\mathbf{H}(\mathbf{y}) = \sum\_{p=0}^{\infty} a\_p \mathbf{y}^p \tag{19}$$

$$\frac{\partial H}{\partial \mathbf{y}} = \sum p a\_p \mathbf{y}^{p-1}$$

$$\frac{\partial^2 H}{\partial \mathbf{y}^2} = \sum p(p-1)a\_p \mathbf{y}^{p-2}$$

From Eq. (18)

$$\sum p(p-1)a\_p y^{p-2} - \sum \left[2p - \left(\frac{\varepsilon}{\alpha} - 1\right)\right] a\_p y^p = 0$$

This expression is valid only when coefficient of each power of *y* is zero. And p = p + 2

$$\sum (p+2)(p+2-1)a\_{p+2}p^{p+2-2} - \sum \left[2p - \left(\frac{e}{a} - 1\right)\right]a\_p p^p = 0$$

$$a\_{p+2}(p+2)(p+1) = a\_p\left[2p - \left(\frac{e}{a}\right) + 1\right]$$

$$a\_{p+2} = \frac{\left[2p - \left(\frac{e}{a}\right) + 1\right]}{(p+2)(p+1)}a\_p\tag{20}$$

We can determine values of all the coefficients in terms of two arbitrary constants a0 and a1

Thus, complete solution of Schrödinger's equation is [26]

$$\begin{aligned} \psi &= e^{-ax^2/2} H\_{(y)} \\ \psi &= e^{-y^2/2} H\_{(y)} \end{aligned}$$

#### **3.4 Energy eigen values**

*<sup>ψ</sup>* <sup>¼</sup> *<sup>e</sup>*�*y*2*=*<sup>2</sup>*H*ð Þ*<sup>y</sup>* of a simple harmonic oscillator will be physically accepted only when y! ∞, the increase in the value of Hermite Polynomial *H*ð Þ*<sup>y</sup>* is more rapid than the decrease in the value of *e*�*y*2*=*<sup>2</sup> value [27].

Value of *<sup>e</sup>*�*y*2*=*<sup>2</sup>*H*ð Þ*<sup>y</sup>* can be zero only when power series for *<sup>H</sup>*ð Þ*<sup>y</sup>* is finite series. Let series be finite for p=n, the Eq. (20) becomes.

$$2\mathbf{n} - \frac{\varepsilon}{\alpha} + \mathbf{1} = \mathbf{0}$$

$$\mathbf{N} = \frac{1}{2} \left(\frac{\varepsilon}{\alpha} - \mathbf{1}\right) \frac{\varepsilon}{\alpha} = 2n + \mathbf{1}\varepsilon = \frac{2mE}{\hbar^2}\alpha = \sqrt{\frac{mk}{\hbar^2}}$$

$$\frac{2mE}{\hbar^2} \sqrt{\frac{m}{\hbar^2}} = 2\mathbf{n} + \mathbf{1}\ \frac{2}{\hbar} \sqrt{\frac{m}{k}}E = 2n + \mathbf{1}$$

$$\mathbf{E} = \frac{2n+1}{2}\hbar \sqrt{\frac{k}{m}}$$

But we know ffiffiffi *k m* q ¼ *ω* (angular frequency)

$$\mathbf{E} = \left(\frac{2n+1}{2}\right)\hbar\nu = \left(n+\frac{1}{2}\right)\hbar\nu\tag{21}$$

Where n = 0, 1, 2, 3, …

The above equation gives the energy levels of a harmonic oscillator [28], where n is a non-negative integer, *h* is reduced Planck constant, *ω* is an angular frequency of the oscillator, and E\_n is the energy of the oscillator in the nth energy level. In quantum mechanics, the energy levels of simple harmonic oscillator are quantized, which means they take on only certain discrete values.

*Schrödinger Wave Equation for Simple Harmonic Oscillator DOI: http://dx.doi.org/10.5772/intechopen.112381*

If n = 0 then *<sup>E</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> 2 *ν* n = 1 then *<sup>E</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup> 2 *ν* n = 2 then *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>5</sup> 2 *ν*

The energy levels of a harmonic oscillator are equally spaced, with the energy of each level separated by an amount *h̅ω*. The ground state of the oscillator, n=0, has the lowest energy level and corresponds to the oscillator's minimum energy state, where the particle is localized at the center of the potential well. As n increases, the energy levels increase and the wave function oscillates with more nodes [27].

The energy of the harmonic oscillator is always positive, and the oscillator can never reach the zero-point energy, which is the minimum possible energy that a quantum mechanical system can have [29].

#### **Author details**

Noor-ul-ain<sup>1</sup> \*, Sadaf Fatima<sup>1</sup> , Mushtaq Ahmad<sup>1</sup> , Muhammad Rizwan Khan1 and Muhammad Aslam<sup>2</sup>

1 Institute of Physics, The Islamia University of Bahawalpur, Pakistan

2 Institute of Physics and Technology, Ural Federal University, Yekaterinburg, Russia

\*Address all correspondence to: noorulain@iub.edu.pk

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[20] Zhang K et al. Simple harmonic oscillation in a non-Hermitian Su-Schrieffer-Heeger chain at the exceptional point. Physical Review A. 2018;**98**(2):022128

*Schrödinger Wave Equation for Simple Harmonic Oscillator DOI: http://dx.doi.org/10.5772/intechopen.112381*

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[29] Fernández FM. On the singular harmonic oscillator. arXiv preprint arXiv:2112.03693, 2021

#### **Chapter 2**

## A Schrödinger Equation for Light

*Daniel R.E. Hodgson*

#### **Abstract**

In this chapter we examine the quantised electromagnetic (EM) field in the context of a Schrödinger equation for single photons. For clarity we consider only a onedimensional system. As a universal tool for calculating the time-evolution of quantum states, a Schrödinger equation must exist that describes the propagation of single photons. Being inherently relativistic, however, critical aspects of both special relativity and quantum mechanics must be combined when quantising the EM field. By taking the approach of a Schrödinger equation for localised photons, we will show how novel and previously overlooked features of the quantised EM field become a necessary part of a complete description of photon dynamics. In this chapter, I shall provide a thorough examination of new features and discuss their significance in topics such as quantum relativity and photon localisation.

**Keywords:** photon localisation, photon wave function, causality, negative frequencies, non-locality

#### **1. Introduction**

Thomas Young's double slit experiment gives a simple but clear demonstration that light is certainly a wave. The appearance of an alternating pattern of dark fringes is evidence of the destructive superposition of waves passing through different slits onto the screen behind. In this classical experiment, the pattern emerging on the screen is generated by the interference between oscillating electromagnetic (EM) waves that are predicted by Maxwell's theory of electromagnetism. The modification from a classical to a quantum theory, however, reinterprets these waves as oscillations of the probabilistic wave function for a collection of photons, the indivisible particles of light. In order to form a complete description of how photons evolve, it is important that we are able to define a wave function for each photon wave packet describing its oscillations through both space and time.

By initially postulating that photons are discrete and countable objects, and that each photon has an energy proportional to its frequency, it is possible to derive complete expressions for the electric and magnetic field observables up to an overall phase [1]. More conventional quantisation methods, however, take the reverse approach. Here the field observables, not the photons, are the main focus of the quantisation process, which are obtained by means of canonical quantisation. See, for example, Ref. [2]. By adopting a Hamiltonian procedure, correspondence with classical physics can be maintained by imposing canonical commutation relations. Moreover, working directly with quantised fields may be viewed as more fundamental than working with particles, which are not

covariant objects. Providing a wave function for the excitations of the field observables, however, has proved exceptionally challenging.

It was proven in 1948 by Newton and Wigner that no position-dependent wave function existed for the photon [3, 4]. More specifically, subject to certain conditions, there was no photon position operator with which to define a basis of localised eigenstates for the wave function. Since this time, the localisation of single photons has been researched extensively [5–12], but there is as yet no unanimous agreement on whether localisation is possible. In the work of Fleming [13, 14], for example, some of the relativistic properties of the Newton-Wigner (NW) operator were clarified, but localisation of the photon was again shown to be impossible. Only more recently, by considering the longitudinal components of spin, Hawton has been able to show that a photon position operator with commuting components conjugate to the momentum operator can be defined [15–17].

It is also unclear how the photon wave function ought to be interpreted. Early wave functions, such as the Landau-Peierls wave function [18–20], for example, were criticised for being non-locally related to the electric and magnetic field observables. Such a relationship emerges due to a disparity between the units for a probabilistic wave function and the field observables. Wave functions locally related to the field observables have been studied in both the first and second quantisation regimes [21–25]. In some schemes, such as those of Knight [26] and Licht [27, 28], a photon is only localised if the electric and magnetic field expectation values are also localised. In this case, however, a single photon cannot be localised if the field observables do not commute [29]. To overcome problems of non-locality, many authors have introduced non-local inner products, which lead to the use of non-standard and non-Hermitian models [30–33].

Localisation in quantum theory is also closely connected to causality. A photon wave packet that is localised to one region, for instance, cannot reach another until a time has elapsed no less than the distance between these regions divided by the speed of light in a vacuum. The theorems of Hegerfeldt [34] and Malament [35] show, however, that non-zero correlations between the position of a wave packet can be generated at speeds exceeding the speed of light. The question that this raises about causality has been a large topic of research [36–40], with a particular interest in causality in the transmission of radiation between two two-level atoms [41–44]. Whilst many insist that only causality in the sense of no-signalling, rather than of strict causality, is necessary, a wave function can only be usefully and properly interpreted if the speed at which it propagates never exceeds the speed of light.

In this chapter we explore a recent quantisation of the one-dimensional free EM field in the position representation [45]. The main focus of this quantisation will be the construction of single-photon wave packets in a basis of localised photonic excitations. We determine an equation of motion for these excitations which leads to a Schrödinger equation for the photon. By focussing on dynamics, new parameters are introduced that were previously neglected or overlooked. This provides us with a fuller description of the quantised EM field. For completeness, expressions for the EM field observables shall be constructed, and a comparison with standard quantisations shall be given.

#### **2. The classical EM field in one dimension**

The equations of motion for light are given by Maxwell's equations. The solutions to these equations provide us with the expected dynamics of the quantised particles of the EM field. The purpose of this section is to review the appropriate equations of motion and their solutions in one dimension, and to determine an expression for the energy of the EM field.

#### **2.1 The dynamics of the EM field**

#### *2.1.1 Maxwell's equations*

Light consists of two real, mutually propagating vector fields: the electric field and the magnetic field. In one dimension, the electric and magnetic fields propagate along a single axis parametrised by a position coordinate *x*. The electric and magnetic fields at a position *x* at a time *t* are denoted **E**(*x*, *t*) and **B**(*x*, *t*) respectively.

Although **E**(*x*, *t*) and **B**(*x*, *t*) are parametrised by a position along the *x*-axis only, the fields are oriented, or polarised, in the plane orthogonal to the direction of propagation. By specifying a right-handed Cartesian coordinate system (*x*, *y*, *z*), the electric and magnetic field have components in the *y* and *z* directions only. The components of the fields oriented along the *y* (*z*)-axis shall be referred to as horizontally (vertically) polarised. The polarisation of the field is specified by a discrete parameter *λ* ¼ H, V.

In a dielectric medium of constant permittivity *ε* and permeability *μ*, and by denoting *<sup>c</sup>* <sup>¼</sup> ð Þ *εμ* �1*=*<sup>2</sup> , the horizontally and vertically polarised components of **E**(*x*, *t*) and **B**(*x*, *t*) satisfy the following simplified forms of Maxwell's equations:

$$\begin{aligned} \frac{\partial}{\partial \mathbf{x}} \mathbf{E}(\mathbf{x}, t) &= \quad \pm \frac{\partial}{\partial t} \mathbf{B}(\mathbf{x}, t) \\ \mathbf{c}^2 \frac{\partial}{\partial \mathbf{x}} \mathbf{B}(\mathbf{x}, t) &= \quad \pm \frac{\partial}{\partial t} \mathbf{E}(\mathbf{x}, t) \end{aligned} \tag{1}$$

In both lines of Eq. (1) above, the electric and magnetic fields have alternate polarisations. The positive (negative) sign applies when the electric and magnetic fields are vertically (horizontally) and horizontally (vertically) polarised respectively.

#### *2.1.2 The wave equation*

Maxwell's equations, Eq. (1) couple together different components of the electric and magnetic field vectors. By combining these equations, we can construct a secondorder differential equation for each of the four field components independently. These equations are

$$\begin{aligned} \left[\frac{\partial^2}{\partial t\mathbf{x}} - \frac{1}{c^2} \frac{\partial^2}{\partial t}\right] \mathbf{E}(\mathbf{x}, t) &= \quad \mathbf{0} \\\\ \left[\frac{\partial^2}{\partial \mathbf{x}} - \frac{1}{c^2} \frac{\partial^2}{\partial t}\right] \mathbf{B}(\mathbf{x}, t) &= \quad \mathbf{0}. \end{aligned} \tag{2}$$

Here we have four identical equations of motion, one for each of the four components of the EM field. The solutions to Eq. (2) will be examined in Section 2.3.

#### **2.2 The energy and momentum of the EM field**

#### *2.2.1 The energy observable*

At each point in space and time, the electric and magnetic fields exert a force on any charged matter present at that point. For this reason, the EM field is able to do mechanical work on charged matter, and must therefore store a certain amount of energy. Taking this into account, explicit expressions for the energy and momentum of light in a dielectric medium can be determined. By considering the work done by the fields on a charge current density in a dielectric medium, one can show that the total energy along the *x*-axis is given by the expression

$$H\_{\text{energy}}(t) = \int\_{-\infty}^{\infty} \text{d}\mathbf{x} \, \frac{A}{2} \left\{ \varepsilon \left| \mathbf{E}(\mathbf{x}, t) \right|^2 + \frac{\mathbf{1}}{\mu} \left| \mathbf{B}(\mathbf{x}, t) \right|^2 \right\}. \tag{3}$$

Here *A* is the area occupied by the field in the *y*-*z* plane.

#### *2.2.2 The Poynting vector*

The energy stored in the EM field in a particular region is carried in the direction of propagation in the form of the Poynting vector **S**. Since in one dimension light can only propagate along the *x*-axis, the only non-zero component of the Poynting vector is the *x* component, which is given by the expression

$$\mathbf{S}(\mathbf{x},t) = \frac{1}{\mu} [\mathbf{E}\_{\mathrm{H}}(\mathbf{x},t)\mathbf{B}\_{\mathrm{V}}(\mathbf{x},t) - \mathbf{E}\_{\mathrm{V}}(\mathbf{x},t)\mathbf{B}\_{\mathrm{H}}(\mathbf{x},t)].\tag{4}$$

In the above the H and V subscripts refer to the horizontally and vertically polarised components of the fields respectively. The expressions above, particularly Eq. (3), will be of importance in Section 4.2.2.

#### **2.3 The solutions to Maxwell's equations**

#### *2.3.1 Left- and right-propagating waves*

The wave equation, Eq. (2) describes the propagation of a wave along the *x*-axis at a constant speed *c*. This is the speed of light in the medium. In only one dimension, the solutions of the wave equation take a simple form. By considering first the components of the electric field E*λ*ð Þ *x*, *t* , one can show that the expressions

$$\mathbf{E}\_{\boldsymbol{\lambda}}(\mathbf{x},t) = \sum\_{\boldsymbol{\kappa}=\pm 1} \mathbf{E}\_{\boldsymbol{\kappa}\boldsymbol{\lambda}}(\mathbf{x},t) \tag{5}$$

satisfy Eq. (2) when E*s<sup>λ</sup>*ð Þ¼ *x*, *t* E*s<sup>λ</sup>*ð Þ *x* � *sct*, 0 . In Eq. (5) above, the parameter *s* ¼ �1 is introduced in order to differentiate between solutions propagating to the left (decreasing *x*) or the right (increasing *x*). In this notation, light characterised by *s* ¼ �1ð Þ þ1 propagates to the left (right). The exact form of E*s<sup>λ</sup>*ð Þ *x*, *t* is determined from the initial conditions of the system.

#### *2.3.2 Complete electric and magnetic field solutions*

The corresponding magnetic field solution to Eq. (2) is not independent of the electric field solution. By using Maxwell's equation, Eq. (1), the magnetic field can be determined directly from the electric field solution (5). After taking into account the sign difference for different polarisations, one may show that

$$\mathbf{E}(\mathbf{x},t) = \sum\_{\mathbf{s}=\pm 1} c \left[ \mathbf{E}\_{\text{i}\mathbf{H}}(\mathbf{x},t)\hat{\mathbf{y}} + \mathbf{E}\_{\text{i}\mathbf{V}}(\mathbf{x},t)\hat{\mathbf{z}} \right] \tag{6}$$

and

$$\mathbf{B}(\mathbf{x},t) = \sum\_{\mathbf{s}=\pm 1} \mathbf{s} \left[ -\mathbf{E}\_{\text{sV}}(\mathbf{x},t)\hat{\mathbf{y}} + \mathbf{E}\_{\text{sH}}(\mathbf{x},t)\hat{\mathbf{z}} \right]. \tag{7}$$

Here *y*^ and ^*z* are unit vectors oriented in the positive *y* and *z* directions respectively.

#### *2.3.3 Energy and momentum*

Since **E**(*x*, *t*) and **B**(*x*, *t*) are both characterised by the solutions E*s<sup>λ</sup>*ð Þ *x*, *t* , the energy and Poynting vector of the field must also be characterised by these solutions. Substituting Eqs. (6) and (7) into Eqs. (3) and (4) one finds that

$$H\_{\text{energy}}(t) = \sum\_{\mathbf{s} = \pm 1} \sum\_{\lambda = \mathbf{H}\_{\mathbf{s}}} \int\_{-\infty}^{\infty} \text{d}\mathbf{x} \, \, \frac{A}{2} \left\{ \varepsilon \left| \mathbf{E}\_{t\lambda}(\mathbf{x}, t) \right|^2 \right\} \tag{8}$$

and

$$S(\mathbf{x},t) = \sum\_{\mathbf{s}=\pm 1} \sum\_{\lambda=\pm 1,\mathbf{V}} \left| \text{sc}(\mu) | \mathbf{E}\_{\text{sl}}(\mathbf{x},t) \right|^2. \tag{9}$$

It is clear from Eq. (9) that a positive Poynting vector indicates propagation to the right whereas a negative Poynting vector indicates propagation to the left.

#### **3. The wave function of the photon**

The equations of motion for light in a homogeneous and isotropic dielectric medium apply the first set of constraints to the particle behaviour of light. For a correct and natural interpretation of the photon wave function, the probability distribution of particles represented by the wave function must evolve identically to the classical wave packets of an EM wave. In this section we construct a Fock space of localised bosonic excitations that provide a basis for constructing single-photon wave packets. By imposing a constraint on the dynamics of these excitations in 1 + 1 dimensional space-time, a Schrödinger equation is formulated for the photon.

