**1. Introduction**

Nonrelativistic quantum mechanics is commonly expressed by using a Hamiltonian function, with the total energy of an atomic system given by the sum of the kinetic and potential energies. If the atom is in an excited state the energy resides at the location of the electron. The electron in an excited state has a potential energy that is converted into a photon as it decays, so emission is a statistical event that occurs at a particular point in time. Because photon creation occurs at a single point in time, the electron must be in two energy states simultaneously. The two states correspond to a single time, so emission is described by non-covariant means even though we know that it is Lorentz invariant. Although nonrelativistic methods are used to describe atomic structure, they are not a necessary requirement of quantum mechanical formalism. As Dirac stated in the first ever paper on quantum electrodynamics, "The theory is nonrelativistic only on account of the time being counted throughout as a c-number [classically], instead of being treated symmetrically with the space coordinates" [1]. The use of a continuous time parameter to describe microscopic phenomena could easily mask the way time functions in atomic structure. The practice should be closely questioned because time is the least understood of the variables in quantum mechanics. Furthermore there are grounds to support a different approach to quantum mechanics using Lagrangian mechanics. It is not

only more fundamental than Hamiltonian mechanics, but if we begin with an action principle by taking the time integral of a Lagrangian, it is in agreement with special relativity theory.

## **2. Excitation**

We begin an attempt to understand the role of continuous time in quantum mechanics by describing emission as a sequence of events that are relativistically correct. Consider a uniform distribution of atomic oscillators immersed in a coherent or partially coherent radiation field with an outer electron that occupies either of two allowable energy states |1i or |2i. The radiation field is conceived of as classical, consisting of many oscillating wave train fields superimposed on the oscillators. The system may be described by a Lagrangian density that is a function of oscillator fields *φ<sup>r</sup>* and radiation fields ∂μ*φ*. If Hamilton's principle is applied macroscopically to an arbitrary classical region of space-time Ω, we obtain the usual expression:

$$S(\mathfrak{Q}) = \int\_{\mathfrak{Q}} L\left(\mathfrak{q}\_r, \partial\_{\mu}\mathfrak{q}\right) d\mathfrak{Q} \tag{1}$$

**3. Decay**

coincident with the steady states.

begin with a Lagrangian density *L ϕi*, *ϕ<sup>i</sup>*,*<sup>μ</sup>*

invariant, the same for all observers:

**21**

*S φ<sup>i</sup>* ½ �¼ ð Þ*t*

*R* ð1 ð*t*1

*L φi*, *φ<sup>i</sup>*,*<sup>μ</sup>* � �*d*<sup>3</sup>

The end points of the electron's path are located on equipotential, space-like surfaces, and the action minimum is not equal to zero as in classical theory, but to Planck's constant h. The action S[φi(t)] in (2) is functional, a function of the values of coordinates on the *discrete* boundary of the space-time surface R2 which is in turn a function of the *continuous* space-time variables of the fields within the surface.

*t*2

*R*2

The first appearance of a Lagrangian in quantum mechanics is in a paper by Dirac. "We ought to consider the classical Lagrangian not as a function of the coordinates and velocities but rather as a function of the coordinates at time t and the coordinates at time t+dt" [5]. Rather than specify emission as an event that occurs at a particular point in time, Dirac is seeking compatibility with relativity theory by calculating the change in action of the electron over a space-time interval. That idea resulted in an interpretation of electron transitions as a "sum over histories" of all possible paths from one diagonalized steady state to another [6]. Although calculations made with the path integral method are accurate, they include serious theoretical problems resulting from the renormalization methods used to deal with infinities that have so far prevented a complete theory from being formulated. At a later point in the same paper, Dirac provides a possible way out of this difficulty by proposing a complementary description of quantum mechanics in terms of a "vibrating medium." The idea begins from a classical vantage point. "We introduce at each point of space-time a Lagrangian density, which must be a function of the coordinates and their first derivatives with respect to x, y, z, and t, corresponding to the Lagrangian in particle theory being a function of coordinates and velocities. The integral of the Lagrangian density over any (four-dimensional) region of space-time must then be stationary for all small variations of the coordinates inside the region, provided the coordinates on the boundary remain invariant." To obtain the quantum analogue, he divided the classical region into a number of very small subregions; however, the idea was never pursued to completion. We shall investigate that line of reasoning in more detail by describing fields over a four-dimensional region of space-time with respect to invariant field boundaries

