3. Solving the eigenvalue equation of the finite well: a historical perspective

The eigenvalue equations for the wave vectors (18)–(20) and (22)–(24) are transcendental equations, and their solutions cannot be written as a finite combination of elementary functions. More than this, till now, they cannot be expressed neither in terms of the special functions of the mathematical physics. There are a large number of papers devoted to this subject, in the last 60 years.

The first one, due to Pitkanen [14], writes the eigenvalue Eqs. (8) and (9) in the simpler form (18)– (20) and (22)–(24), providing an interesting visualization of the solutions. The second one, due to Cantrell [15] (who does not cite [14], producing a delay in the circulation of this paper), also proposes the replacement of Eqs. (8) and (9) with (18)–(20) and (22)–(24)—in fact, a repetition of Pitkanen's contribution—and notices that the eigenvalue equation for odd states is also the eigenvalue equation for a particle moving in a semi-infinite well, i.e., in a potential given by

$$\mathcal{U}\mathcal{U}(\mathbf{x}<0) = \ast, \quad \mathcal{U}(0<\mathbf{x}a) = \mathbf{0} \tag{30}$$

Graphical solutions are proposed by Guest [16], who made visible the similarities between the bound-state energies in a finite well and the modes of a metallic wave guide ([17]; fig. (8.14)); actually, both the electrodynamic and quantum mechanical problems are equivalent forms of the same Sturm-Liouville problem [18]. Aronstein and Stroud [19] wrote the eigenvalue equation as

$$\frac{ka}{2} + \arcsin\frac{ka/2}{P} = \frac{n\pi}{2} \tag{31}$$

<sup>δ</sup> <sup>¼</sup> <sup>ℏ</sup>

tractable, algebraic equations. For instance, we can use the approximations:

tan <sup>x</sup><sup>≃</sup> <sup>0</sup>:45<sup>x</sup> <sup>1</sup> � <sup>2</sup><sup>x</sup> π

will be actually inverted. We shall consider rsn <sup>≃</sup> <sup>n</sup> � <sup>1</sup>

that the ascendant part of this parabola is given by the equation:

<sup>y</sup> <sup>¼</sup> <sup>4</sup> <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> 2 � �π<sup>3</sup>

where the pair of constants can be chosen as

4. The parabolic approximation

has the coordinates ð Þ y; x :

[25], § 24.

as the magnitude of the domain outside the well, where the wave function can penetrate significantly, decreasing however exponentially. This concept is similar to the concept of skin depth in electromagnetism [17] or to the concept of viscous penetration depth in fluids

In the context of various approximations, it is worth to mention the "algebraization" of trigonometric functions, proposed by de Alcantara Bonfim and Griffiths [26], which transforms the transcendental equations for the eigenvalues of the finite well in approximate,

; cos <sup>x</sup><sup>≃</sup> <sup>1</sup> � ð Þ <sup>2</sup>x=<sup>π</sup>

To solve the eigenvalue equations, or—more generally—Eq. (16), with y≷0, means, as already mentioned, to obtain the inverse of the function y xð Þ defined by (16), i.e., to obtain the function x yð Þ: Geometrically, the inverse of the function y xð Þ, plotted as a curve whose generic point is ð Þ x; y , is its symmetric with respect of the first bisectrix. A generic point of the inverse function

Clearly, only the monotonic functions can be inverted; for instance, in our case, the function sin x=x must be replaced with its restriction on their intervals of monotony, and this restriction

function sin x=x on the interval 2ð Þ nπ;ð Þ 2n þ 1 π with a segment of parabola. It is easy to see

<sup>4</sup> � <sup>x</sup> � <sup>2</sup><sup>n</sup> <sup>þ</sup>

π2

2nπ < x < 2n þ

2

1 2 � �<sup>π</sup>

1 2 � �<sup>π</sup>

� �<sup>2</sup> ( ), (35)