#### **3.1 The parameter space of single-photon wave packets**

#### *3.1.1 Unitary time evolution*

Consider the propagation of a photon wave packet through the dielectric medium along the *x*-axis. At an initial time *t* ¼ 0, we may represent this wave packet in the Hilbert space by a state vector ∣*ψ*1ð Þi 0 . After a time *t* has passed, the photon wave packet is now found in the time-evolved state

$$|\psi\_1(t)\rangle = U(t,0)|\psi\_1(0)\rangle \tag{10}$$

where *U*(*t*, 0) is the unitary time-evolution operator from time *t* ¼ 0 to time *t*. As we have determined that light must propagate along the *x*-axis at a speed *c*, the unitary operator *U*(*t*, 0) transports the left- and right-moving components of the wave packet to the left or the right by an exact distance *ct*.

#### *3.1.2 A complete parameter space*

If we consider two single-photon wave packets that are entirely distinguishable from each other, then their corresponding state vectors must be orthogonal. When we localise two photons to different points along the *x*-axis, they are distinguishable from each other. Localised photon states, therefore, are orthogonal to one another, and it is natural to characterise them by their position along the *x*-axis. In the same way, photons with different polarisations are distinguishable and their state vectors orthogonal. Photon states are therefore also characterised by a polarisation *λ*. In addition to this, states describing propagation in opposite directions must be orthogonal to one another, and must be characterised by the discrete parameter *s* ¼ �1.

To see that this last parametrisation must be so, consider the setup illustrated in **Figure 1** showing two identical single-photon wave packets propagating in opposite directions. We denote the state vectors for the left- and right-hand systems ∣*ψ*1ð Þi *x*, *t* and ∣*ψ*2ð Þi *x*, *t* respectively where ∣*ψ*1ð Þi ¼ *x*, 0 ∣*ψ*2ð Þi *x* þ 2*a*, 0 . At an initial time *t* ¼ 0, the photon in the left-hand diagram is localised to a position *x* ¼ �*a* whereas the photon in the right-hand diagram is localised to the position *x* ¼ *a*. Since the two wave packets occupy separate regions of the *x*-axis, their state vectors must be orthogonal:

$$
\langle \psi\_1(\mathbf{x}, \mathbf{0}) | \psi\_2(\mathbf{x}, \mathbf{0}) \rangle = \langle \psi\_1(\mathbf{x}, \mathbf{0}) | \psi\_1(\mathbf{x} - 2\mathbf{a}, \mathbf{0}) \rangle = \mathbf{0}.\tag{11}
$$

At a later time *t* ¼ *a=c*, both photons will have travelled a distance *a* to the left or the right of their initial positions. After taking into account the direction of propagation of each of the photons, at this later time *t* ¼ *a=c*, both wave packets will coincide with each other perfectly at the origin. When parametrised by only a position and

**Figure 1.**

*The diagram illustrates the propagation of two localised wave packets. In the left-hand diagram, a single photon propagates to the right from an initial position x* ¼ �*a. In the right-hand diagram, a single photon of identical shape propagates to the left from an initial position x* ¼ *a: At a later time both wave packets reach the origin. Here the two wave packets are completely identical with respect to their position.*

polarisation, therefore, their corresponding state vectors will no longer be orthogonal. Hence, the inner product

$$<\langle \psi\_1(\mathbf{x},t) | \psi\_2(\mathbf{x},t) \rangle = \langle \psi\_1(\mathbf{x},t) | \psi\_1(\mathbf{x} - 2\mathbf{a},t) \rangle \tag{12}$$

will necessarily be non-zero. Given that the states ∣*ψ*1ð Þi *x*, *t* and ∣*ψ*2ð Þi *x*, *t* evolve unitarily according to Eq. (10), however, the inner product between the two states at a time *t* is given by

$$\begin{split} \langle \boldsymbol{\Psi}\_{1}(\mathbf{x},t) | \boldsymbol{\Psi}\_{2}(\mathbf{x},t) \rangle &= \left\langle \boldsymbol{\Psi}\_{1}(\mathbf{x},\mathbf{0}) | \boldsymbol{U}^{\dagger}(t,\mathbf{0}) \boldsymbol{U}(t,\mathbf{0}) | \boldsymbol{\Psi}\_{2}(\mathbf{x},\mathbf{0}) \right\rangle \\ &= \langle \boldsymbol{\Psi}\_{1}(\mathbf{x},\mathbf{0}) | \boldsymbol{\Psi}\_{2}(\mathbf{x},\mathbf{0}) \rangle \end{split} \tag{13}$$

where † denotes hermitian conjugation. This inner product must therefore be constant with respect to time. The assumption that our two state vectors are initially orthogonal, therefore, is inconsistent with unitary time evolution and we reach a contradiction. The resolution to this problem is to ensure that wave packets propagating in different directions remain orthogonal at all times. To properly differentiate between states propagating in different directions, therefore, single-photon wave packets must also be parametrised by *s* ¼ �1 in addition to position and polarisation.

#### **3.2 Local photons**

#### *3.2.1 Creation and annihilation operators*

During the interactions between light and matter, atoms absorb and emit light on the level of single photons. The appropriate Hilbert space for the free EM field, therefore, is a Fock space of identical and non-interacting bosonic particles. From what we have determined in the previous section, the localised photonic excitations of the EM field are characterised at any one time by a coordinate *x*∈ ð Þ �∞, ∞ , a polarisation *λ* ¼ H, V and a direction of propagation *s* ¼ �1. From now on we shall refer to such excitations as blips, which is the acronym for bosons localised in position.

As is usual for a system of identical particles we may define a collection of blip annihilation operators that remove a single blip from the system. The blip annihilation operator is denoted *asλ*ð Þ *x*, *t* in the Heisenberg picture and *asλ*ð Þ *x*, 0 in the Schrödinger picture. A state containing only a single blip is defined

$$
\langle \mathbf{1}\_{\iota\lambda}(\mathbf{x}, t) \rangle = a\_{\iota\lambda}^{\dagger}(\mathbf{x}, t) \,|\mathbf{0}\rangle. \tag{14}
$$

The operator *a*† *<sup>s</sup><sup>λ</sup>*ð Þ *x*, *t* is termed the blip creation operator and generates a single blip characterised by the parameters ð Þ *x*, *t*, *λ*, *s* . In Eq. (14) above, ∣0i is the vacuum state containing precisely zero blips. The vacuum state satisfies the property

$$a\_{s\boldsymbol{\lambda}}(\mathbf{x}, t) \,\vert\mathbf{0}\rangle = \mathbf{0} \tag{15}$$

for all *x*, *t*, *s* and *λ*.

#### *3.2.2 Commutation relations*

Although the state defined in Eq. (14) contains only one blip, states containing an arbitrary number of blips can be generated by repeatedly applying the blip creation

operators to the vacuum state. Since blips are bosons, the resulting state must be unchanged through any reordering of the blips' positions. Consequently, the ordering of these creation operators must be insignificant and they must commute with one another. Hence

$$\left[a\_{\iota\lambda}^{\dagger}(\mathbf{x},t), a\_{\iota'\lambda'}^{\dagger}(\mathbf{x'},t')\right] = \mathbf{0} = \left[a\_{\iota\lambda}(\mathbf{x},t), a\_{\iota'\lambda'}(\mathbf{x'},t')\right] \tag{16}$$

for any *x*, *x*<sup>0</sup> , *t*, *t* 0 , *λ*, *λ*<sup>0</sup> , *s* and *s* 0 .

In Section 3.1.2 it was discussed how blips located at different positions, carrying different polarisations or propagating in opposite directions must be perfectly distinguishable from one another. As a consequence, the states that represent them must also be orthogonal. Taking this into account, we specify the following inner product for two single-blip states:

$$
\langle \mathbf{1}\_{\boldsymbol{\lambda}\boldsymbol{\lambda}}(\mathbf{x},t) | \mathbf{1}\_{\boldsymbol{\lambda}'\boldsymbol{\lambda}'}(\mathbf{x}',t) \rangle = \delta\_{\boldsymbol{\lambda}\boldsymbol{\lambda}'} \delta\_{\boldsymbol{\lambda},\boldsymbol{\lambda}'} \delta(\mathbf{x} - \mathbf{x}'). \tag{17}
$$

Using Eqs. (14) and (15), and expressing the inner product (17) in terms of blip creation and annihilation operators, it can be shown that at any fixed time *t*

$$\left[a\_{\imath\lambda}(\mathfrak{x},t), a\_{\imath'\lambda'}^{\dagger}(\mathfrak{x'},t)\right] = \delta\_{\mathfrak{s},\mathfrak{s}'}\delta\_{\lambda,\mathfrak{s}'}\delta(\mathfrak{x}-\mathfrak{x}').\tag{18}$$

This is the fundamental commutation relation for blips.

#### *3.2.3 The photon wave function*

In the context of linear optics experiments [46, 47], it is usual to talk about single photons when referring to particles whose state vectors ∣1ð Þi *t* can be expressed ∣1ð Þi ¼ *t <sup>a</sup>*†ð Þ*<sup>t</sup>* <sup>∣</sup>0<sup>i</sup> where *<sup>a</sup>*(*t*) is an annihilation operator satisfying the commutation relation

$$\left[a(t), a(t)^{\dagger}\right] = \mathbf{1}.\tag{19}$$

The blip states defined in Eq. (14) are not normalisable, but when superposed over a region of the *x*-axis can provide a localised basis for normalised single-photon wave packets. Taking this into account, the annihilation operator for a single-photon wave packet can be defined in the following way:

$$a(t) = \sum\_{s=\pm 1} \sum\_{\lambda \in \mathcal{H}, \mathcal{V}} \int\_{-\infty}^{\infty} \mathbf{dx} \boldsymbol{\psi}\_{s\lambda}^{\*}(\boldsymbol{x}) a\_{s\lambda}(\boldsymbol{x}, t) \tag{20}$$

where ∗ denotes complex conjugation. In Eq. (20), the operator *a*(*t*) is properly normalised and satisfies Eq. (19) when

$$\sum\_{\boldsymbol{\omega} = \pm 1} \sum\_{\boldsymbol{\lambda} = \mathbf{H}, \mathbf{V}} \int\_{-\infty}^{\infty} \mathbf{d} \boldsymbol{\omega} \cdot |\boldsymbol{\nu}\_{\boldsymbol{\kappa}}(\boldsymbol{\kappa})|^{2} = \mathbf{1}. \tag{21}$$

The function *ψs<sup>λ</sup>*ð Þ *x*, *t* in Eq. (20) represents the probability amplitude for finding a photon with polarisation *λ* propagating in the *s* direction at a position *x*. More specifically, the transition probability between the single-photon state ∣1ð Þi *t* and the state ∣1*s<sup>λ</sup>*ð Þi *x*, *t* is given by the expression

*A Schrödinger Equation for Light DOI: http://dx.doi.org/10.5772/intechopen.112950*

$$\left| \left< \mathbf{0} \middle| a\_{s\lambda}(\mathbf{x}, t) a^{\dagger}(t) \middle| \mathbf{0} \right> \right|^{2} = \left| \psi\_{s\lambda}(\mathbf{x}) \right|^{2}. \tag{22}$$

Hence *ψs<sup>λ</sup>*ð Þ *x*, *t* has the correct properties to be correctly interpreted as a singlephoton wave function in the position representation.

#### **3.3 A Schrödinger equation for light**

#### *3.3.1 A Hamiltonian constraint*

In order to calculate the dynamics of a quantum system, it is usual to first determine the Hamiltonian for that system. In a closed system, the Hamiltonian would be given by the energy observable. Once found, the Hamiltonian is used to construct a Schrödinger equation for state vectors in the Hilbert space. So far, an energy observable has not been constructed for the blip states. Moreover, there is no immediate choice for this observable, as, having complete uncertainty in their frequency, blips are not the eigenstates of the energy observable. Fortunately, however, the dynamics of single blips have already been determined. They are given by the solutions to Maxwell's equations, Eq. (5).

Blips are characterised by both a coordinate *x* in space and a coordinate *t* in time. A single blip, therefore, may exist at one position at one moment in time, and then at a different position at another moment in time. The classical dynamics of light in the medium places a constraint on which positions the blip may take from one moment to the next. Being more specific, in order to satisfy Maxwell's equations, the expectation value of a localised blip must propagate at a speed *c* along the *x*-axis without any dispersion. These dynamics are imposed by the constraint h i *asλ*ð Þ *x*, *t* ¼ h i *asλ*ð Þ *x* � *sct*, 0 . Since this applies for any time-independent state, we can determine the general constraint

$$a\_{s\boldsymbol{\lambda}}(\mathbf{x},t) = a\_{s\boldsymbol{\lambda}}(\mathbf{x} - \mathbf{s}ct, \mathbf{0}). \tag{23}$$

When allowed to propagate freely, a blip found at *x* at a time *t* will be found at a position *x* � *sct* at the time *t* ¼ 0.

The constraint on the dynamics, Eq. (23), enables us to define an equation of motion for the blip operators *asλ*ð Þ *x*, *t* . More specifically, by taking the time derivative of Eq. (23) it can be shown that

$$
\left[\frac{\partial}{\partial t} + s\varepsilon \frac{\partial}{\partial \mathbf{x}}\right] a\_{s\boldsymbol{\lambda}}(\mathbf{x}, t) = \mathbf{0}.\tag{24}
$$

The equation above takes the form of a Wheeler-deWitt equation, and defines a stationary or "timeless" state of the system [48]. In the system considered here, this equation confines the trajectories of blips to the boundaries of the light cone. By relating a change in time to a change in the position of a blip in this way, we obtain a Schrödinger equation for blips:

$$i\hbar\frac{\partial}{\partial t}|\mathbf{1}\_{\imath\dot{\lambda}}(\varkappa,t)\rangle = -i\hbar\mathbf{s}c\frac{\partial}{\partial \mathbf{x}}|\mathbf{1}\_{\imath\dot{\lambda}}(\varkappa,t)\rangle. \tag{25}$$

#### *3.3.2 The dynamical Hamiltonian*

In the context of a Schrödinger equation, the motion of the blip given by the righthand side of Eq. (25) is generated by the Hamiltonian for this system. It is very convenient to determine this Hamiltonian as it provides a basis for introducing

interactions in more complex models. To this end, by using the Schrödinger equation for blips (25), we can determine exactly the Hamiltonian for the free propagation of light in a one-dimensional dielectric medium. Here we shall denote this operator *H*dynð Þ*t* with the subscript "dynamical" to distinguish it as the Hamiltonian operator present in the Schrödinger equation.

Looking again at the right-hand side of Eq. (25), it can be seen that the number of blips, their polarisation and their direction of propagation are all preserved as they evolve in time. It is only their position that changes. Taking this into account, a suitable ansatz for the dynamical Hamiltonian *H*dynð Þ*t* would be

$$H\_{\rm dyn}(t) = \sum\_{s=\pm 1} \sum\_{l=\pm 1, \rm V} \int\_{-\infty}^{\infty} \mathrm{d}x \int\_{-\infty}^{\infty} \mathrm{d}x \,\, i\hbar \mathrm{s} \mathbf{f}\_{s\boldsymbol{\lambda}}(\mathbf{x}, \mathbf{x}') a\_{s\boldsymbol{\lambda}}^{\dagger}(\mathbf{x}, t) a\_{s\boldsymbol{\lambda}}(\mathbf{x}', t) \tag{26}$$

where *fs<sup>λ</sup> x*, *x*<sup>0</sup> ð Þ is a function to be determined. This operator takes the form of an exchange operator that annihilates a blip at one position and replaces it with an identical blip at a different position. To ensure that *H*dynð Þ*t* is hermitian, *fs<sup>λ</sup> x*, *x*<sup>0</sup> ð Þ¼�*fs<sup>λ</sup> x*<sup>0</sup> ð Þ , *x* .

In the Heisenberg picture, the dynamics of a blip operator *asλ*ð Þ *x*, *t* can be equivalently expressed through Heisenberg's equation of motion:

$$\frac{\partial}{\partial t} a\_{\imath\lambda}(\mathbf{x}, t) = -\frac{i}{\hbar} \left[ a\_{\imath\lambda}(\mathbf{x}, t), H\_{\text{dyn}}(t) \right]. \tag{27}$$

Hence by substituting the Hamiltonian (26) into Heisenberg's Eq. (27), making use of the commutation relations (16) and (18), and ensuring equivalence to Eq. (24), it can be shown that

$$f\_{s\lambda}(\mathbf{x}, \mathbf{x}) = -\frac{\partial}{\partial \mathbf{x}} \delta(\mathbf{x} - \mathbf{x}) \tag{28}$$

and therefore

$$H\_{\rm dyn}(t) = -i \sum\_{\varepsilon=\pm 1} \sum\_{\lambda=\pm 1,\lambda} \int\_{-\infty}^{\infty} \text{d}\mathbf{x} \,\,\hbar \mathbf{c} \, a\_{\imath \lambda}^{\dagger}(\mathbf{x}, t) \frac{\partial}{\partial \mathbf{x}} a\_{\imath \lambda}(\mathbf{x}, t) . \tag{29}$$

This Hamiltonian is hermitian and therefore a generator of unitary dynamics. It should also be noted that the Hamiltonian for right-propagating blips takes the negative value of the Hamiltonian for left-propagating blips. This demonstrates that a right-propagating blip behaves identically to a left-propagating blip when the direction of time is reversed.

#### **4. Field observables in the position representation**

The approach to quantisation taken here differs from usual procedures by focussing on the particle character of the EM field rather than the quantised field observables. It is by taking this point of view that the field observables do not require a direct relationship to the wave function of the photon. This view is also held in Ref. [49]. The field observables remain, however, the fundamental observables from which we

may derive expressions for the energy and momentum of the EM field. The purpose of this section is to construct the electric and magnetic field observables in the position representation acting on the extended blip Hilbert space. By insisting that blips are the localised excitations of the EM field, the field observables obtain unique characteristics that are crucial for a fuller understanding of many quantum effects.

#### **4.1 The EM field observables**

#### *4.1.1 An ansatz for the EM field observables*

The electric and magnetic field observables **E**(*x*, *t*) and **B**(*x*, *t*) are a linear and hermitian superposition of the creation and annihilation operators for the photonic excitations of the system. In the position representation, these are the blip operators *a*† *<sup>s</sup><sup>λ</sup>*ð Þ *x*, *t* and *asλ*ð Þ *x*, *t :* Although this superposition is linear, there is no reason to assume that this superposition must be local. In other words, the field observables at a position *x* do not need to be a superposition of blip operators defined at that same point only. For this reason, it is useful to introduce the notation

$$\mathcal{R}[a\_{\imath\dot{\imath}}](\mathbf{x},t) = \int\_{-\infty}^{\infty} \mathbf{dx}' \,\, \mathcal{R}\_{\imath\dot{\imath}}(\mathbf{x},\mathbf{x}') \, a\_{\imath\dot{\imath}}(\mathbf{x}',t). \tag{30}$$

Here R is referred to as the regularisation operator and R *x*, *x*<sup>0</sup> ð Þ is a distribution over the *x* axis.