*Lagrangian Quantum Mechanics: A Fully Relativistic Theory of Atomic Structure*

*DOI: http://dx.doi.org/10.5772/intechopen.90168*

The excited states of an atom should be described discretely in time. As Dirac pointed out, discrete time is symmetric with the spatial coordinates allowing the emission and absorption of radiation to be described relativistically. A relativistic formulation is desirable in order to make quantum mechanics compatible with special relativity theory. To formulate a fully relativistic description of emission, we

their first derivatives. Then the action integral of the Lagrangian density over a particular region of space-time must be stationary for all small variations of the continuous coordinates in the region provided the discrete coordinates on the boundary remain invariant. Let three-dimensional surfaces R1 and R2 representing the diagonalized coordinates be erected and then used together with the discrete time period *t*<sup>2</sup> � *t*<sup>1</sup> as field boundaries to define a region of space-time between |2i and |1i. Integrating from |2i to |1i yields a relativistic formulation of emission that is

� � that is a function of the coordinates and

*xdt* ¼ *h* (2)

The behavior of the radiation field may be described by a Lagrangian density microscopically as well. The amplitude of the field at the local level fluctuates randomly, either reinforcing or canceling, thereby causing the electron to oscillate with respect to the nucleus. If the fields are of insufficient intensity to raise the electron from |1i to |2i, the uncertainty relations do not apply and the electron oscillates without radiating. However, if they are sufficiently intense, the electron will resonate at the excitation energy emitting a photon with each amplitude of the wave. When this happens radiation is emitted with double the frequency of the incident wave (see **Figure 1**).

Emission by atomic oscillators is referred to as "nonlinear" if excitation is thought to be quantum mechanical or "spontaneous" if it is thought to be governed by the macroscopic Maxwell's equations [2]. The use of microscopic Maxwell's equations in a Lagrangian density allows emission to be described by a distribution of locally superposed fields that includes continuous, statistical, and quantum mechanical properties in a single model.

Due to the generality of the assumptions, examples of frequency doubling should be common in nature occurring at many energy levels. This is in fact the case, and it is most clearly evident when laser light is passed through a crystal [3]. It is also readily observed using incoherent light in scattering experiments as secondary radiation [4]. On the other hand, when an energy quantum is completely absorbed, an electron will be expelled due to the photoelectric effect, and no frequency change is observed. We see in these examples evidence that the Lagrangian provides a versatile model for the description of electromagnetic phenomena.

**Figure 1.** *Frequency doubling.*

*Lagrangian Quantum Mechanics: A Fully Relativistic Theory of Atomic Structure DOI: http://dx.doi.org/10.5772/intechopen.90168*

## **3. Decay**

only more fundamental than Hamiltonian mechanics, but if we begin with an action principle by taking the time integral of a Lagrangian, it is in agreement with special

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

We begin an attempt to understand the role of continuous time in quantum mechanics by describing emission as a sequence of events that are relativistically correct. Consider a uniform distribution of atomic oscillators immersed in a coherent or partially coherent radiation field with an outer electron that occupies either of two allowable energy states |1i or |2i. The radiation field is conceived of as classical, consisting of many oscillating wave train fields superimposed on the oscillators. The system may be described by a Lagrangian density that is a function of oscillator fields *φ<sup>r</sup>* and radiation fields ∂μ*φ*. If Hamilton's principle is applied macroscopically to an arbitrary classical region of space-time Ω, we obtain the usual expression:

*S*ð Þ¼ Ω

ð

*L φr*, *∂μφ* � �*d*Ω (1)

Ω

The behavior of the radiation field may be described by a Lagrangian density microscopically as well. The amplitude of the field at the local level fluctuates randomly, either reinforcing or canceling, thereby causing the electron to oscillate with respect to the nucleus. If the fields are of insufficient intensity to raise the electron from |1i to |2i, the uncertainty relations do not apply and the electron oscillates without radiating. However, if they are sufficiently intense, the electron will resonate at the excitation energy emitting a photon with each amplitude of the wave. When this happens radiation is emitted with double the frequency of the