In the following, the operators **E**(*x*, *t*) and **B**(*x*, *t*) shall denote the complex part of the electric and magnetic field observables respectively. The total real fields are given by the hermitian superposition **<sup>O</sup>** <sup>þ</sup> **<sup>O</sup>**† � �*=*2 where **<sup>O</sup>** <sup>¼</sup> **<sup>E</sup>**,**B***:* Taking this into account, an appropriate ansatz for the complex field observables is

$$\mathbf{E}(\mathbf{x},t) = \sum\_{s=\pm 1} \mathcal{c} \left[ \mathcal{R}[a\_{s\mathcal{H}}](\mathbf{x},t)\mathbf{j} + \mathcal{R}[a\_{s\mathcal{V}}](\mathbf{x},t)\hat{\mathbf{z}} \right] \tag{31}$$

and

$$\mathbf{B}(\mathbf{x},t) = \sum\_{\mathbf{s}=\pm 1} \mathbf{s} \left[ -\mathcal{R}[a\_{\mathrm{sV}}](\mathbf{x},t)\hat{\mathbf{y}} + \mathcal{R}[a\_{\mathrm{sH}}](\mathbf{x},t)\hat{\mathbf{z}} \right]. \tag{32}$$

It may be noted here that all components of the real electric and magnetic field observables commute.

#### *4.1.2 The regularisation operator*

The regularisation operator R provides a relationship between the field observables and the blip operators. Whilst in many quantisations photon wave packets must be locally related to the field observables, for a general choice of R *x*, *x*<sup>0</sup> ð Þ, blips at one position may contribute to the field observables at another position. In fact, we shall see later in this chapter that a single blip contributes to the field observables at all positions along the *x*-axis. Notwithstanding this, the function R *x*, *x*<sup>0</sup> ð Þ must satisfy several general conditions.

Like the blip operators, the expectation values of the field observables must satisfy Maxwell's equations with respect to any time-independent state. Taking into account the orientation of the field components in Eqs. (31) and (32), this condition implies that the regularised blip operators R½ � *as<sup>λ</sup>* ð Þ *x*, *t* must satisfy Eq. (24). Since this equation is also satisfied by the blip operators, using Eq. (30) it can be demonstrated that R*s<sup>λ</sup> x*, *x*<sup>0</sup> ð Þ must be position invariant; that is, R*s<sup>λ</sup> x*, *x*<sup>0</sup> ð Þ¼ R*s<sup>λ</sup> x* � *x*<sup>0</sup> ð Þ. What is more, since the medium is homogeneous and isotropic, the regularisation must be symmetric, R*s<sup>λ</sup> x* � *x*<sup>0</sup> ð Þ¼ R*s<sup>λ</sup> x*ð Þ <sup>0</sup> � *x* , and independent of *s* and *λ*, R*s<sup>λ</sup> x* � *x*<sup>0</sup> ð Þ¼ R *x* � *x*<sup>0</sup> ð Þ*:*

#### **4.2 Energy in the position representation**

#### *4.2.1 The energy observable*

Now that we have a pair of expressions for the electric and magnetic field observables, it is possible to determine the energy observable for the free field in onedimension. To do so we substitute the field observables (31) and (32) into the classical expression for the energy determined in Eq. (3). In return we find that

$$H\_{\text{energy}}(t) = \sum\_{\iota=\pm 1} \sum\_{\lambda=\pm 1,\,\mathbf{V}} \int\_{-\infty}^{\infty} d\mathbf{x} \, \frac{Aec^2}{4} \left\{ \mathcal{R}[a\_{\lambda i}](\mathbf{x}, t) + H.c \right\}^2 \tag{33}$$

Due to the square in the integrand, this observable is strictly positive as would be expected for an energy. This result, however, implies that the energy observable cannot be equal to the dynamical Hamiltonian. Whereas the energy of a single blip is always positive, the left- and right-moving components of the dynamical Hamiltonian have opposite signs.

#### *4.2.2 Energy conservation*

In a closed system, energy is always conserved. In standard quantisations, when the dynamical Hamiltonian is equivalent to the energy observable, conservation of energy is guaranteed automatically as a consequence of Heisenberg's equation. We have seen, however, that in this quantisation the dynamical Hamiltonian and the energy observable are not equal. Energy conservation is only guaranteed, therefore, if *H*energyð Þ*t* and *H*dynð Þ*t* commute. Using the expressions for *H*dynð Þ*t* and *H*energyð Þ*t* given in Eqs. (26) and (33) respectively, and by taking into account that *fs<sup>λ</sup> x* � *x*<sup>0</sup> ð Þ is an odd function and R *x* � *x*<sup>0</sup> ð Þ an even function, it can be shown that the dynamical Hamiltonian and the energy observable commute with each other. Hence, the energy of the free EM field is conserved.

#### **4.3 Non-local contributions to the field observables**

#### *4.3.1 Monochromatic excitations*

It can be seen from Eq. (33) that the regularisation operator plays an important role in determining the energy of a photon. Since the energy of a photon is determined by its frequency, it is convenient to express the energy observable (33) in a basis of monochromatic excitations. Such a set of excitations can be constructed by

considering the Fourier transform of the localised blip operators. We introduce, therefore, the operators

$$a\_{s\lambda}(k,t) = \int\_{-\infty}^{\infty} \frac{\mathbf{d}k}{\sqrt{2\pi}} \ e^{-ik\mathbf{x}} a\_{s\lambda}(\mathbf{x},t) \tag{34}$$

which, using Eq. (18), can be shown to satisfy the equal-time commutation relation

$$\left[a\_{s\boldsymbol{\lambda}}(\boldsymbol{k},\boldsymbol{t}), a\_{\boldsymbol{\lambda}'\boldsymbol{\lambda}'}^{\dagger}(\boldsymbol{k}',\boldsymbol{t})\right] = \delta\_{\boldsymbol{a}'}\delta\_{\boldsymbol{\lambda}\boldsymbol{\lambda}'}\delta(\boldsymbol{k}-\boldsymbol{k}').\tag{35}$$

All annihilation operators commute amongst themselves, as do the creation operators.

#### *4.3.2 The energy of a photon*

Expressed in terms of the monochromatic operators *asλ*ð Þ *k*, *t* and their hermitian conjugates, the energy observable (33) takes the alternative form

$$H\_{\text{energy}}(t) = \sum\_{s=\pm 1} \sum\_{k=\text{H}\_{\text{s}}\text{V}} \int\_{-\infty}^{\infty} \text{d}k \, \frac{A \epsilon \pi c^{2}}{2} \|\mathcal{R}(k)a\_{i\ell}(k,t) + \mathcal{R}^{\*}(-k) \, a\_{i\ell}^{\dagger}(-k,t)\|^{2} . \tag{36}$$

In the expression above, Rð Þ*k* is the Fourier transform of the regularisation function R *x* � *x*<sup>0</sup> ð Þ defined earlier. By taking into account the commutation relation (35), we can show that the energy expectation value of a single monochromatic excitation of angular frequency *ω* ¼ *skc* is

$$
\langle \mathbf{1}\_{\boldsymbol{\omega}\boldsymbol{k}}(\boldsymbol{k},\boldsymbol{t}) | \boldsymbol{H}\_{\text{energy}}(\boldsymbol{t}) | \mathbf{1}\_{\boldsymbol{\omega}\boldsymbol{k}}(\boldsymbol{k},\boldsymbol{t}) \rangle = \boldsymbol{A}\boldsymbol{\epsilon}\boldsymbol{\pi}\boldsymbol{\epsilon}^{2} \left| \mathcal{R}(\boldsymbol{k}) \right|^{2} \delta(\mathbf{0}).\tag{37}
$$

In the above the state ∣1*s<sup>λ</sup>*ð Þi *k*, *t* is defined analogously to the blip state in Eq. (14). The delta function appearing in Eq. (37) is due only to the infinite normalisation of the monochromatic states.

When an atom with transition energy ℏ∣*ω*∣ emits a photon, exactly one excitation is generated oscillating with an angular frequency *ω*. If the total energy is carried away by the photon, an excitation of frequency *ω* must have an energy ℏ∣*ω*∣. Equating the energy of the monochromatic excitation, therefore, with the expectation value (37), we can determine, up to an overall phase, an expression for the function Rð Þ*k* . Doing so we find that

$$\mathcal{R}(k) = \sqrt{\frac{\hbar|k|}{A\epsilon\pi c}}\tag{38}$$

#### *4.3.3 A non-local regularisation*

Now that we have determined Rð Þ*k* , we can calculate R *x* � *x*<sup>0</sup> ð Þ explicitly, thus providing us with a relationship between blips and the field observables in the position representation. Taking the Fourier transform of Eq. (38), we find that

$$\mathcal{R}(\mathbf{x} - \mathbf{x}') = \int\_{-\infty}^{\infty} \frac{\mathbf{d}k}{2\pi} \ e^{i\mathbf{k}(\mathbf{x} - \mathbf{x}')} \sqrt{\frac{2\hbar|\mathbf{k}|}{e\mathbf{A}c}}.\tag{39}$$

When *x* 6¼ *x*<sup>0</sup> , this expression can be given in the alternative form

#### **Figure 2.**

*The figure illustrates the contribution of a single blip (the blue spot) to the electric field observable. The blip in the diagram contributes to measurements of the electric field observable at all points along the x axis. The magnitude of the contribution decreases with the distance of the measurement from the blip.*

$$\mathcal{R}(\mathbf{x} - \mathbf{x}') = -\sqrt{\frac{\hbar}{4\pi\epsilon A c}} \frac{1}{|\mathbf{x} - \mathbf{x}'|^{3/2}}.\tag{40}$$

The distribution R *x* � *x*<sup>0</sup> ð Þ is non-vanishing everywhere, and decreases away from the origin with the negative three halves power of distance.

Whilst the blips represent the localised particles of the system, and interact only locally with their surroundings, they contribute to the measurement of the electric and magnetic field observables in a highly non-local way. One way to think about this relationship is that the measurement of the electric and magnetic fields takes into account the behaviour of blips at all positions in space. An alternative perspective would be to treat each blip as a "carrier" of a non-local field. This interpretation is visualised in **Figure 2** showing the contribution of a single blip to the electric field. In a similar manner to as one may think about the gravitational field about the earth, one may think of a single blip as being surrounded by and carrying with it a non-local electromagnetic field.

#### **5. Conclusions**

In standard treatments of the quantised EM field in one dimension, single photon wave packets are characterised by a wave vector *k* and a polarisation *λ*. By demanding that photons can be localised, however, and that they must propagate at the speed of light along the *x*-axis, we have shown that an additional discrete parameter *s* ¼ �1 must be introduced, thus doubling the size of the usual Hilbert space. Whilst canonical quantisation overlooks these degrees of freedom, by taking the viewpoint of a Schrödinger equation, these additional degrees of freedom arise naturally. What is more, by maintaining the perspective of a Schrödinger equation, a Hamiltonian operator *H*dynð Þ*t* was constructed that generates the time-evolution of state vectors in the Hilbert space. Unlike other quantisations, however, our Hamiltonian necessarily possess both positive and negative eigenvalues. This can be most clearly seen by

expressing the dynamical Hamiltonian in the momentum representation. In terms of the monochromatic operators defined in Eq. (34), *H*dynð Þ*t* is given by the expression

$$H\_{\rm dyn}(t) = \sum\_{s=\pm 1} \sum\_{\lambda = \pm \mathbf{l}\_{\lambda} \mathbf{V}} \int\_{-\infty}^{\infty} \mathbf{d}k \,\,\hbar ck \,\mathbf{k} \, a\_{\lambda \lambda}^{\dagger}(k, t) a\_{\lambda \lambda}(k, t) . \tag{41}$$

This Hamiltonian is almost identical to the usual Hamiltonian of the free EM field, but the eigenvalues ℏ*sck* can take both positive and negative values.

In standard quantisations, the Hamiltonian is always positive. The Hamiltonian above is positive only when the signs of *s* and *k* coincide, or equivalently when the sign of *k* indicates direction of propagation. Under this condition the dynamical Hamiltonian (41) and the energy observable (36) are equivalent. In this quantisation we therefore go beyond current assumptions by demanding that the Hamiltonian can be both positive and negative. This idea is not new, however. In the work of Hegerfeldt and others, it was shown that a wave packet cannot propagate causally when the Hamiltonian is bounded from below [37]. Mostafazadeh and Zamani [50], therefore, introduced a new inner product that enabled the use of negative frequency states. More recently, with similar justifications, Hawton considers real EM field excitations that necessarily contain both positive and negative frequency contributions [51–53].

By extending the Hilbert space to include both positive- and negative- frequency photons, it is possible to localise a photon not only in space, but also in time. In quantum physics, it has been a significant problem to define a time operator [48]. Whereas the position of a particle is associated with a position observable, time is only a parameter of the system. Since in our quantisation all wave packets propagate causally, an operator can be defined that determines the time at which a particle will reach a particular position. In investigations into a "quantum relativity", where both space and time are placed on an equal quantum footing, a key concept is that of a closed and stationary universe [54]. The dynamics of these systems are constrained by a Wheeler-deWitt equation. As the dynamical constraint for blips (24) is of this form, and defines a stationary state in the global Hilbert space of excitations in space-time, the blip quantisation may provide a useful and insightful scheme for the modelling of quantum clocks and the study of a quantum time.

One of the most significant features of this quantisation is the non-locality between the blip operators and the field observables. This result is a direct consequence of the frequency-dependence of energy carried by the EM field. In many schemes, localisation of a photon is synonymous with localisation of the field observables. To fix the disparity between a probabilistic wave function and an energy carrying field, however, non-local or non-hermitian inner products are introduced. Although often elegant, such inner products can be cumbersome, and may depend upon the boundary conditions of the system. In the view of this quantisation, however, an excitation of the field is only localised if it satisfies the orthogonality relation (17), and such nonlocal contributions must therefore exist. Whilst this viewpoint may raise questions regarding instantaneous contributions to the field observables, the non-locality of the field observable is an important feature of the EM field. For instance, in Ref. [55], it is shown that these non-local field contributions are responsible for the Casimir effect between two parallel conducting plates.

The Schrödinger equation for light gives an alternative perspective on the particle behaviour of the free EM field. By returning to principles of wave-particle duality, the existing theory of the EM field has been shown to fall short with regard to the

propagation of localised particles, and as a result new physics has been unearthed. The new quantisation that has followed provides a major change in our perspective on the interactions of photons and the way we describe them. For instance, using this quantisation one can construct a local interaction Hamiltonian for a double-sided semi-transparent mirror; something that was not previously possible [56]. Moreover, an investigation of blips in a cavity leads to a new perspective on the origin of the Casimir effect [55]. In future, this quantisation may contribute towards a fuller understanding of well-known quantum effects, and may also provide the tools necessary for studying new ones.

### **Acknowledgements**

The author acknowledges financial support from the UK Engineering and Physical Sciences Research Council EPSRC [grant number EP/W524372/1]. The author also acknowledges the contributions of M. Basil Altaie, Almut Beige, Rob Purdy and Jake Southall to the original research that is presented and discussed within this chapter.

#### **Conflict of interest**

The author declares no conflict of interest.

### **Author details**

Daniel R.E. Hodgson University of Leeds, Leeds, United Kingdom

\*Address all correspondence to: phydreh@leeds.ac.uk

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A Schrödinger Equation for Light DOI: http://dx.doi.org/10.5772/intechopen.112950*

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#### **Chapter 3**

### Path Integral of Schrödinger' s Equation

*Hocine Boukabcha, Salah Eddin Aid and Amina Ghobrini*

#### **Abstract**

The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.

**Keywords:** radial propagator, space–time transformation, modified Pöschl-Teller potential, energy spectrum, wave functions

#### **1. Introduction**

In this chapter, the Schrödinger solutions of potential problem have been evaluated using the Feynman path integral formulation of quantum mechanics; an appropriate space-time transformation has been applied to Green's function associated with the problem, which made it an integrable function. Also, the energy spectrum in a non-relativistic regime with normalized wave functions for potential, is obtained using path integral formalism of quantum mechanics; the results are evaluated for any state due to the use of an approximation scheme for centrifugal term 1/r<sup>2</sup> , the constructed propagator associated with the Schrödinger equation of the problem was treated by space-time transformation trick that made it integrable, and energy eigenvalues for some exceptional cases of potential were also presented to compare our solutions with those obtained in previous studies. The organization of this chapter is as follows: in Section 1, we formulate the radial propagator and its corresponding Green's function associated with a nonrelativistic particle in the presence of a potential where we use an approximation to the centrifugal term. In Section 2, we treat Green's function of Generalized inverse quadratic Yukawa potential by performing a nontrivial space-time transformation to pass from the actual complex problem to another already solved one, which is a Pöschl-Teller (PT) potential problem. In Section 3, energy eigenvalues and corresponding eigenfunctions are extracted from the poles and residues of the aforementioned solved Green's function. Section 4 discusses special cases of Deng Fun potentiel, Generalized inverse quadratic Yukawa

potential as Kratzer potential, Yukawa potential, inversely quadratic Yukawa potential, and Coulomb potential.

#### **2. Propagator and Schrödinger equation**

The Schrödinger equation is a fundamental equation in quantum physics that describes the behavior of quantum systems. It was formulated by Erwin Schrödinger in 1925. The Schrödinger equation describes the time evolution of the wave function of a quantum system is governed by the equation:

$$i\hbar\frac{d}{dt}|\psi\left(t\right)\rangle = \hat{H}|\psi\left(t\right)\rangle,\tag{1}$$

which integrates in the particular case of a time-independent Hamiltonian:

$$|\psi\left(t'\right)\rangle = \exp\left[-\frac{i}{\hbar}\hat{H}\left(t'-t\right)\right]|\psi\left(t\right)\rangle,\tag{2}$$

where *U t* ^ <sup>0</sup> ð Þ¼ , *<sup>t</sup>* exp � *<sup>i</sup>* <sup>ℏ</sup> *H t* ^ ð Þ <sup>0</sup> � *<sup>t</sup>* � � represents the evolution operator. Let us consider a particle in a potential, the Hamiltonian is written as:

$$
\hat{H} = \frac{\hat{P}^2}{2m} + V(\hat{\mathfrak{x}}).\tag{3}
$$

In the position representation *x*, the evolution equation becomes:

$$
\langle \mathbf{x'} | \boldsymbol{\nu} \left( t' \right) \rangle = \left\langle \mathbf{x'} | \exp \left( \frac{-i}{\hbar} \hat{H} (t' - t) \right) | \boldsymbol{\nu} \left( t \right) \right\rangle. \tag{4}
$$

Let us use the position closure relation *x*

$$\int d\mathbf{x} |\mathbf{x}\rangle\langle\mathbf{x}| = \mathbf{1}.\tag{5}$$

Eq. (4) becomes:

$$
\langle \mathbf{x'} | \boldsymbol{\nu}(t') \rangle = \int d\mathbf{x} \left\langle \mathbf{x'} | \exp\left(\frac{-i}{\hbar} \hat{H}(t'-t)\right) | \boldsymbol{\nu}(t) \right\rangle. \tag{6}
$$

Thus, it can be written as

$$
\psi\left(\mathbf{x}',t'\right) = \int K\left(\mathbf{x}',t';\mathbf{x},t\right)\psi\left(\mathbf{x},t\right)d\mathbf{x}.\tag{7}
$$

The propagator *K x*<sup>0</sup> , *t* <sup>0</sup> ð Þ¼ ; *x*, *t x*<sup>0</sup> <sup>j</sup> exp �*<sup>i</sup>* <sup>ℏ</sup> *H t* ^ ð Þ <sup>0</sup> � *<sup>t</sup>* � �j*<sup>x</sup>* � � allows us to evaluate the transition amplitude between the two states. Let us consider an initial state localized at *x*<sup>0</sup>

$$
\psi(\mathbf{x}, t\_0) = \delta(\mathbf{x} - \mathbf{x}\_0), \tag{8}
$$

then,

$$
\Psi(\mathbf{x}',t') = K(\mathbf{x}',t';\mathbf{x}\_0,t\_0). \tag{9}
$$

the probability amplitude of finding the particle at position *x*<sup>0</sup> at time *t* 0 *:*

#### **3. Transition from propagator to Green's function**

As we saw earlier, the propagator can be expressed in terms of the time-evolution operator as follows [1, 2]:

$$K(\mathbf{x''},t'';\mathbf{x'},t')=\langle \mathbf{x''}|U(t'',t')|\mathbf{x'}\rangle,\tag{10}$$

where

$$U(t'',t') = \exp\left[\frac{-i}{\hbar}\hat{H}(t''-t')\right],\tag{11}$$

with *T* ¼ *t* <sup>00</sup> � *t* 0 *:* Moreover, it is possible to extract the energy spectrum as well as the wave function corresponding to a given physical system, from Green's function, the latter being none other than the Fourier transform of the propagator. In effect [3],

$$G(\mathbf{x''}, \mathbf{x'}; E) = \frac{i}{\hbar} \int\_0^\infty dT e^{i(E + i\epsilon)T/\hbar} K(\mathbf{x''}, \mathbf{x'}; T), \tag{12}$$

where *ε*, is a positive constant

$$(\hat{H} - E)G(\mathfrak{x}'', \mathfrak{x}'; E) = \delta(\mathfrak{x}'' - \mathfrak{x}'). \tag{13}$$

Formula (10) allows us to write:

$$\mathcal{G}\left(\mathbf{x}'',\mathbf{x}';E\right) = \left\langle \mathbf{x}'' \vert \frac{\mathbf{1}}{\hat{H} - E - i\varepsilon} \vert \mathbf{x}' \right\rangle. \tag{14}$$

By introducing the closure relation on position ∣*n*, ℓi, we can express the probability amplitude (11) as follows:

$$G(\mathbf{x}'', \mathbf{x}'; E) = \frac{i}{\hbar} \int\_0^\infty \sum\_{n, \ell} dT \langle \mathbf{x}''|n, \ell' \rangle \left\langle n, \ell' | e^{\frac{i(E+i\epsilon)T}{\hbar}} e^{-\frac{i\ell T}{\hbar}} | \mathbf{x}' \right\rangle,\tag{15}$$

or alternatively,

$$G\left(\mathbf{x}^{\prime\prime},\mathbf{x}^{\prime};E\right) = \sum\_{n,\ell}^{\infty} \frac{\chi^{\*}\_{n,\ell}\left(\mathbf{x}^{\prime}\right)\chi\_{n,\ell}\left(\mathbf{x}^{\prime}\right)}{E\_{n,\ell} - E - i\varepsilon}.\tag{16}$$

where *χ<sup>n</sup>*,<sup>ℓ</sup> ð Þ *x* is the wave function corresponding to the eigenenergy *En*,<sup>ℓ</sup>, and it is also possible to arrive at *χ<sup>n</sup>*,*<sup>l</sup>*ð Þ *x* and *En*,<sup>ℓ</sup> from the relation.

$$K(\mathbf{x}'',t'';\mathbf{x}',t') = \sum\_{n,\ell}^{\infty} \chi\_{n,\ell}^\*(\mathbf{x}')\chi\_{n,\ell}(\mathbf{x}'')e^{-\frac{i\mathbf{E}\_{n,\ell}T}{\hbar}}.\tag{17}$$

#### **4. Path integral in spherical coordinates**

In quantum mechanics, rotational symmetry is crucial in finding the wave functions and corresponding energies of physical systems. Spherical coordinates transform from the Schrödinger equation of rotational symmetry. Therefore, we can separate this equation into an angular part expressed in terms of spherical harmonics, whose solutions are known, and a radial part that contains specific information about the dynamical systems.

In the path integral, this coordinate transformation is possible, but initially, things become complicated. One of these complexities arises when studying the presence of a centrifugal barrier, which eliminates the possibility of "time slicing."

The following relation represents the formula for the three-dimensional (3D) propagator [4, 5]:

$$\begin{split} K\left(\overrightarrow{r'},t';\overrightarrow{r'},t'\right) &= \int\_{\overrightarrow{r'}}^{\overrightarrow{\overrightarrow{\eta'}}} D\overrightarrow{r}(t) \exp\left[\frac{i}{\hbar}\left(\frac{m}{2}\left(\Delta\overrightarrow{r'\_{j}}\right)^{2} - V\left(\overrightarrow{r}\right)\right)\right] dt\\ &= \lim\_{N\to\infty} \prod\_{j=1}^{N+1} \left(\frac{m}{2i\pi\hbar e}\right)^{\frac{1}{2}} \left[\prod\_{i=1}^{N} \int\_{\mathbb{R}^{3}} d\overrightarrow{r\_{j}}\right] \exp\left[\frac{i}{\hbar}\mathbf{S}\_{N}\right],\end{split} \tag{18}$$

with

$$\begin{aligned} t'' &= t\_{N+1}; \ t' = t\_0\\ \overrightarrow{r'} &= \overrightarrow{r}\_{N+1}; \ \overrightarrow{r'} &= \overrightarrow{r}\_0 \end{aligned}$$

and the total action:

$$\mathbf{S}\_{N} = \sum\_{j=1}^{N+1} \mathbf{S}\_{j} = \sum\_{j=1}^{N+1} \left( \frac{m}{2\varepsilon} \left( r\_{j}^{2} + r\_{j-1}^{2} - 2\overrightarrow{r}\_{j} \cdot \overrightarrow{r}\_{j-1} \right) - \varepsilon V(\overrightarrow{r\_{j}}) \right). \tag{19}$$

Using the spherical coordinate system ð Þ *r*, *θ*, *φ* defined as:

$$\begin{cases} x = r \sin \theta \cos \varphi \\ y = r \sin \theta \sin \varphi, \\ z = r \cos \theta \end{cases} \tag{20}$$

with *r*>0; 0≤ *θ* <*π* and 0≤*φ* <2*π:*

Where the volume element is expressed in spherical coordinates as:

$$d\overrightarrow{r\_j} = r\_j^2 \sin\theta\_j dr\_j d\theta\_j d\rho\_j,\tag{21}$$

the propagator (18) can be rewritten in spherical coordinates as:

*Path Integral of Schrödinger's Equation DOI: http://dx.doi.org/10.5772/intechopen.112183*

$$\begin{split} K\left(\overrightarrow{r''},t'';\overrightarrow{r'},t'\right) &= \lim\_{N\to\infty} \prod\_{j=1}^{N+1} \left(\frac{m}{2i\pi\hbar\epsilon}\right)^{\frac{1}{2}} \prod\_{j=1}^{N} \left[\int\_{0}^{\infty} \left[\int\_{0}^{\pi}\int\_{0}^{2\pi} r\_{j}^{2} \sin\theta\_{j} dr\_{j} d\theta\_{j} d\phi\_{j}\right]\right. \\ &\quad \times \prod\_{j=1}^{N+1} \left[\exp\left(\frac{i}{\hbar}S\_{j}\right)\right], \end{split} \tag{22}$$

The elemental action is:

$$S\_{\dot{\jmath}} = \frac{m}{2\varepsilon} \left( r\_{\dot{\jmath}}^2 + r\_{\dot{\jmath}-1}^2 - 2r\_{\dot{\jmath}}r\_{\dot{\jmath}-1}\cos\Theta\_{\dot{\jmath},\dot{\jmath}-1} \right) - \varepsilon V(r\_{\dot{\jmath}}),\tag{23}$$

where Θ*j*,*j*�<sup>1</sup> ¼ *r* ! *<sup>j</sup>*, *r* ! *j*�1 � �,

with the angle between two vectors in spherical coordinates being:

$$\cos\Theta\_{j\dot{j}-1} = \cos\theta\_{\dot{j}}\cos\theta\_{\dot{j}-1} + \sin\theta\_{\dot{j}}\sin\theta\_{\dot{j}-1}\cos\left(\varphi\_{\dot{j}} - \varphi\_{\dot{j}-1}\right),\tag{24}$$

and the measurement takes the form:

$$\prod\_{j=1}^{N+1} \left(\frac{m}{2i\pi\hbar e}\right)^{\frac{3}{2}} \prod\_{j=1}^{N} d\vec{r} = \left(\frac{m}{2i\pi\hbar e}\right)^{\frac{3}{2}(N+1)} \prod\_{j=1}^{N} \left[\int\_{0}^{\infty} \int\_{0}^{\pi} \int\_{0}^{2\pi} r\_{j}^{2} \sin\theta\_{j} dr\_{j} d\theta\_{j} d\rho\_{j}\right].\tag{25}$$

The previous expression of the propagator is not appropriate for integration due to the presence of the term �*<sup>i</sup>* ℏ *m <sup>ε</sup> rjrj*�<sup>1</sup> cos Θ*j*, *<sup>j</sup>*�<sup>1</sup> � �, and this term is separable into a radial part and an angular part.

For an explicit evolution of the angular part of the propagator, we will use the following formula [6]:

$$\begin{split} \exp(i z \cos \phi) &= \quad 2^{\frac{1}{2}} \Gamma \Big( \frac{1}{2} \Big) \sum\_{k=0}^{+\infty} \left( k + \frac{1}{2} \right) i^k(z)^{\frac{-1}{2}} f\_{k + \frac{1}{2}}(z) P\_k(\cos \phi) \\ &= \quad \sqrt{\frac{\pi}{2x}} \sum\_{k=0}^{+\infty} (2k+1) i^k f\_{k + \frac{1}{2}} P\_k(\cos \phi). \end{split} \tag{26}$$

*Jn*ð Þ *x* is the Bessel function, which is given by:

$$J\_n(\infty) = \sum\_{P=0}^{+\infty} \frac{\left(-1\right)^P}{P!(n+P)!} \left(\frac{\chi}{2}\right)^{2P+n},\tag{27}$$

if

$$
\mathfrak{x} = -U,\tag{28}
$$

then

$$\begin{split} J\_n(-U) &= \sum\_{P=0}^{+\infty} \frac{(-1)^P}{P!(n+P)!} \left(\frac{-U}{2}\right)^{2P+n} \\ &= \sum\_{P=0}^{+\infty} \frac{(-1)^P}{P!(n+P)!} \left(\frac{U}{2}\right)^{2P+n} (-1)^n. \end{split} \tag{29}$$

we have

$$J\_n(i\mathbf{x}) = i^\text{\textquotedblleft}I\_n\text{,}\tag{30}$$

where *In* is the modified Bessel function. We define

$$\mathbf{y} = \mathbf{i}\mathbf{z},\tag{31}$$

According to (26) and (30), we can deduce that

$$\begin{split} e^{\mathcal{Y}^{\text{cos}}\cdot\varphi} &= \quad \sqrt{\frac{-\pi}{2\dot{\mathcal{Y}}}} \sum\_{k=0}^{+\leftrightarrow} (2k+1) i^k f\_{k+\frac{1}{2}} P\_k(\cos\varphi) \\ &= \quad (-1)^{\frac{1}{2}} \sqrt{\frac{\pi}{2\dot{\mathcal{Y}}}} \sum\_{k=0}^{+\leftrightarrow} (2k+1) i^k (-1)^{k+\frac{1}{2}} I\_{k+\frac{1}{2}}(\dot{\mathcal{Y}}) P\_k(\cos\varphi) \\ &= \quad (-1)^{\frac{1}{2}} \sqrt{\frac{\pi}{2\dot{\mathcal{Y}}}} \sum\_{k=0}^{+\leftrightarrow} (2k+1) i^k (-1)^{k+\frac{1}{2}} (i)^{k+\frac{1}{2}} I\_{k+\frac{1}{2}}(\dot{\mathcal{Y}}) P\_k(\cos\varphi) \\ &= \quad \sqrt{\frac{\pi}{2\dot{\mathcal{Y}}}} \sum\_{k=0}^{+\leftrightarrow} (2k+1) I\_{k+\frac{1}{2}}(\dot{\mathcal{Y}}) P\_k(\cos\varphi), \end{split} \tag{32}$$

where *Pk*ð Þ cos *φ* are the Legendre polynomials.

We arrive at the following expression for the propagator, by substituting formula (32) in (22):

$$\begin{split} K\left(\overrightarrow{r'},t';\overrightarrow{r'},t'\right) &=& \lim\_{N\to\infty} \prod\_{j=1}^{N+1} \left(\frac{m}{2i\pi\hbar\varepsilon}\right)^{\frac{1}{2}} \prod\_{i=1}^{N} \left[\int\_{0}^{\omega} \left[\int\_{0}^{2\varepsilon} r\_{j}^{2} \sin\theta\_{j} dr d\theta\_{j} d\theta\_{j}\right] \right] \\ & \quad \times \prod\_{j=1}^{N+1} \left[\exp\left[\frac{i}{\hbar} \left(\frac{m}{2\varepsilon}\left(r\_{j}^{2} + r\_{j-1}^{2}\right) - \epsilon V(r\_{j})\right)\right] \right] \\ & \quad \times \prod\_{j=1}^{N+1} \left(\frac{i\pi\hbar\varepsilon}{2mr\_{j}r\_{j-1}}\right)^{\frac{1}{2}} \sum\_{l\_{i},l\_{k+1=0}}^{+\infty} (2l\_{j} + 1) \\ & \quad \times I\_{l\_{j}+\frac{1}{2}}(\frac{mr\_{j}r\_{j-1}}{i\hbar\varepsilon}) P\_{l}(\cos\Theta\_{j;l-1}), \\ & \quad \quad l\_{j} = 1, \forall j = 1, \ldots, N+1 \end{split} (33)$$

where else

$$\begin{split} K\left(\overrightarrow{r'},r'',\overrightarrow{r'},t'\right) &= \lim\_{N\to\infty} \left(\frac{m}{2i\pi\hbar\epsilon}\right)^{\frac{1}{2}(N+1)} \sum\_{l-l\mu\_{l+1}}^{+\infty} \left[\prod\_{j=1}^{N} \int\_{0}^{\alpha} r\_{j}^{2} dr\_{j}\right] \\ &\times \quad \left[\prod\_{j=1}^{N} \int\_{0}^{\alpha} \left(2l\_{j}+1\right) I\_{l\_{j}+1} \left(\frac{mrr\_{j-1}}{i\hbar\epsilon}\right) P\_{l\_{l}}(\cos\Theta\_{j-1}) \sin\theta\_{j} d\theta\_{j} d\phi\_{j}\right] \\ &\quad \times \quad \left[(2l\_{N+1}+1)I\_{l\_{N+1}+\frac{1}{2}} \left(\frac{mrr\_{j}r\_{j-1}}{i\hbar\epsilon}\right) P\_{l\_{N+1}}(\cos\theta\_{j-1})\right] \\ &\quad \times \quad \prod\_{j=1}^{N+1} \left[\exp\left[\frac{i}{\hbar} \left(\frac{m}{2\epsilon} \left(r\_{j}^{2} + r\_{j-1}^{2}\right) - \epsilon V(r\_{j})\right)\right]\right] \\ &\quad \times \quad \prod\_{j=1}^{N+1} \left(\frac{in\hbar\epsilon}{2mr\_{j}r\_{j-1}}\right)^{\frac{1}{2}}, \end{split} \tag{34}$$

we can use the following expression

$$\prod\_{j=1}^{N} r\_j = \frac{1}{\sqrt{r'r'}} \prod\_{j=1}^{N+1} \left(r\_j r\_{j-1}\right)^{\frac{1}{2}},\tag{35}$$

by substituting Expression (35) in (34), we obtain

$$\begin{split} K\left(\overrightarrow{r'},t';\overrightarrow{r'},t'\right) &= \lim\_{N\to\infty} \left(\frac{m}{2i\pi\hbar\varepsilon}\right)^{(N+1)} \left(\frac{1}{\sqrt{r'}r'}\right) \sum\_{\ell\_{-}=\ell\_{N+1}}^{+\infty} \left[\prod\_{j=1}^{N} \left[\int\_{0}^{+\infty} r\_j dr\_j\right] \right. \\\\ &\times \quad \prod\_{j=1}^{N+1} \left[\exp\left[\frac{i}{\hbar}\left(\frac{m}{2\epsilon}\left(r\_j^2 + r\_{j-1}^2\right) - \varepsilon V\left(r\_j\right)\right)\right]\right] \\\\ &\times \quad \prod\_{j=1}^{N} \left[\int\_{0}^{\pi} \left(2\ell\_j + 1\right) I\_{\ell\_j + \frac{1}{2}} \left(\frac{mr\_j r\_{j-1}}{i\hbar\epsilon}\right) P\_{\ell\_j}(\cos\Theta\_{j;-1}) \sin\theta\_j d\rho\_j\right] \\\\ &\times \quad \left[\left(2\ell\_{N+1} + I\_{\ell\_{N+1} + \frac{1}{2}}\left(\frac{mr\_j r\_{j-1}}{i\hbar\epsilon}\right) P\_{\ell\_{N+1}}(\cos\Theta\_{j;-1})\right)\right]. \end{split} \tag{36}$$

The Legendre polynomials can be decomposed into spherical harmonics

$$P\_{\ell}(\cos \theta) = \left(\frac{4\pi}{2\ell + 1}\right) \sum\_{m=-\ell} Y\_{\ell,m}(\theta\_N, \varphi\_N) Y\_{\ell,m}^\*(\theta\_{N-1}, \varphi\_{N-1}),\tag{37}$$

where *P*ℓð Þ cos *θ* represents the Legendre polynomial of order ℓ and degree *m*, *θ* is the polar angle, and *ϕ* is the azimuthal angle. *Y*<sup>ℓ</sup>,*<sup>m</sup>*ð Þ *θ*, *φ* corresponds to the associated spherical harmonic of order ℓ and degree *m*.