Emission by atomic oscillators is referred to as "nonlinear" if excitation is thought to be quantum mechanical or "spontaneous" if it is thought to be governed by the macroscopic Maxwell's equations [2]. The use of microscopic Maxwell's equations in a Lagrangian density allows emission to be described by a distribution of locally superposed fields that includes continuous, statistical, and quantum

Due to the generality of the assumptions, examples of frequency doubling should be common in nature occurring at many energy levels. This is in fact the case, and it is most clearly evident when laser light is passed through a crystal [3]. It is also readily observed using incoherent light in scattering experiments as secondary radiation [4]. On the other hand, when an energy quantum is completely absorbed, an electron will be expelled due to the photoelectric effect, and no frequency change is observed. We see in these examples evidence that the Lagrangian provides a versa-

relativity theory.

**2. Excitation**

incident wave (see **Figure 1**).

**Figure 1.** *Frequency doubling.*

**20**

mechanical properties in a single model.

tile model for the description of electromagnetic phenomena.

The first appearance of a Lagrangian in quantum mechanics is in a paper by Dirac. "We ought to consider the classical Lagrangian not as a function of the coordinates and velocities but rather as a function of the coordinates at time t and the coordinates at time t+dt" [5]. Rather than specify emission as an event that occurs at a particular point in time, Dirac is seeking compatibility with relativity theory by calculating the change in action of the electron over a space-time interval. That idea resulted in an interpretation of electron transitions as a "sum over histories" of all possible paths from one diagonalized steady state to another [6]. Although calculations made with the path integral method are accurate, they include serious theoretical problems resulting from the renormalization methods used to deal with infinities that have so far prevented a complete theory from being formulated. At a later point in the same paper, Dirac provides a possible way out of this difficulty by proposing a complementary description of quantum mechanics in terms of a "vibrating medium." The idea begins from a classical vantage point. "We introduce at each point of space-time a Lagrangian density, which must be a function of the coordinates and their first derivatives with respect to x, y, z, and t, corresponding to the Lagrangian in particle theory being a function of coordinates and velocities. The integral of the Lagrangian density over any (four-dimensional) region of space-time must then be stationary for all small variations of the coordinates inside the region, provided the coordinates on the boundary remain invariant." To obtain the quantum analogue, he divided the classical region into a number of very small subregions; however, the idea was never pursued to completion. We shall investigate that line of reasoning in more detail by describing fields over a four-dimensional region of space-time with respect to invariant field boundaries coincident with the steady states.

The excited states of an atom should be described discretely in time. As Dirac pointed out, discrete time is symmetric with the spatial coordinates allowing the emission and absorption of radiation to be described relativistically. A relativistic formulation is desirable in order to make quantum mechanics compatible with special relativity theory. To formulate a fully relativistic description of emission, we begin with a Lagrangian density *L ϕi*, *ϕ<sup>i</sup>*,*<sup>μ</sup>* � � that is a function of the coordinates and their first derivatives. Then the action integral of the Lagrangian density over a particular region of space-time must be stationary for all small variations of the continuous coordinates in the region provided the discrete coordinates on the boundary remain invariant. Let three-dimensional surfaces R1 and R2 representing the diagonalized coordinates be erected and then used together with the discrete time period *t*<sup>2</sup> � *t*<sup>1</sup> as field boundaries to define a region of space-time between |2i and |1i. Integrating from |2i to |1i yields a relativistic formulation of emission that is invariant, the same for all observers:

$$S[\varphi\_i(t)] = \int\_{R\_1}^{R\_1} \int\_{\mathbf{r}\_2} L(\rho\_i, \rho\_{i,\mu}) d^3 \mathbf{x} dt = h \tag{2}$$

The end points of the electron's path are located on equipotential, space-like surfaces, and the action minimum is not equal to zero as in classical theory, but to Planck's constant h. The action S[φi(t)] in (2) is functional, a function of the values of coordinates on the *discrete* boundary of the space-time surface R2 which is in turn a function of the *continuous* space-time variables of the fields within the surface.

Solving for the action, we obtain solutions for localized energy or equivalently, photon creation; due to a transformation of field:

$$\mathbf{E}\mathbf{\tau} = \mathbf{h} \tag{3}$$

and the Lagrangian is similarly given by

*DOI: http://dx.doi.org/10.5772/intechopen.90168*

existing mathematical models.

and for all times.