This formula establishes a connection between Legendre polynomials and spherical harmonics, providing an expansion in terms of angles for functions or phenomena with spherical symmetry,

where

$$Y\_{\ell,m}(\theta,\varphi) = (-1)^m \left[ \frac{(2\ell+1)}{4\pi} \times \frac{(\ell-m)!}{(\ell+m)!} \right] P\_{\ell}^m(\cos\theta) \exp(im\varphi),\tag{38}$$

Formula (32) is as follows:

$$\mathfrak{e}^{\text{y}\cos\varphi} = 2\pi \sqrt{\frac{2\pi}{\mathcal{Y}}} \sum\_{\ell=0}^{+\infty} (2\ell+1) I\_{\ell+\frac{1}{2}}(\mathfrak{y}) \sum\_{m=-\ell}^{\ell} Y\_{\ell,m}(\theta\_N, \varphi\_N) Y\_{\ell,m}^\*(\theta\_{N-1}, \varphi\_{N-1}), \tag{39}$$

By inserting the last formula into the propagator expression (36)

*Schrödinger Equation – Fundamentals Aspects and Potential Applications*

$$\begin{split} K\left(\overrightarrow{r'},t';\overrightarrow{r'},t'\right) &=& \lim\_{N\to\infty} \left(\frac{m}{i\hbar c}\right)^{(N+1)} \left(\frac{1}{\sqrt{r'}r'}\right) \sum\_{\ell\_{-},\ell\_{N+1}=0}^{+\infty} \left[\prod\_{j=1}^{N} \int\_{0}^{\infty} r\_{j} d\eta\right] \\ &\times \quad \prod\_{j=1}^{N+1} \left[\exp\left[\frac{i}{\hbar}\left(\frac{m}{2\varepsilon}\left(r\_{j}^{2} + r\_{j-1}^{2}\right) - eV(r\_{j})\right)\right]\right] \\ &\quad \times \quad \prod\_{j=1}^{N} \left[I\_{\ell\_{j}+\frac{1}{2}}(\frac{mr\_{j}r\_{j-1}}{i\hbar c})I\_{\ell\_{N+1}+\frac{1}{2}}(\frac{mr\_{j}r\_{j-1}}{i\hbar c})\right] \\ &\quad \times \quad \sum\_{m\_{j}=-\ell\_{j}}^{\ell\_{j}} \int\_{0}^{\pi} \mathbf{Y}\_{\ell\_{j},m\_{j}}\left(\theta\_{j},\rho\_{j}\right) Y\_{\ell\_{j},m\_{j}}^{\*}\left(\theta\_{j-1},\rho\_{j-1}\right) \sin\theta\_{j} d\theta\_{j} d\rho\_{j} \\ &\quad \times \quad \sum\_{m=-\ell\_{j}}^{\ell} Y\_{\ell\_{N+1,m}}(\theta\_{N+1},\rho\_{N+1}) Y\_{\ell\_{N+1,m}}^{\*}(\theta\_{N},\rho\_{N}). \end{split} \tag{40}$$

Using the orthogonality relation of spherical harmonics, which is described by the following equation

$$\int\_0^\pi d\rho \int\_0^{2\pi} Y\_{\ell,m}(\theta,\rho) Y\_{\ell,m}^\*(\theta,\rho) \sin\theta d\theta = \delta\_{\ell,\ell'}, \delta\_{m,m'},\tag{41}$$

thus we find the following expression for the propagator

$$K\left(\overrightarrow{r''},t'';\overrightarrow{r'},t'\right) = \sum\_{\ell=0}^{+\infty} \frac{(2\ell+1)}{4\pi} K\_{\ell}(r'',t'';r',t')P\_{\ell}(\cos\Theta),\tag{42}$$

where the radial propagator *K*<sup>ℓ</sup> *r*00, *t* <sup>00</sup>;*r*<sup>0</sup> , *t* <sup>0</sup> ð Þ is also expressed as

$$\begin{split} K\_{\ell}(r'',t'';r',t') &= \lim\_{N \to \infty} \left(\frac{m}{i\hbar\varepsilon}\right)^{(N+1)} \left(\frac{1}{\sqrt{r''r'}}\right) \lim\_{N \to \infty} \left[\prod\_{j=1}^{N} \int\_{0}^{\infty} r\_j dr\_j\right] \\ &\times \prod\_{j=1}^{N+1} \left[I\_{\ell\_j + \frac{1}{2}} \binom{mr\_jr\_{j-1}}{i\hbar\varepsilon}\right] \\ &\times \prod\_{j=1}^{N+1} \left[\exp\left[\frac{i}{\hbar} \left(\frac{m}{2\varepsilon} \left(r\_j^2 + r\_{j-1}^2\right) - \varepsilon V(r\_j)\right)\right]\right], \end{split} \tag{43}$$

Indeed, considering the asymptotic behavior of the modified Bessel functions, [6].

$$I\_{\dagger}\left(\frac{z}{\varepsilon}\right) \underset{\varepsilon \to 0}{\longrightarrow} \left(\frac{\varepsilon}{2\pi\varpi}\right)^{\frac{1}{2}} \exp\left\{\frac{z}{\varepsilon} - \frac{1}{2}\frac{\varepsilon}{z}\left(\upsilon^{2} - \frac{1}{4}\right)\right\},\tag{44}$$

then

$$I\_{\ell+\frac{1}{2}}\left(\frac{\frac{mr\_{\mathcal{T}-1}}{i\hbar}}{\varepsilon}\right) \underset{\varepsilon\to 0}{\to} \left(\frac{\epsilon i\hbar}{2\pi mr\_{\mathcal{T}}r\_{j-1}}\right)^{\frac{1}{2}} \exp\left\{\frac{mr\_{\mathcal{T}}r\_{j-1}}{\varepsilon i\hbar} - \left(\frac{\epsilon i\hbar}{2mr\_{\mathcal{T}}r\_{j-1}}\right) (\ell'(\ell+1))\right\},\tag{45}$$

then we arrive at the formulation of the radial propagator in spherical coordinates and as a function of the effective potential *Veff rj* � �:

$$\begin{split} K\_{\ell}(r'',t'';r',t') &= \quad \left(\frac{1}{r''r'}\right) \lim\_{N\to\infty} \left(\frac{m}{2\pi i\hbar e}\right)^{\frac{(N+1)}{2}} \times \left[\prod\_{j=1}^{N} \int\_{0}^{\infty} dr\_{j}\right] \\ &\quad \times \quad \exp\frac{i}{\hbar} \sum\_{j=1}^{N+1} \left\{\frac{m}{2\varepsilon} \left(\Delta r\_{j}\right)^{2} - \varepsilon V(r\_{j}) - \frac{\ell'(\ell+1)\hbar^{2}\varepsilon}{2mr\_{j}r\_{j-1}}\right\}, \end{split} \tag{46}$$

where the effective potential is defined by the following expression:

$$\mathcal{V}\_{\ell\hat{\mathcal{V}}}(r\_{\hat{\jmath}}) = \mathcal{V}(r\_{\hat{\jmath}}) + \frac{\hbar^2 \ell(\ell'+1)}{2mr\_{\hat{\jmath}}r\_{\hat{\jmath}-1}},\tag{47}$$

So the propagator (46) becomes:

$$\begin{split} K\_{\ell}(r'',t'';r',t') &= \ \left(\frac{1}{r''r'}\right)\lim\_{N\to\infty} \left(\frac{m}{2\pi i\hbar e}\right)^{\frac{(N+1)}{2}} \times \left[\prod\_{j=1}^{N}\Big|\_{0}^{\infty} dr\_{j}\right] \\ &\times \quad \exp\left[\frac{i}{\hbar}\sum\_{j=1}^{N+1} \left\{\frac{m}{2\varepsilon}\left(\Delta r\_{j}\right)^{2} - \varepsilon V\_{\mathcal{G}\mathcal{T}}(r\_{j})\right\}\right]. \end{split} \tag{48}$$

The specific form of the radial propagator will depend on the potential energy term *V*(*r*) in the radial Schrödinger equation, which corresponds to the particular physical system being studied. Different potential energy functions will lead to different solutions and, consequently, different forms of the radial propagator.

#### **5. Feynman propagator**

The propagator related to a central potential *V*(*r*) between two space-time points *r* !0 , *t* 0 � � and *<sup>r</sup>* !00, *<sup>t</sup>* <sup>00</sup> � �, in spherical coordinates is written as [4, 5]:

$$K\left(\overrightarrow{r'},t'';\overrightarrow{r'},t'\right) = \frac{1}{4\pi r''r'}\sum\_{\ell=0}^{\infty} (2\ell+1) \times K\_{\ell}(r'',t'';r',t')P\_{\ell}(\cos\Theta),\tag{49}$$

where *P*<sup>ℓ</sup> ð Þ cos Θ is the Legendre polynomial and Θ � *r* !00, *<sup>r</sup>* !0 � � with

$$K\_{\ell}(r'',t'';r',t') = \lim\_{N \to \infty} \left[ \prod\_{j=1}^{N} \exp\left[\frac{i}{\hbar} \mathbf{S}\_{j}\right] \times \prod\_{j=1}^{N} \left[\frac{m}{2\pi i\hbar e}\right]^{\frac{1}{2}} \prod\_{j=1}^{N-1} dr\_{j} \right] \tag{50}$$

where

$$\mathcal{S}\_{\circ} = \frac{m}{2\varepsilon} \left(\Delta r\_{\circ}\right)^{2} - \varepsilon V\_{\text{eff}}\left(r\_{\circ}\right),\tag{51}$$

here

Δ*rj* ¼ *rj* � *rj*�1, *ε* ¼ Δ*tj* ¼ *tj* � *tj*�1, *t* <sup>0</sup> ¼ *t*<sup>0</sup> and *t* <sup>00</sup> ¼ *tN*, and effective potential *Veff* is defined by the relation as:

$$V\_{eff}(r) = \frac{\hbar^2}{2m} \frac{\ell(\ell+1)}{r^2} + V(r),\tag{52}$$

Thus, the condensed form is given by:

$$K\_{\ell}(r'',t'';r',t') = \mathfrak{D}r(t)\exp\left[\int\_{t'}^{t''} \left(\frac{m}{2}\dot{r}^2 - V\_{\ell f}(r)\right)dt\right],\tag{53}$$

#### **5.1 Pöschl-Teller potential**

This potential is an important diatomic molecular potential. Many applications of the analytical and approximate technique in the current literature have been made to establish eigensolutions and thermodynamic properties [5, 7]. Another example of this potential used as an effective model is as a reference potential manifested to elaborate on the reliability of the order ambiguity parameters.

In the present chapter, the Pöschl-Teller potential of hyperbolic form [5] has been used and is given by:

$$V^{PT}(r) = \left[\frac{A}{\sinh^2(ar)} - \frac{B}{\cosh^2(ar)}\right], \quad r \ge 0,\tag{54}$$

and

$$\begin{cases} A = \frac{\hbar^2 a^2}{2m} \eta(\eta - \mathbf{1}) \\ B = \frac{\hbar^2 a^2}{2m} \lambda(\lambda + \mathbf{1}) \end{cases},\tag{55}$$

where *α*, *η*, *λ* are positive constants.

### *5.1.1 s-states* ð Þ ℓ ¼ 0

For ℓ ¼ 0, taking into account (50), the propagator of the Pöschl-Teller potential (54) becomes:

$$\begin{split} K\_{\ell}(r'',r',s) &= \int \mathfrak{D}r(s) \exp\left[\frac{i}{\hbar} \int\_{0}^{r} \left(\frac{m}{2}\dot{r}^2 - V\_{\ell\overline{\mathcal{G}}}(r)\right) ds\right] \\ &= \int \mathfrak{D}r(s) \exp\left[\frac{i}{\hbar} \int\_{0}^{r} \left(\frac{m}{2}r^2 - \frac{\hbar^2}{2m} \left[\frac{\eta(\eta-1)}{\sinh^2(r)} - \frac{\lambda(\lambda+1)}{\cosh^2(r)}\right]\right) ds\right] \\ &= \int \mathfrak{D}r(s) \mu\_{\lambda/q} [\sinh(r)\cosh(r)] \exp\left(\frac{im}{2\hbar} \int\_{0}^{r} \dot{r}^2 ds\right) \\ &= \lim\_{N\to\infty} \left(\frac{m}{2\pi i\epsilon}\right)^{\frac{N}{2}} \prod\_{j=1}^{N-1} \int\_{0}^{\pi} dr^{(j)} \prod\_{j=1}^{N} \mu\_{\lambda/q} \left[\sinh(r)^{(j)} \cosh(r)^{(0)}\right] \exp\left(\frac{im}{2\epsilon\hbar} \left(\gamma - r\_{j-1}\right)^2\right), \end{split} \tag{56}$$

**46**

*Path Integral of Schrödinger's Equation DOI: http://dx.doi.org/10.5772/intechopen.112183*

we use the notation sinhd<sup>2</sup> ð Þ*<sup>θ</sup>* ð Þ*<sup>j</sup>* <sup>¼</sup> sinh ð Þ*<sup>θ</sup>* ð Þ*<sup>j</sup>* sinh ð Þ*<sup>θ</sup>* ð Þ *<sup>j</sup>*�<sup>1</sup> , when the functional measure *μλ η* given by [4, 5]:

$$\begin{split} \mu\_{\boldsymbol{\lambda},\boldsymbol{\eta}}[\sinh(\boldsymbol{\omega}\boldsymbol{\sigma}),\cosh(\boldsymbol{\omega}\boldsymbol{\sigma})] &=& \lim\_{N\to\infty} \prod\_{j=1}^{N} \mu\_{\boldsymbol{\lambda},\boldsymbol{\eta}}\Big[\sinh\left(\boldsymbol{\omega}\boldsymbol{\sigma}\right)^{(j)},\cosh\left(\boldsymbol{\omega}\boldsymbol{\sigma}\right)^{(j)}\Big] \\ &=& \lim\_{N\to\infty} \left(\frac{2\pi m}{\epsilon\hbar}\right)^{N} \prod\_{j=1}^{N} \sinh\left(\boldsymbol{\omega}\boldsymbol{\sigma}\right)^{(j)} \cosh\left(\boldsymbol{\omega}\boldsymbol{\sigma}\right)^{(j)} \\ & \quad \times \quad \exp\left(\frac{-m}{i\epsilon\hbar} \left(\sinh^{2}(\boldsymbol{\omega}\boldsymbol{\sigma})^{(j)} - \cosh^{2}(\boldsymbol{\omega}\boldsymbol{\sigma})^{(j)}\right)\right) \\ & \quad \times \quad I\_{\eta-\frac{1}{4}}\Big(\frac{m}{i\epsilon\hbar}\sinh^{2}(\boldsymbol{\sigma}\boldsymbol{\sigma})\right) \times I\_{\lambda-\frac{1}{2}}\Big(\frac{im}{\epsilon\hbar}\cosh^{2}(\boldsymbol{\sigma}\boldsymbol{\sigma})\Big), \end{split} \tag{57}$$

This is a known solved problem.

Adapting Frank and Wolf's notion, the solution of the path integral reads 2*S* ¼ *η η*ð Þ � 1 , � 2*C* ¼ *λ λ*ð Þ þ 1 , and by introducing the numbers *k*1, *k*<sup>2</sup> which are defined as a function of *C* and *S* [5], by setting,

$$\begin{cases} k\_1 = \frac{1}{2} \left[ 1 \pm \left( \frac{1}{4} - 2C \right)^{\frac{1}{2}} \right] \\\\ k\_2 = \frac{1}{2} \left[ 1 \pm \left( \frac{1}{4} + 2S \right)^{\frac{1}{2}} \right] \end{cases} \tag{58}$$

The propagator *K*<sup>ℓ</sup> *r*00,*r*<sup>0</sup> ð Þ , *T* contains discrete and continuous terms, becomes:

$$\begin{split} K\_{\ell'}(r'',r',T) &= \sum\_{n=0}^{N\_M} \exp\left(\frac{-is''E\_{n,\ell}^{PT}}{\hbar}\right) \Psi\_{\ell,n}^{\*(k\_1,k\_2)}(r') \Psi\_{\ell,n}^{(k\_1,k\_2)}(r'') \\ &+ \int\_0^\infty dk \, \exp\left(-is''\frac{\hbar k^2}{2m}\right) \Psi\_k^{\*(k\_1,k\_2)}(r') \Psi\_k^{(k\_1,k\_2)}(r''), \end{split} \tag{59}$$

we have *NM* indicate the maximum number of states with 0, 1, … , *n* ≤ *NM* <*k*<sup>1</sup> � *<sup>k</sup>*<sup>2</sup> � <sup>1</sup> <sup>2</sup> *:* The signs depend on the boundary conditions for*r* ! 0 and *r* ! ∞, respectively.

The bound states are explicitly given by [4, 5]:

$$\begin{split} \Psi^{(k\_1,k\_2)}\_{\ell,n}(r) &=& N\_n^{(k\_1,k\_2)} (\sinh(wr))^{2k\_1-\frac{1}{2}} (\cosh(wr))^{-2k\_1+\frac{1}{2}} \\ &\quad \times\_2 \quad F\_1 (-k\_1+k\_2+k\_,-k\_1+k\_2-k+1; 2k\_2; -\sinh^2(wr)) \\ &=& \left( \frac{2n!(2k\_1-1)\Gamma(2k\_1-n-1)}{\Gamma(2k\_2+n)\Gamma(2k\_1-2k\_2-n)} \right)^{\frac{1}{2}} (\sinh(wr))^{2k\_2-\frac{1}{2}} (\cosh(wr))^{2n-2k\_1+\frac{3}{2}} \\ &\quad \times \quad P\_n^{[2k\_1-1,2(k\_1-k\_2-n)-1]} \left( \frac{1-\sinh^2(wr)}{\cosh^2(wr)} \right), \end{split} \tag{60}$$

and

*Schrödinger Equation – Fundamentals Aspects and Potential Applications*

$$N\_n^{(k\_1, k\_2)} = \frac{1}{\Gamma(2k\_2)} \left( \frac{(2k - 1)\Gamma(k\_1 + k\_2 - k)\Gamma(k\_1 + k\_2 + k - 1)}{\Gamma(k\_1 - k\_2 + k)\Gamma(k\_1 - k\_2 - k + 1)} \right). \tag{61}$$

the energy spectrum is also obtained by:

$$E\_n^{PT} = -\left(\frac{\hbar^2 \alpha^2}{2m}\right) \left(2k - 1\right)^2 = -\left(\frac{\hbar^2 \alpha^2}{2m}\right) \left[2(k\_1 - k\_2 - n) - 1\right]^2. \tag{62}$$

*5.1.2* ℓ*-states* ð Þ ℓ 6¼ 0

Usually, we find that the effective potential is not exactly solvable for ℓ-states <sup>00</sup><sup>ℓ</sup> 6¼ <sup>0</sup><sup>00</sup> ð Þ, To deal with the centrifugal term <sup>1</sup> *r*2 � �, we need to find a better approximate expression for this term and such approximations have been proposed as a general approximation similar to the type of Pöschl-Teller potential [8]:

$$\frac{1}{r^2} \approx F(r) = a^2 \left( \frac{1}{\Im \cosh^2(ar)} + \frac{1}{\sinh^2(ar)} \right),\tag{63}$$

Moreover, these approximations are only valid for small values of the parameter *α* and collapse for large *α*. This choice is useful and allows us to treat this hyperbolic potential.

Substituting (63) into (52) we find:

$$\mathcal{V}\_{\mathcal{G}\overline{\mathcal{V}}}(r) = \frac{\hbar^2 \alpha^2}{2m} \left[ \frac{\eta\_1(\eta\_1 - \mathbf{1})}{\sinh^2(ar)} - \frac{\lambda\_1(\lambda\_1 + \mathbf{1})}{\cosh^2(ar)} \right] + \mathcal{C}\_1,\tag{64}$$

with

$$\begin{cases} \eta\_1(\eta\_1 - 1) = \left[ \frac{2\hbar^2 a^2}{m} d\_1 \left[ \left( \ell + \frac{D}{2} - \mathbf{1} \right)^2 - \frac{\mathbf{1}}{4} \right] + \eta(\eta - \mathbf{1}) \right] \\\ \lambda\_1(\lambda\_1 + 1) = \left[ -\frac{\hbar^2 a^2}{m} d\_0 \left[ \left( \ell + \frac{D}{2} - \mathbf{1} \right)^2 - \frac{\mathbf{1}}{4} \right] + \lambda(\lambda + 1) \right], \\\ C\_1 = \frac{2\hbar^2 a^2}{m} d\_2 \left[ \left( \ell + \frac{D}{2} - \mathbf{1} \right)^2 - \frac{\mathbf{1}}{4} \right] \end{cases} \tag{65}$$

with the bound states being explicitly given by [5]:

$$\begin{array}{rcl} \Psi^{(k\_1,k\_2)}\_{\ell,n}(r) &=& N^{(k\_1,k\_2)}\_{n}(\sinh(\operatorname{or}))^{2k\_1-\frac{1}{2}}(\cosh(\operatorname{or}))^{-2k\_1+\frac{1}{2}}\\ &\times& F\_1(-k\_1+k\_2+k\_,-k\_1+k\_2-k+1;2k\_2;-\sinh^2(\operatorname{or})) \\ &=& \left(\frac{2n!(2k\_1-1)\Gamma(2k\_1-n-1)}{\Gamma(2k\_2+n)\Gamma(2k\_1-2k\_2-n)}\right)^{\frac{1}{2}}(\sinh(\operatorname{or}))^{2k\_1-\frac{1}{2}}(\cosh(\operatorname{or}))^{2n-2k\_1+\frac{1}{2}} \\ &\times& P\_n^{[2k\_1-1,2(k\_1-k\_2-n)-1]}\left(\frac{1-\sinh^2(\operatorname{or})}{\cosh^2(\operatorname{or})}\right), \end{array} \tag{66}$$

*Path Integral of Schrödinger's Equation DOI: http://dx.doi.org/10.5772/intechopen.112183*

and

$$N\_n^{(k\_1, k\_2)} = \frac{1}{\Gamma(2k\_2)} \left( \frac{(2k - 1)\Gamma(k\_1 + k\_2 - k)\Gamma(k\_1 + k\_2 + k - 1)}{\Gamma(k\_1 - k\_2 + k)\Gamma(k\_1 - k\_2 - k + 1)} \right). \tag{67}$$

the energy spectrum is also obtained by:

$$E\_{n, \epsilon'}^{PT} = -\left(\frac{\hbar^2 a^2}{2m}\right) (2k - 1)^2 = -\left(\frac{\hbar^2 a^2}{2m}\right) \left[2(k\_1 - k\_2 - n) - 1\right]^2. \tag{68}$$

with

$$\begin{cases} k\_1 = \frac{1}{2} \left[ \mathbf{1} \pm \left( \frac{\mathbf{1}}{4} + \lambda\_1 (\lambda\_1 + \mathbf{1}) \right)^{\frac{1}{2}} \right] \\\\ k\_2 = \frac{1}{2} \left[ \mathbf{1} \pm \left( \frac{\mathbf{1}}{4} + \eta\_1 (\eta\_1 - \mathbf{1}) \right)^{\frac{1}{2}} \right] \end{cases} \tag{69}$$

The energy spectrum is obtained from Eq. (69), namely

$$E\_{n,\ell}^{PT} = -\left(\frac{\hbar^2 \alpha^2}{2m}\right) \left[ 2 \left[ \begin{array}{c} \frac{1}{2} \left[ 1 \pm \left( \frac{1}{4} + \left( \frac{2m}{\hbar^2 a^2} \left[ -\frac{\hbar^2 \alpha^2}{6m} \frac{\left[ \left( l + \frac{D}{2} - 1 \right)^2 - \frac{1}{4} \right]}{3} + \frac{B}{q} \right] \right) \right)^{\frac{1}{2}} \\\\ -\frac{1}{2} \left[ 1 \pm \left( \frac{1}{4} + \left( \frac{2m}{\hbar^2 a^2} \left[ \frac{\hbar^2 \alpha^2}{2m} \left[ \left( l + \frac{D}{2} - 1 \right)^2 - \frac{1}{4} \right] + \frac{A}{q} \right] \right) \right)^{\frac{1}{2}} \right] - 1 \end{array} \right]^2,\tag{70}$$

#### **6. Duru-Kleinert method**

We often introduce a coordinate transformation followed by a local time transformation to make the study much more accessible.

Let us perform the following space and time changes [9]:

$$\begin{cases} \quad r = f(q) \\ dt = f'^2(q)ds \end{cases} \tag{71}$$

These transformations allow us to transform a difficult propagator to calculate into a more manageable form.

Moreover, Green's function relative to a given propagator allows us to derive from its poles the spectrum of energies and the corresponding wave functions from the residues at the poles. This function is obtained from the Fourier transform of the propagator *K*<sup>ℓ</sup> *r*00,*r*<sup>0</sup> ð Þ ; *T* as follows:

$$K\_{\ell}(r'',r';T) = \frac{1}{2\pi\hbar} \left[ G\_{\ell}(r'',r';E) \exp\left(\frac{-iET}{\hbar}\right) dE,\tag{72}$$

with

$$\mathcal{G}\_{\ell'}(r'',r';E) = \frac{i}{\hbar} \left[f'(q')f'(q'')\right]^{\frac{1}{\hbar}} \int\_0^\infty \hat{\mathcal{K}}\_{\ell'}(q'',q';s'')ds'',\tag{73}$$

and

$$\hat{\mathcal{K}}\_{\ell}(q'',q';s'') = \int\_{q'} \mathfrak{D}q(s) \exp\left[\frac{i}{\hbar} \left[ \left(\frac{m}{2}\dot{q}^2 - f'^2(q)\left[V\_{\text{eff}}(q) - E\right] - \Delta V(q)\right)ds\right],\tag{74}$$

and the quantum correction Δ*V* is given by:

$$
\Delta V(q) = \frac{\hbar^2}{8m} \left[ 3 \frac{\left[ f''(q) \right]^2}{\left[ f'(q) \right]^2} - 2 \frac{f'''(q)}{f'(q)} \right] \tag{75}
$$

#### **7. Energy spectrum and wave functions**

#### **7.1 Shifted Deng-Fan Oscillator potential**

Another important empirical potential of diatomic molecules is the Shifted Deng-Fan Oscillator potential [7]. It was proposed since more than half century ago, but has attracted much interest lately, and this potential is the form

$$V\_{SDF}(r) = D\_1 \left( 1 - \frac{b}{e^{ar} - 1} \right)^2 - D\_2, \ b = e^{ar\_\varepsilon} - 1,\tag{76}$$

where *D*<sup>2</sup> is the dissociation energy, *re* is the position of the minimum, and *α* denotes the radius of the potential.

Here, we use for this potential a different approximation obtained using a power series decomposition [10, 11].

$$\frac{1}{r^2} \simeq \frac{1}{r\_\epsilon^2} \left[ C\_0 + \frac{C\_1}{e^{ar} - 1} + \frac{C\_2}{\left(e^{ar} - 1\right)^2} \right],\tag{77}$$

,

where *re* is the minimum of the potential (76) and

$$\begin{cases} C\_0 = \frac{\left(1 - \frac{\left(1 - \eta\right)^2}{u^2} \left(\frac{4u}{1 - \eta} - \left(3 + u\right)\right)\right)}{u^2} \\ C\_1 = \frac{\left(\exp(u) \left(1 - \eta\right)^2\right)}{u^3} \\ C\_2 = \frac{\left(\exp(2u) \left(1 - \eta\right)^4\right)}{u^4} \left(3 + u - \frac{2u}{1 - \eta}\right) \end{cases}$$

where *u* ¼ 2*α re*, and *η* ¼ expð Þ �*u* . Substituting Eqs. (77) and (78) into Eq. (52), we find

$$V\_{\rm eff}(r) = \frac{\hbar^2}{2m} \left[ C\_0 + \frac{C\_1 \exp(-2ar)}{1 - \exp(-2ar)} + \frac{C\_2 \exp(-2ar)}{\left(1 - \exp(-2ar)\right)^2} \right] + D\_1 \left(1 - \frac{b}{\exp(ar) - 1}\right)^2 - D\_2,\tag{78}$$

In *a* more compact for me, it reads

$$V\_{\rm eff}(r) = -A\coth(ar) + \frac{B}{\sinh^2(ar)} + C,\tag{79}$$

where

$$\begin{cases} A = \frac{\hbar^2}{2m} \ell(\ell+1)\alpha^2 \left(\frac{\mathcal{C}\_2}{2} - \frac{\mathcal{C}\_1}{2}\right) + D\_1 b + D\_1 \frac{b^2}{2} \\ B = \frac{\hbar^2}{2m} \ell(\ell+1)\alpha^2 \frac{\mathcal{C}\_2}{4} + D\_1 \frac{b^2}{4} \\ C = \frac{\hbar^2}{2m} \ell(\ell+1)\alpha^2 \left(\mathcal{C}\_0 - \frac{\mathcal{C}\_1}{2} + \frac{\mathcal{C}\_2}{2}\right) + D\_1 + D\_1 b + D\_1 \frac{b^2}{2} - D\_2 \end{cases}, \tag{80}$$

Thus, the condensed form is given by:

$$K\_{\ell}(r'',t'';r',t') = \mathfrak{D}r\,(t)\exp\left[\int\_{t'}^{t''} \left(\frac{m}{2}\dot{r}^2 - V\_{\ell\ell}(r)\right)dt\right],\tag{81}$$

the potential given by (79) is similar to the Manning-Rosen, a direct path integration is not possible, the problem can be solved with the help of the folowing spacetime transformation

$$\begin{cases} r = F(q) = \frac{1}{\alpha} \operatorname{arccoth} \left( 2 \coth^2(q) - 1 \right) \\ dt = \left[ F'(q) \right]^2 d\mathfrak{s} \end{cases} , \tag{82}$$

According to [7], the wave function is given by

$$\begin{split} \chi^{\Sigma J F(k,k\_{2})}\_{n,\ell}(r) &= \sqrt{a} N\_{n}^{(k\_{1},k\_{2})} (1-u)^{1/2-k\_{1}+n} (u)^{k\_{1}-1-\varepsilon/2-n} \\ &\quad \times {}\_{2}F\_{1} \left( -n, 2k\_{1}-n-1; s+1; \frac{1}{1-u} \right) \\ &= \left[ \frac{a(2k\_{1}-1)n! \Gamma(2k\_{1}-n-1)}{\Gamma(n+s+1)\Gamma(2k\_{1}-s-n-1)} \right]^{1/2} \left( 1-e^{-2r} \right)^{k\_{1}} \exp\left[ -2r \left( k\_{1}-\frac{s}{2}-n-1 \right) \right] \\ &\quad \times P\_{n}^{(2k\_{1}-2n-s-2,\varepsilon)} (1-2e^{-2r}), \end{split} \tag{83}$$

where *<sup>P</sup>*ð Þ *<sup>α</sup>*,*<sup>β</sup> <sup>n</sup>* denotes the Jacobi polynomials and *<sup>u</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ½ � 1 � tanh 2ð Þ *αr* , where *k* ¼ *k*<sup>1</sup> � *k*<sup>2</sup> � *n:* and

*Schrödinger Equation – Fundamentals Aspects and Potential Applications*

$$k\_1 = \frac{1}{2} \left[ \left( 1 + \frac{1}{2} (s + 2n + 1) \right) + \frac{2mA}{a^2 \hbar^2 (s + 2n + 1)} \right],\tag{84}$$

$$k\_2 = \frac{1}{2} \left[ 1 + \sqrt{1 + \frac{8mB}{\alpha^2 \hbar^2}} \right] \equiv \frac{1}{2} (1 + s),\tag{85}$$

The energy spectrum is obtained from the poles of the Green function, Eq. (82), namely

$$E\_{n,l}^{SDF} = -\left[\frac{a^2\hbar^2(s+2n+1)^2}{8m} + \frac{2mA^2}{a^2\hbar^2(s+2n+1)^2}\right] + \text{C.}\tag{86}$$

#### **7.2 Generalized inverse quadratic Yukawa potential**

The generalized inverse quadratic Yukawa potential extends this concept by introducing additional parameters or modifications to the potential. These modifications can include terms that account for different types of interactions or other physical phenomena, depending on the specific context or application.

The general form of Generalized Inverse Quadratic Yukawa Potential is:

$$V\_{GIQY}(r) = -a - b\frac{e^{-ar}}{r} - c\frac{e^{-2ar}}{r^2},\tag{87}$$

which means that the effective potential becomes

$$W\_{\rm eff}(r) = -a - b\frac{e^{-ar}}{r} - c\frac{e^{-2ar}}{r^2} + \frac{\hbar^2\ell(\ell+1)}{2r^2}.\tag{88}$$

First of all, we deal with the centrifugal terms using the approximation [10, 11].

$$\frac{1}{r^2} = \frac{4a^2e^{-2ar}}{\left(1 - e^{-2ar}\right)^2},\tag{89}$$

and

$$\frac{1}{r} = \frac{2ae^{-ar}}{1 - e^{-2ar}},\tag{90}$$

putting these considerations together, we find the following:

$$W\_{\rm eff}(r) = -b \frac{2ae^{-2ar}}{1 - e^{-2ar}} - c \frac{4a^2 e^{-4ar}}{\left(1 - e^{-2ar}\right)^2} + \frac{\hbar^2 \ell(\ell + 1)}{2} \frac{4a^2 e^{-2ar}}{\left(1 - e^{-2ar}\right)^2} - a,\tag{91}$$

*Veff*ð Þ*r* can be reformulated as

$$V\_{\rm eff}(r) = A \coth(ar) + \frac{B}{\sinh^2(ar)} + C,\tag{92}$$

with

*Path Integral of Schrödinger's Equation DOI: http://dx.doi.org/10.5772/intechopen.112183*

$$\begin{cases} A = a(2ca - b); \\ B = a^2(\frac{\hbar^2}{2m}\ell(\ell + 1) - c); \\ C = -2ca^2 + ba - a. \end{cases} \tag{93}$$

Since the difficulties of doing the integration of Eq. (53) straightforwardly, we perform a space-time transformation depending on the Duru-Kleinert method [4, 9], so we do a nontrivial change of variable ð Þ *r* ! *q* followed by time local transformation ð Þ *t* ! *s*

$$\begin{cases} r = h(q) = \frac{1}{\alpha} \text{argcoth} \left( 2 \text{coth}^2(q) - 1 \right); \\ \qquad t \to s \Leftrightarrow dt = \left[ h'(q(s)) \right]^2 ds. \end{cases} \tag{94}$$

Putting these considerations together, we find the new Green's function

$$G\_{\ell}(q\_{b}, q\_{a}; E) = \frac{i}{\hbar} \left[ h'(q\_{a}) h'(q\_{b}) \right]^{\frac{1}{2}} \Big|\_{0}^{\infty} P\_{\ell}(q\_{b}, q\_{a}; \mathcal{S}) d\mathcal{S}, \tag{95}$$

where *h*<sup>0</sup> is the derivative of *h* with respect to *q*, and the new form of the promotor is

$$P\_{\ell}(q\_{b}, q\_{a}; \mathbf{S}) = \int Dq(\mathbf{s}) \exp\left[\frac{i}{\hbar} \int\_{0}^{\mathbf{S}} \left\{\frac{m}{2}\dot{q}^{2} - h^{2} \left(V\_{\text{eff}}(q) - E\right) - \Delta V(q)\right\} d\mathbf{s}\right],\tag{96}$$

the quantum correction Δ*V q*ð Þ [4] is given by

$$
\Delta V(q) = \frac{\hbar^2}{8m} \left( 3\frac{h''^2}{h'^2} - 2\frac{h'''}{h'} \right) = \frac{\hbar^2}{8m} \left( \frac{1}{\cosh^2(q)} + \frac{1}{\sinh^2(q)} \right),
\tag{97}
$$

and the transformed effective potential is

$$\mathcal{V}\_{\rm eff}(q) = A \left( 2 \coth^2(q) - 1 \right) + 2B \left( 2 \coth^2(q) - 2 \right) \coth^2(q) + \mathcal{C},\tag{98}$$

therefore

$$\begin{split} h^{2} \left( V\_{\text{eff}}(q) - E \right) + \Delta V(q) &= \frac{\hbar^{2}}{2m} \left( \frac{\frac{8mB}{a^{2}\hbar^{2}} + \frac{3}{4}}{\sinh^{2}(q)} + \frac{2m}{\cosh^{2}(q)} \frac{E + A - C}{\cosh^{2}(q)} \right) \\ &- \frac{1}{a^{2}} (E - A - C). \end{split} \tag{99}$$

And using the following abbreviations

$$\begin{cases} D = \frac{1}{\alpha^2} (E - A - C); \\ \eta^2 - \frac{1}{4} = \frac{8mB}{\alpha^2 \hbar^2} + \frac{3}{4}; \\\ v^2 - \frac{1}{4} = -\frac{2m}{\alpha^2 \hbar^2} (E + A - C) - \frac{1}{4}, \end{cases} \tag{100}$$

which means that

*Schrödinger Equation – Fundamentals Aspects and Potential Applications*

$$\begin{cases} \;D = \frac{1}{\alpha^2} (E - A - C); \\ \;\eta = \pm \sqrt{1 + \frac{8mB}{\alpha^2 \hbar^2}}; \\ \;\nu = \pm \sqrt{-\frac{2m}{\alpha^2 \hbar^2} (E + A - C)}, \end{cases} \tag{101}$$

we can rewrite the promotor as follows:

$$\begin{split} P\_{\ell}(q\_{b},q\_{a};s) &= \int [Dq(s)\exp\left[\frac{i}{\hbar}\int\_{0}^{S}\left\{\frac{m}{2}\dot{q}^{2} - \frac{\hbar^{2}}{2m}\left(\frac{\eta^{2}-\frac{1}{4}}{\sinh^{2}(q)} + \frac{\nu^{2}-\frac{1}{4}}{\cosh^{2}(q)}\right)\right\}ds\right] \\ &\quad \times \quad \exp\left[\frac{i}{\hbar}DS\right], \end{split} \tag{102}$$

which is nothing but a promotor formula corresponding to a system with modified Pöschl-Teller potential and energy *D* [12], and accordingly, the integration over time *S* enables us to obtain directly the radial Green's function related to this system

$$\mathcal{G}\_{\ell}(q\_b, q\_a; \mathcal{D}) = \int\_0^\infty P\_{\ell}(q\_b, q\_a; \mathcal{S}) d\mathcal{S},\tag{103}$$

thus

$$\begin{split} \mathbf{G}\_{\ell}(q\_{b},q\_{a};D) &= \int\_{0}^{\infty} d\mathbf{S} \exp\left[\frac{i}{\hbar}D\mathbf{S}\right] \\ &\times \quad \int Dq(\mathbf{s}) \exp\left[\frac{i}{\hbar}\int\_{0}^{\mathbf{S}} \left\{\frac{m}{2}\dot{q}^{2} - \frac{\hbar^{2}}{2m}\left(\frac{\eta^{2}-\frac{1}{4}}{\sinh^{2}(q)} + \frac{\nu^{2}-\frac{1}{4}}{\cosh^{2}(q)}\right)\right\}ds\right], \end{split} \tag{104}$$