**23**

**4.3 Matrix mechanics**

**4.2 The path integral formulation**

L ¼ T þ ε–V (5)

where T represents the bound electron's energy, ε represents the energy of a "captured" photon, and V represents the potential energy due to the nucleus/proton. Each of the three field sources possesses a unique vector field, that is, a field with definite field geometry that is delimited from the others by field boundaries, where plus and minus signs indicate the linear superposition of delimited fields. Eqs. (4) and (5) contain the essence of quantum mechanics as a three-body problem in real space as opposed to current descriptions based on a two-body system in abstract space. The use of an abstract space is necessary for nonrelativistic descriptions of atomic structure since the photon is not treated as an independent particle. The equations revert to their classical form when the influence of ε is negligible or equivalently when field boundaries are no longer determinable. To see whether the model accurately describes atomic structure, we shall compare it to the

*Lagrangian Quantum Mechanics: A Fully Relativistic Theory of Atomic Structure*

From (5), it is postulated that the contribution of two energies is summed and one is subtracted to give the transition energy. In the path integral formulation, there are in fact two contributions that are summed, one determined by the paths and the other by the phase. In Feynman's words, "The paths contribute equally in magnitude, but the phase of their contribution is the classical action (in units of ђ)" [9]. The computation of the total energy is not complete, however, until contributions to the potential V due to self-energy are subtracted away by renormalizing. Therefore the mathematical structure of the Lagrangian in nonrelativistic quantum mechanics indicates the existence of three contributions and is in agreement with (5). It differs fundamentally from the fully relativistic Lagrangian method described here in its interpretation of space-time. Eq. (2) treats space and time equivalently as real parameters for both the integration limits and the region between them. On the other hand, the path integral formulation uses abstract forms of space and time to describe the region between the steady states since the paths follow all trajectories

In quantum mechanics, observables are determined by pairs of states, while in classical theory they refer to the same state. This is especially evident in matrix mechanics which describes the atom as a twofold infinite, denumerable array of virtual oscillators, where observables are vectors in Hilbert space whose magnitude defines a spectral line intensity, or transition amplitude, and whose direction corresponds to either an absorption or an emission. Although the physical model consists of two ideal particles, the virtual harmonic oscillators, the matrix elements include three field components: the fields of the two ideal particles and a spectral line intensity due to photon superpositions. All three components of the modified Hamiltonian given by Eq. (4) are present but with respect to classical space and time. In nonrelativistic theory, photons are singularities that correspond to pairs of states, whereas conformance with relativity theory requires that the photon be spatially and temporally extended. Fields are localized by exact four-dimensional

Solutions of Equation (3) are determinations of energy and time between exact four-dimensional field boundaries, so they are also exact. In the case of time periods, this has been confirmed to the limits of experimental accuracy by atomic clocks that can operate for many billions of years without significant error [7]. Thus field energy from a laser is absorbed by the lattice of ytterbium atoms and localized within a four-dimensional field boundary. The emission and absorption energies E21 and E12 have also long been assumed to be exact by astronomers when employed for the measurement of distant star composition. Emission and absorption spectra, together with a red shift, often require billions of years before they are observed, thereby reflecting the precise role of energy and time in natural phenomena.

From (2), a model of atomic structure may be constructed. The three field sources present in excited atomic states, electron, proton, and photon, superpose linearly and are momentarily stabilized in steady states. Although force is an unnatural concept in quantum mechanics, it may be interpreted with respect to the field boundaries that separate point sources by a careful consideration of (2). Thus the force on a bound electron due to the potential is equal to the continuously distributed excitation energy divided by the distance between field boundaries. In order for quantum mechanical forces to agree with relativity theory, we need only require that the action integral be invariant for all potentials both free and bound.

Whereas the quantum mechanical force of bound states is due to well-defined field boundaries, the force due to instantaneous exchanges of momentum, such as occurs in the Compton effect, may be interpreted as a result of the encounter of a single, exact field boundary (xo,yo,zo,to) with a material point. Wave properties, on the other hand, occur in free space when field boundaries have no reference point, so they cannot be observed at all. Thus we interpret complementarity as the different ways that fields and its field boundaries interact with matter.