The energy spectrum is obtained from the poles of Green's function which leads us to

$$D = -\frac{\hbar^2}{2m}(2n + \eta - \nu + 1)^2,\tag{105}$$

therefore

$$\begin{aligned} E\_{n,\ell} &= -\frac{\hbar^2 \alpha^2}{8m} \left( 2n + \mathbf{1} + \sqrt{1 + \frac{8mB}{\alpha^2 \hbar^2}} \right)^2 \\ &- \frac{2A^2}{\frac{\hbar^2 \alpha^2}{m} \left( 2n + \mathbf{1} + \sqrt{\mathbf{1} + \frac{8mB}{\alpha^2 \hbar^2}} \right)^2} + C, \end{aligned}$$

the energy spectrum is thus

*Path Integral of Schrödinger's Equation DOI: http://dx.doi.org/10.5772/intechopen.112183*

$$\begin{array}{rcl} E\_{n,\ell} &=& -\frac{\hbar^2 a^2}{8m} \left( 2n + \mathbf{1} + \sqrt{1 + 4\ell(\ell + 1) - \frac{8m}{\hbar^2}c} \right)^2 \\\\ &- \frac{2(2ca - b)^2}{\hbar^2 \left( 2n + \mathbf{1} + \sqrt{1 + 4\ell(\ell + 1) - \frac{8m}{\hbar^2}c} \right)^2} - 2ca^2 + ba - a, \end{array} \tag{106}$$

On the other hand, the associated wave functions can be displayed as

$$\begin{array}{rcl} \psi\_{\pi}(r) &=& \left[ \left( a - \frac{4mA}{a\hbar^2(\omega + 2n + 1)^2} \right) \frac{(2k\_1 - 2n - \omega - 2)n! \Gamma(2k\_1 - n - 1)}{\Gamma(n + \omega + 1)\Gamma(2k\_1 - \omega - n - 1)} \right]^{1/2} \\ & \times \quad (1 - \exp(-2ar))^{\frac{\omega + 1}{2}} \exp(k\_1 - \omega/2 - n - 1) \\ & \times \quad P\_n^{(2k\_1 - 2n - \omega - 2, \alpha)}(1 - 2\exp(-2ar)), \end{array}$$

where *P*ð Þ <sup>2</sup>*k*1�2*n*�*ω*�2,*<sup>ω</sup> <sup>n</sup>* are Jacobi polynomials with the notations

$$\begin{cases} k\_1 = \frac{1}{2} \left( \mathbf{1} + \frac{1}{2} (\boldsymbol{\omega} + \boldsymbol{2} \boldsymbol{n} + \mathbf{1}) - \frac{2m\boldsymbol{A}}{a^2 \hbar^2 (\boldsymbol{\omega} + \boldsymbol{2} \boldsymbol{n} + \mathbf{1})} \right); \\\\ k\_2 = \frac{1}{2} (\mathbf{1} + \boldsymbol{\omega}), \end{cases} \tag{107}$$

and

$$
\omega = \sqrt{1 + \frac{8mB}{a^2 \hbar^2}}.\tag{108}
$$

#### **7.3 Modified screened Coulomb plus inversely quadratic Yukawa potential**

The Modified Screened Coulomb plus Inversely Quadratic Yukawa potential (MSC-IQY) is a combined potential energy function that incorporates both the screened Coulomb potential and the inversely quadratic Yukawa potential. This modified potential is often used in various areas of physics to describe interactions between charged particles, taking into account both screening effects and long-range Coulombic interactions. For *a* ¼ 0, the GIQY potential reduces to Modified Screened Coulomb Plus Inversely Quadratic Yukawa potential (MSC-IQY) of the form

$$V\_{GIQY}(r) = -b\frac{e^{-ar}}{r} - c\frac{e^{-2ar}}{r^2},\tag{109}$$

and the associated energy eigenvalues are obtained as

$$E\_{n, \ell}^{\rm GlQY} = -\frac{\hbar^2 a^2}{8m} \left( 2n + 1 + \sqrt{1 + 4\ell(\ell + 1) - \frac{8m}{\hbar^2}c} \right)^2$$

$$-\frac{2(2ca - b)^2}{\frac{\hbar^2}{m} \left( 2n + 1 + \sqrt{1 + 4\ell(\ell + 1) - \frac{8m}{\hbar^2}c} \right)^2} - 2ca^2 + ba. \tag{110}$$

#### **7.4 Kratzer potential**

The Kratzer potential [13] is a mathematical model used to describe the interaction between a particle and a central force field. It is commonly employed to study molecular systems and the vibrational motion of diatomic molecules. For

*<sup>α</sup>* <sup>¼</sup> 0, *<sup>a</sup>* <sup>¼</sup> 0, *<sup>b</sup>* <sup>¼</sup> <sup>2</sup>*Dere*, and *<sup>c</sup>* <sup>¼</sup> *Der*<sup>2</sup> *<sup>e</sup>*, the GIQY potential (87) reduces to the Kratzer potential of the form

$$V\_K(r) = -2D\_\epsilon \left(\frac{r\_\epsilon}{r} - \frac{1}{2}\frac{r\_\epsilon^2}{r^2}\right),\tag{111}$$

where *re* is the equilibrium bond length and *De* is the dissociation energy. The energy eigenvalues of the Kratzer potential are obtained as

$$E\_{n, \ell}^{\mathbb{K}} = -\frac{b^2}{\frac{\hbar^2}{2m} \left(2n + \mathbf{1} + \sqrt{\mathbf{1} + 4\ell(\ell + \mathbf{1}) - \frac{8m}{\hbar^2}c}\right)^2},\tag{112}$$

thus

$$E\_{n, \ell}^{K} = -\frac{\left(2D\_{\ell}r\_{\ell}\right)^{2}}{\frac{\hbar^{2}}{2m}\left(2n+1+\sqrt{1+4\ell'(\ell'+1)-\frac{8m}{\hbar^{2}}D\_{\ell}r\_{\ell}^{2}}\right)^{2}}.\tag{113}$$

#### **7.5 Yukawa potential**

The Yukawa potential, also known as the screened Coulomb potential or the Debye-Hückel potential, is a mathematical model used to describe the interaction between charged particles with an exponential decay due to screening effects. It is commonly employed in physics to study phenomena such as electromagnetic interactions, nuclear forces, and scattering processes.

The Yukawa potential is given by the following equation (setting *a* ¼ 0 and *c* ¼ 0, Eq. (88) takes the form)

$$V\_Y(r) = -b\frac{e^{-ar}}{r},\tag{114}$$

which is known as Yukawa potential, its corresponding energy eigenvalues achieved are

$$\begin{split} E\_{n,\ell}^{\mathrm{V}} &= -\frac{\hbar^2 a^2}{8m} \left( 2n + 1 + \sqrt{1 + 4\ell(\ell + 1)} \right)^2 \\ &- \frac{2b^2}{\hbar^2 \left( 2n + 1 + \sqrt{1 + 4\ell(\ell + 1)} \right)^2} + ba,\end{split} \tag{115}$$

or equivalently

$$E\_{n,\ell}^Y = -\frac{\hbar^2 a^2}{2m} (n+1+\ell)^2 - \frac{b^2}{\frac{\hbar^2}{m} 2(n+1+\ell)^2} + ba,\tag{116}$$

#### **7.6 Inversely quadratic Yukawa potential**

The Inversely Quadratic Yukawa potential (IQY) is a modified version of the Yukawa potential that takes into account an additional inverse square term. As *a* ¼ 0 and *b* ¼ 0, Eq. (88) reduces to the Inversely Quadratic Yukawa potential (IQY) of the form

$$V\_{IQY}(r) = -c \frac{e^{-2ar}}{r^2},\tag{117}$$

the energy eigenvalue equation becomes

$$\begin{array}{rcl} E\_{n,\ell}^{\mathrm{QQ}} &=& -\frac{\hbar^2 \alpha^2}{8m} \left( 2n + 1 + \sqrt{1 + 4\ell(\ell + 1) - \frac{8m}{\hbar^2}c} \right)^2 \\\\ &- \frac{2(2ca)^2}{\hbar^2 \left( 2n + 1 + \sqrt{1 + 4\ell(\ell + 1) - \frac{8m}{\hbar^2}c} \right)^2} - 2ca^2. \end{array} \tag{118}$$

#### **7.7 Coulomb potential**

The Coulomb potential is used to calculate important properties such as the electric potential, electric field, and electrostatic forces in systems involving charged particles. It forms the basis for understanding phenomena such as the behavior of ions in solutions, the interaction between charged particles in plasmas, and the structure of atoms and molecules.

When *a* ¼ 0, *α* ¼ 0, and *c* ¼ 0, Eq. (88) reduces to the Coulomb potential of the form

$$V\_C(r) = -\frac{b}{r},\tag{119}$$

the energy eigenvalues of the Coulomb potential are obtained as

$$E\_{n, \ell}^C = -\frac{b^2}{\frac{\hbar^2}{2m} \left(2n + 1 + \sqrt{1 + 4\ell(\ell' + 1)}\right)^2},\tag{120}$$

hence

$$E\_{n, \ell}^C = -\frac{2m}{\hbar^2} \frac{b^2}{2(n + \ell + 1)^2},\tag{121}$$

#### **8. Conclusions**

We have presented a rigorous treatment using the path integral approach of Feynman. We affirm that this formalism is an efficient and powerful tool for finding the propagator associated with several problems in quantum physics, particularly

nonrelativistic problems. Most of these problems cannot be treated exactly, and practically no physical system can be studied without approximation methods.

In this chapter, we have adopted a two-step approach to study exponentially shaped potentials. In the first step, by introducing a judicious approximation to handle the centrifugal term, we were able to transition from solving a problem related to ℓstates to that of the *s-*state. The other step involves adapting a spatio-temporal transformation by Duru-Kleinert. The use of this transformation was revisited in the reasoning process to reduce the unsolvable relative propagator to the effective potential of several potentials, specifically to the modified Pöschl-Teller potential. This problem is well known and was previously addressed within the framework of the Schrödinger formulation and the path integral. The energy spectrum and wave functions are determined in this case.

In conclusion, our method is effective in solving this type of potential. We hope to continue developing the path integral formalism not only for exponential-type potentials but also for other types and more general forms, and in other domains of physics.

#### **Acknowledgements**

The authors would like to thank the **LESI** laboratory of the University of Khemis Miliana for its help in carrying out this study.

#### **Author details**

Hocine Boukabcha<sup>1</sup> \*†, Salah Eddin Aid2† and Amina Ghobrini<sup>3</sup>

1 Laboratory of Energy and Intelligent Systems, University of Khemis Miliana, Khemis Miliana, Algeria

2 Laboratory of Mechanics and Energy, Hassiba Benbouali University of Chlef, Chlef, Algeria

3 Laboratory of Coatings, Materials and Environment University Mhamed Bougara of Boumerdes, Boumerdes, Algeria

\*Address all correspondence to: h.boukabcha@univ-dbkm.dz

† These authors contributed equally.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Path Integral of Schrödinger's Equation DOI: http://dx.doi.org/10.5772/intechopen.112183*

#### **References**

[1] Messiah A. Mécanique Quantique. Paris: Dunod; 1964

[2] Landau L, Lifchitz E. Mécanique Quantique. Tome III. Editions Mir: Moscou; 1967

[3] Ince P. Ordinary differential equations. New York: Dover publications INC; 1966

[4] Kleinert H. Path integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets. World Scientific: Singapore; 2009

[5] Grosche C. Path integral solution of a class of potentials related to the Pöschl-Teller potential. Journal of Physics A: Mathematical and General. 1989;**22**:5073. DOI: 10.1088/0305-4470/22/23/012

[6] Nikiforov AF, Uvarov VB. Special Functions of Mathematical Physics. Basel: Birkhauser; 1988

[7] Boukabcha H, Hachama M, Diaf A. Ro-vibrational energies of the shifted Deng-Fan oscillator potential with Feynman path integral formalism. Applied Mathematics and Computation. 2018;**321**:121-129

[8] Badawi R, Bessis N, Bessis G. On the introduction of the rotation-vibration coupling in diatomic molecules and the factorization method. Journal of Physics B: Atomic and Molecular Physics. 1972;**5**: L157-L161. DOI: 10.1088/0022-3700/5/8/ 004

[9] Duru IH, Kleinert H. Solution of the path integral for the H-atom. 1979;**84**: 185. DOI: 10.1016/0370-2693(79) 90280-6

[10] Greene RL, Aldrich C. Variational wave functions for a screened Coulomb potential. Physical Review A. 1976;**14**: 2363. DOI: 10.1103/PhysRevA.14.2363

[11] Gönül Bözer O, Canelik Y, Koak M. Hamiltonian hierarchy and the Hulthèn potential. Physics Letters A. 2000;**275**: 238-243

[12] Aid SE, Boukabcha H, Benzaid D. Non-relativistic treatment of generalized inverse quadratic Yukawa potential via path integral approach. Indian Journal of Physics. 2023;**97**(7):1989-1995

[13] Kratzer A. Die ultraroten rotations spektren der halogenwasserstoffe. Zeitschrift fr Physik. 1920;**3**:289-307

#### **Chapter 4**

## The Inverse of the Discrete Momentum Operator

*Armando Martínez-Pérez and Gabino Torres-Vega*

#### **Abstract**

In the search of a quantum momentum operator with discrete spectrum, we obtain some properties of the discrete momentum operator for nonequally spaced spectrum. We find the inverse operator. We use the matrix representation of these operators, and we find that there is one more eigenvalue and eigenfunction than the dimension of the matrix. We apply the results to obtain the discrete adjoint of the momentum operator. We conclude that we can have discrete operators which can be self-adjoint and that it is possible to define a self-adjoint extension of the corresponding Hilbert space. These results help us understand the quantum time operator.

**Keywords:** discrete quantum mechanics, discrete momentum operator, inverse of the momentum operator, nonstandard finite differences derivative, exact discrete integration

#### **1. Introduction**

Nonstandard finite difference derivatives help determine the discrete versions of some differential equations and their solutions [1–10]. This method uses nonstandard expressions of the finite differences derivative in such a way that they give the exact result when applied to a particular function.

Another benefit of nonstandard finite difference for the derivative of a function is that it can be used as a discrete quantum operator to deal with quantum mechanical operators with discrete spectrum [11, 12]. Since some quantum operators have a discrete spectrum, a discrete derivative can be very useful in quantum mechanics theory [11, 12].

In Section 2, we define and obtain some properties of the discrete derivative operator from a global point of view, i.e., considering all the values of a function on all the points of a mesh at once. This is done by defining a matrix that collects the derivatives for each mesh point when applied to a given vector. We find the eigenvalues and eigenvectors of the derivative matrix. We also discuss the commutation properties between the derivative and coordinate matrices. The canonical commutator is satisfied only along some directions.

The summation by parts theorem and the adjoint of the momentum operator are found in Section 3. We introduce the discrete symmetric operator definition similar to continuous variables functions in a Hilbert space.

An interesting result is that the considered matrices have more eigenvalues and corresponding *almost* eigenvectors (the last entry of the eigenvector is null) beside the usual number due to their dimension of them. For a semi-infinite matrix, the last entry is of little effect, and such additional eigenvalues will belong to the matrix spectrum when seen as an operator. Such additional eigenvalues and eigenvectors are common to all the considered matrices. With these results, we can say that there are also self-adjoint discrete operators and that we can also have discrete selfadjoint extensions in the corresponding Hilbert space. These results are beneficial when dealing with the question of the existence of a time operator in quantum mechanics [12].

We introduce the discrete inverse matrix of the discrete derivative operator in Section 4. The difference between the scheme we address in this work with other proposals for a discrete derivative is a modification in the derivative matrix for the final point of a grid of points, which causes the derivative matrix to have an inverse.

We can deal with any mesh without asking for equidistant points. At the end of this paper, there are some concluding remarks.

#### **2. Discrete derivation**

Let us consider a partition P ¼ *q*0, *q*1, *q*2, … , *qN* � � of the interval *<sup>q</sup>*0, *qN* � � and vectors **f** ¼ *f* <sup>0</sup>, *f* <sup>1</sup>, … , *f <sup>N</sup>* � �*<sup>T</sup>* , and **g** ¼ *g*0, *g*1, … , *gN* � �*<sup>T</sup>* associated to this partition. The distances <sup>Δ</sup>*<sup>j</sup>* <sup>¼</sup> *qj*þ<sup>1</sup> � *qj* , for each *j*, are not supposed to be equal.

The finite differences derivative matrix **D** is defined as

$$\mathbf{D} = \begin{pmatrix} -\frac{1}{\tilde{\xi}\_0} & \frac{1}{\tilde{\xi}\_0} & 0 & \dots & 0 & 0\\ 0 & -\frac{1}{\tilde{\xi}\_1} & \frac{1}{\tilde{\xi}\_1} & \dots & 0 & 0\\ 0 & 0 & -\frac{1}{\tilde{\xi}\_2} & \dots & 0 & 0\\ \vdots\\ 0 & 0 & 0 & \dots & -\frac{1}{\tilde{\xi}\_{N-1}} & \frac{1}{\tilde{\xi}\_{N-1}}\\ 0 & 0 & 0 & \dots & 0 & -\frac{1}{\tilde{\xi}\_N} \end{pmatrix},\tag{1}$$

where

$$\xi\_j = \Delta\_j e^{-ip\Delta\_j/2} \text{sinc}\left(\frac{\Delta\_j}{2}p\right), \quad j = 0, \ldots, N - 1,\tag{2}$$

$$
\xi\_N = -\frac{i}{p}.\tag{3}
$$

The function sincð Þ*<sup>z</sup>* is the entire function equal to one at *<sup>z</sup>* <sup>¼</sup> 0 and *<sup>z</sup>*�<sup>1</sup> sin *<sup>z</sup>* otherwise. The continuous parameter *p* in this expression is related to the conjugate variable to the discrete variable *qj* , see Eq. (11) below. The choice of *ξ<sup>j</sup>* ensures that the finite differences derivative (d-derivative) delivers the exact result when acting on the complex exponential function *e*�*ipq*.

In case it is needed, for small Δ*<sup>j</sup>* we have the power series expansion

$$
\xi\_j \approx \Delta\_j - i \frac{p}{2} \Delta\_j^2 - \frac{p^2}{6} \Delta\_j^3, \quad 0 \le j < N. \tag{4}
$$

We see that *ξ<sup>j</sup>* is similar to the difference Δ*<sup>j</sup>* of the usual finite differences derivative. However, we will only consider the case where Δ*<sup>j</sup>* has a finite value.

Let us discuss some properties of the d-derivative matrix. The action of the dderivative matrix **D** when acting to the left, on the vector **f** *<sup>T</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>, *<sup>f</sup>* <sup>1</sup>, … , *<sup>f</sup> <sup>N</sup>* � �, results in

$$\mathbf{f}^T \mathbf{D} = \left( -\frac{f\_0}{\xi\_0}, -(\mathcal{D}\mathbf{f})\_1, -(\mathcal{D}\mathbf{f})\_2, \dots, -(\mathcal{D}\mathbf{f})\_N \right), \tag{5}$$

where

$$(\mathcal{D}\mathbf{f})\_{\circ} = \frac{f\_j}{\mathfrak{xi}\_j} - \frac{f\_{j-1}}{\mathfrak{xi}\_{j-1}},\tag{6}$$

is a finite differences approximation to the derivative of a function extended to the complex plane. These improved increments *ξ<sup>j</sup>* are defined over the complex plane. For a small difference Δ*j*, we have that

$$(\mathcal{D}\mathbf{f})\_{\dot{j}} \approx \left(\frac{f\_{j+1}}{\Delta\_{j+1}} - \frac{f\_j}{\Delta\_j}\right) + i\frac{p}{2}\left(f\_{j+1} - f\_j\right) + \frac{p^2}{12}\left(f\_{j+1} - f\_{j+1}\right)\Delta\_{j+1}.\tag{7}$$

We see that we have another discrete approximation to the derivative of a function.

Now, the action to the right of the derivative matrix on a vector is:

$$\mathbf{D}\mathbf{g} = \left( (\mathbf{D}\mathbf{g})\_0, (\mathbf{D}\mathbf{g})\_1, \dots, (\mathbf{D}\mathbf{g})\_{N-1}, -\frac{f\_N}{\xi\_N} \right)^T,\tag{8}$$

where

$$(\mathbf{Dg})\_j = \frac{\mathbf{g}\_{j+1} - \mathbf{g}\_j}{\mathfrak{E}\_j},\tag{9}$$

is a modified finite differences derivative of *g q*ð Þ at *qj* . In case Δ*<sup>j</sup>* is small, we have that

$$\Delta \left( \mathbf{D} \mathbf{g} \right)\_j \approx \frac{\mathbf{g}\_{j+1} - \mathbf{g}\_j}{\Delta\_j} + i \frac{p}{2} \left( \mathbf{g}\_{j+1} - \mathbf{g}\_j \right) - \frac{p^2}{12} \Delta\_j \left( \mathbf{g}\_{j+1} - \mathbf{g}\_j \right). \tag{10}$$

The first term in this approximation is the usual finite differences derivative of a function.

Note that for in the limiting case, Δ*<sup>j</sup>* ! 0, both nonstandard finite differences (Eq. 5) and (Eq. 8) reduce to the usual forward finite difference approximation to the derivative.

The eigenvalues of the derivative matrix **D** are *λ<sup>j</sup>* ¼ �1*=ξN*, � 1*=ξN*�1, … , � 1*=ξ*0, and the corresponding eigenvectors are

$$\begin{Bmatrix} \begin{pmatrix} \xi\_{N}^{N} \\ \prod\_{n=0}^{N-1} (\xi\_{N} - \xi\_{n}) \\ \vdots \\ \prod\_{n=1}^{N-1} (\xi\_{N} - \xi\_{n}) \\ \prod\_{n=1}^{N-1} (\xi\_{N} - \xi\_{n}) \\ \hline \\ \prod\_{n=2}^{N-2} (\xi\_{N} - \xi\_{n}) \\ \vdots \\ \prod\_{n=2}^{N-1} (\xi\_{N} - \xi\_{n}) \\ \vdots \\ \prod\_{n=2}^{N-3} (\xi\_{N-1} - \xi\_{n}) \\ \vdots \\ \end{pmatrix}, \begin{Bmatrix} \xi\_{N-1}^{N-1} \\ \hline \\ \prod\_{n=1}^{N-2} (\xi\_{N-1} - \xi\_{n}) \\ \hline \\ \xi\_{N-1}^{N-3} \\ \hline \\ \prod\_{n=2}^{N-3} (\xi\_{N-1} - \xi\_{n}) \\ \vdots \\ \hline \\ \end{Bmatrix} \end{Bmatrix}, \dots, \begin{Bmatrix} \mathbf{1} \\ \mathbf{0} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ \end{Bmatrix} \end{Bmatrix}, \dots, \begin{Bmatrix} \mathbf{1} \\ \mathbf{0} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ \end{Bmatrix} \tag{11}$$

Note that, due to the operator character of the matrix, there is an additional eigenvector, the exponential function **<sup>e</sup>** <sup>¼</sup> *<sup>e</sup>*�*ipq*<sup>0</sup> ,*e*�*ipq*<sup>1</sup> ,*e*�*ipq*<sup>2</sup> , … ,*e*�*ipqN* � �, with eigenvalue �*ip*,

$$\mathbf{De} = -ip\,\mathbf{e}.\tag{12}$$

#### **3. The adjoint of the discrete derivative**

A sesquilinear form between vectors **f** and **g** is defined with the help of the summation matrix:

$$\mathbf{S} = \begin{pmatrix} \xi\_0 & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \xi\_1 & \mathbf{0} & \dots & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \xi\_2 & \dots & \mathbf{0} & \mathbf{0} \\ \vdots & & & & \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \xi\_{N-1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} & \mathbf{0} \end{pmatrix}, \tag{13}$$

obtaining

$$\begin{aligned} &\mathbf{f}^T \mathbf{S} \mathbf{D} \mathbf{g} \\ &= -f\_0 \mathbf{g}\_0 + f\_0 \mathbf{g}\_1 - f\_1 \mathbf{g}\_1 + f\_1 \mathbf{g}\_2 - f\_2 \mathbf{g}\_2 + f\_2 \mathbf{g}\_3 - f\_3 \mathbf{g}\_3 + \dots + f\_N \mathbf{g}\_N \\ &= \mathbf{g}^T (\mathbf{B} - \mathbf{S} \tilde{\mathbf{D}}) \mathbf{f}, \end{aligned} \tag{14}$$

where

$$
\tilde{\mathbf{D}} = \begin{pmatrix}
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & -\frac{1}{\tilde{\xi}\_2} & \frac{1}{\tilde{\xi}\_2} & \dots & \mathbf{0} & \mathbf{0} \\
\vdots & & & & \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \frac{1}{\tilde{\xi}\_{N-1}} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} & \mathbf{0}
\end{pmatrix}.\tag{15}
$$

and

$$\mathbf{B} = \begin{pmatrix} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ \vdots & & & & \\ 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & \dots & 1 & 0 \end{pmatrix}. \tag{16}$$

Eq. (13) is the summation by parts equality in matrix form. We call the matrix **D**~ the d-adjoint of the discrete derivative matrix **D**.

A row of the summation by parts matrix equality is:

$$\sum\_{n=0}^{N-1} \xi\_n \mathbf{f}\_n (\mathbf{D} \mathbf{g})\_n + \sum\_{n=1}^{N-1} \xi\_n (\mathbf{\tilde{D}} \mathbf{f})\_n \mathbf{g}\_n = f\_{N-1} \mathbf{g}\_N - f\_0 \mathbf{g}\_0,\tag{17}$$

which is the discrete version of the integration by parts theorem of the calculus of continuous variables.

The previous results are useful in quantum mechanics theory when considering the momentum or the Hamiltonian operators with a discrete spectrum.

We define the discrete momentum operator at *qj* as

$$
\hat{P}\_j = -i(\mathbf{D})\_j, \quad \mathbf{0} \le j < N,\tag{18}
$$

and its adjoint

$$
\hat{P}\_j^\dagger = -i \left(\tilde{\mathbf{D}}\right)\_j, \quad \mathbf{0} < j < N. \tag{19}
$$

The summation by parts provides the adjoint of the momentum operator and its symmetry property. Explicitly, Eq. (16) is rewritten as

$$\sum\_{n=0}^{N-1} \xi\_n \mathbf{f}\_n^\* \left(-i\mathbf{D}\mathbf{g}\right)\_n - \sum\_{n=1}^{N} \xi\_n \left[\left(-i\tilde{\mathbf{D}}^\* \mathbf{f}\right)\_n\right]^\* \mathbf{g}\_n = -i f\_{N-k}^\* \mathbf{g}\_N + i f\_0^\* \mathbf{g}\_0,\tag{20}$$

This equality yields

$$
\langle \mathbf{f} | \hat{\mathbf{P}} \mathbf{g} \rangle = \langle \tilde{\mathbf{P}} \mathbf{f} | \mathbf{g} \rangle = -i \mathbf{f}\_{N-\mathbf{1}}^{\*} \mathbf{g}\_{N} + i \mathbf{f}\_{0}^{\*} \mathbf{g}\_{0}. \tag{21}
$$

Thus, we say that the discrete momentum operator *<sup>P</sup>*^ is d-symmetric, if *<sup>f</sup> <sup>N</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>e</sup><sup>i</sup><sup>θ</sup><sup>f</sup>* <sup>0</sup> and *gN* <sup>¼</sup> *<sup>e</sup><sup>i</sup><sup>θ</sup>g*0, as is the case for continuous variables operators.

It is also possible to consider self-adjoint extensions for the discrete momentum operator, as it is done for the case of the continuous variable momentum operator [13].

#### **3.1 Commutator between the d-derivative and the coordinate**

In general, a discrete canonical commutation relationship ½ �¼ *A*, *B I* is not possible for finite-dimensional matrices *A* and *B* because the trace of this relationship results in a contradiction ((0 = 1) [14]. However, there are some directions in which the commutator evaluates to a constant different from zero: the directions pointed at by its eigenvectors, for example. In addition, the matrix can be considered as an operation with additional eigenfunctions.

If we call **Q** ¼ diag *qj* � � to the coordinate matrix, the usual commutator between the d-derivative and coordinate matrices is:

$$[\mathbf{D}, \mathbf{Q}] = \begin{pmatrix} \mathbf{0} & \frac{\Delta\_0}{\xi\_0} & \mathbf{0} & \dots & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \frac{\Delta\_1}{\xi\_1} & \dots & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} & \dots & \mathbf{0} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} & \dots & \frac{\Delta\_{N-1}}{\xi\_{N-1}} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} & \dots & \mathbf{0} \end{pmatrix}. \tag{22}$$

This matrix shifts and rescales the vector entries on which it acts. This matrix approaches an identity matrix when Δ*<sup>j</sup>* ! 0.

For a finite Δ*j*, we look for the eigenvectors of the commutator matrix to obtain a diagonal matrix. The eigenvalues of the commutator (21), considered as a matrix, are all zero with multiplicity *<sup>N</sup>* <sup>þ</sup> 1. The eigenvectors are 1, 0, ð Þ … , 0 *<sup>T</sup>* and 0, ð Þ … , 0 *<sup>T</sup>* with multiplicity *N*. In addition to considering the eigenvectors of this commutator matrix to obtain a diagonal matrix, we can take advantage of rescaling to cancel shifting and return to the original vector. Then, the commutator matrix (21) has the additional eigenvector

$$\mathbf{h}^{T} = \left(\frac{\mathbf{1}}{\lambda^{N-1}} \prod\_{j=0}^{N-2} \frac{\Delta\_{j}}{\xi\_{j}}, \frac{\mathbf{1}}{\lambda^{N-2}} \prod\_{j=1}^{N-2} \frac{\Delta\_{j}}{\xi\_{j}}, \dots, \frac{\lambda \xi\_{N-1}}{\Delta\_{N-1}}\right)^{T},\tag{23}$$

with an eigenvalue *λ*. The action of the commutator matrix on these vectors results in the same vector with the last entry equal to zero, which is almost an eigenvector. Still another eigenvector, with eigenvalue one, is

$$\tilde{\mathbf{h}}^T = \left( \mathbf{1}, \frac{\xi\_0}{\Delta\_0}, \dots, \prod\_{n=0}^{N-1} \frac{\xi\_n}{\Delta\_n} \right)^T,\tag{24}$$

The commutator is equal to one along this direction. Then, the canonical commutation relationship is also valid in this direction.

Thus, along the mentioned directions, the d-derivative has similar properties as its continuous variable counterpart.

#### **4. The inverse of the d-derivative**

The d-derivative matrix that we use can be inverted. The determinant of the d-derivative matrix is

*The Inverse of the Discrete Momentum Operator DOI: http://dx.doi.org/10.5772/intechopen.112376*

$$|\mathbf{D}| = \frac{1}{\xi\_0 \xi\_1 \xi\_2 \xi\_3 \dots \xi\_N}. \tag{25}$$

The inverse of the d-derivative matrix **D** is the negative of the progressive discrete integration matrix

$$\mathbf{I} = \begin{pmatrix} \xi\_0 & \xi\_1 & \xi\_2 & \xi\_3 & \dots & \xi\_N \\ \mathbf{0} & \xi\_1 & \xi\_2 & \xi\_3 & \dots & \xi\_N \\ \mathbf{0} & \mathbf{0} & \xi\_2 & \xi\_3 & \dots & \xi\_N \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \xi\_3 & \dots & \xi\_N \\ \vdots & & & & \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} \end{pmatrix}. \tag{26}$$

We discuss some properties of the d-integration matrix **I**. When the d-integration matrix **I** is applied to the left to a vector **f** *<sup>T</sup>* results in

$$\mathbf{f}^T \mathbf{I} = (\mathcal{I}\_0 \mathbf{f}, \mathcal{I}\_1 \mathbf{f}, \mathcal{I}\_2 \mathbf{f}, \dots, \mathcal{I}\_N \mathbf{f}),\tag{27}$$

where

$$\mathcal{T}\_j \mathbf{f} = \xi\_j \begin{pmatrix} f\_0 + f\_1 + \dots + f\_j \end{pmatrix}, \quad j \le N. \tag{28}$$

The entries of the resulting vector are the progressive discrete integrations of **f** when the subintervals are of equal length *ξj*. When the d-integration matrix is applied to the right, we get

$$\mathbf{I}\mathbf{g} = \begin{pmatrix} \mathbf{I}\_0 \mathbf{g}, \mathbf{I}\_1 \mathbf{g}, \mathbf{I}\_2 \mathbf{g}, \dots, \mathbf{I}\_N \mathbf{g} \end{pmatrix},\tag{29}$$

where

$$\mathbf{J}\_{j}\mathbf{g} = \mathbf{g}\_{j}\boldsymbol{\xi}\_{j} + \mathbf{g}\_{j+1}\boldsymbol{\xi}\_{j+1} + \dots + \mathbf{g}\_{N}\boldsymbol{\xi}\_{N}, \quad \mathbf{0} \le j \le N. \tag{30}$$

This result is the progressive discrete integration of **g** when the subintervals are of different lengths.

The eigenvalues of **I** are *ξ*0, … , *ξN*, and its eigenvectors are the same as for **D**, Eq. (10). But, there is the additional eigenvector **<sup>e</sup>** <sup>¼</sup> *<sup>e</sup>*�*ipq*<sup>0</sup> , … ,*e*�*ipqN* � �*<sup>T</sup>* ,

$$\mathbf{I}\mathbf{e} = \frac{i}{p}\mathbf{e}.\tag{31}$$

The d-derivative and its inverse are constant along the same directions. The domain of the d-derivative and d-integration is the same.

Now, the commutator between **S** and **Q** is

$$\begin{aligned} \mathbf{[Q}, \mathbf{I}] = \begin{pmatrix} \mathbf{0} & \xi\_1 (q\_1 - q\_0) & \xi\_2 (q\_2 - q\_0) & \xi\_3 (q\_3 - q\_0) & \dots & -\xi\_N (q\_N - q\_0) \\ \mathbf{0} & \mathbf{0} & \xi\_2 (q\_2 - q\_1) & \xi\_3 (q\_3 - q\_1) & \dots & -\xi\_N (q\_N - q\_1) \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \xi\_3 (q\_3 - q\_2) & \dots & -\xi\_N (q\_N - q\_2) \\ \vdots & & & & \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & -\xi\_N (q\_N - q\_{N-1}) \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} \end{pmatrix}, \end{aligned} \tag{32}$$

which is the progressive discrete integral of *g q*ð Þ *q* � *qj* � � when acting on the vector **<sup>g</sup>**.

#### **5. Conclusions**

We have found another property of the d-derivative matrix: its inverse. The inverse of the d-derivative has the right properties; the properties of the continuous variable integration.

We discussed some of the properties of the discrete momentum operator when considering all of a subset of the spectrum points at once and its associated discrete integration matrices. The matrices are related by a common eigenvector for continuous variable functions. These results give us confidence that our choice is a good candidate for the discrete quantum momentum operator.

We also found that the matrices associated with the discrete derivative and the discrete integration have an additional eigenvalue and eigenvector, in contrast with the usual behavior of standard matrices. We have increased the number of eigenvalues and eigenvectors of a matrix by using it as an operator.

These operators are of help in defining a time operator and its eigenvalues and eigenvectors for use in nonrelativistic quantum mechanics [12]. They can also be used when the angular momentum on a circle is considered [15–17].

These results imply that we can deal with discrete quantum operators in almost the same way as for continuous variable operators case, including deficiency indices and self-adjoint extensions [13].

We have considered the exact discrete derivative for the complex exponential function, but these results are also valid for the real exponential function *e*�*pq* with the replacements

$$
\xi\_N = \frac{1}{p},
\tag{33}
$$

$$\xi\_j = \frac{\mathbf{1} - e^{-p\Delta\_j}}{p}, \quad j < N. \tag{34}$$

#### **Acknowledgements**

A. Martínez-Pérez would like to acknowledge the support from the UNAM Postdoctoral Program (POSDOC).

*The Inverse of the Discrete Momentum Operator DOI: http://dx.doi.org/10.5772/intechopen.112376*

#### **Author details**

Armando Martínez-Pérez1† and Gabino Torres-Vega<sup>2</sup> \*†

1 IIMAS, UNAM, México City, México

2 Departamento de Fisica, Cinvestav, México City, México

\*Address all correspondence to: gabino.torres@cinvestav.mx; gabino@fis.cinvestav.mx

† These authors contributed equally.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 2