2 <sup>1</sup> <sup>þ</sup> cx<sup>2</sup> ð Þ<sup>s</sup> , <sup>0</sup> <sup>⩽</sup>x<sup>⩽</sup>

s ¼ 1=2, c ¼ 0:212 or s ¼ 1, c ¼ 0:101 (34)

ð Þ <sup>2</sup>m Vð Þ <sup>0</sup> � <sup>E</sup> <sup>1</sup>=<sup>2</sup> <sup>=</sup><sup>32</sup> (32)

Quantum Wells and Ultrathin Metallic Films http://dx.doi.org/10.5772/intechopen.74150 91

π

� �π and approximate the bump of the

<sup>2</sup> (33)

This elegant form had been already given in the first edition of Landau's textbook of quantum mechanics, in the late 1940s of the twentieth century (for the English version of a more recent edition, see [20]) but remained unknown to Western physicists—a minor but significant consequence of the poor circulation of scientific information during the Cold War.

A completely different approach was proposed by Siewert [21], who obtained an exact solution in an integral form; unfortunately, it is very complicated and of limited practical use. Recently, Siewert's solutions were discussed in the context of generalized Lambert functions [22], a subject under intense investigation.

Among the papers which provide approximate analytical solutions of the eigenvalue Eqs. (18)–(20) and (22)–(24), the most popular one, authored by Barker et al. [23], is essentially a low-order algebraic approximation of sin x, cos x. Another interesting contribution is that of Garrett [24], who introduced an intuitive physical concept, the characteristic depth δ of a finite well, for a bound electron with energy E:

$$\delta = \frac{\hbar}{\left(2m(V\_0 - E)\right)^{1/2}} \text{ /32} \tag{32}$$

as the magnitude of the domain outside the well, where the wave function can penetrate significantly, decreasing however exponentially. This concept is similar to the concept of skin depth in electromagnetism [17] or to the concept of viscous penetration depth in fluids [25], § 24.

In the context of various approximations, it is worth to mention the "algebraization" of trigonometric functions, proposed by de Alcantara Bonfim and Griffiths [26], which transforms the transcendental equations for the eigenvalues of the finite well in approximate, tractable, algebraic equations. For instance, we can use the approximations:

$$
\tan x \simeq \frac{0.45\pi}{1 - \frac{2x}{\pi}}; \quad \cos x \simeq \frac{1 - \left(2x/\pi\right)^2}{\left(1 + cx^2\right)^s}, \quad 0 \leqslant x \leqslant \frac{\pi}{2} \tag{33}
$$

where the pair of constants can be chosen as

3. Solving the eigenvalue equation of the finite well: a historical

The eigenvalue equations for the wave vectors (18)–(20) and (22)–(24) are transcendental equations, and their solutions cannot be written as a finite combination of elementary functions. More than this, till now, they cannot be expressed neither in terms of the special functions of the mathematical physics. There are a large number of papers devoted to this

The first one, due to Pitkanen [14], writes the eigenvalue Eqs. (8) and (9) in the simpler form (18)– (20) and (22)–(24), providing an interesting visualization of the solutions. The second one, due to Cantrell [15] (who does not cite [14], producing a delay in the circulation of this paper), also proposes the replacement of Eqs. (8) and (9) with (18)–(20) and (22)–(24)—in fact, a repetition of Pitkanen's contribution—and notices that the eigenvalue equation for odd states is also the eigenvalue equation for a particle moving in a semi-infinite well, i.e., in a potential given by

Graphical solutions are proposed by Guest [16], who made visible the similarities between the bound-state energies in a finite well and the modes of a metallic wave guide ([17]; fig. (8.14)); actually, both the electrodynamic and quantum mechanical problems are equivalent forms of the same Sturm-Liouville problem [18]. Aronstein and Stroud [19] wrote the eigenvalue equa-

<sup>2</sup> <sup>þ</sup> arcsin ka=<sup>2</sup>

This elegant form had been already given in the first edition of Landau's textbook of quantum mechanics, in the late 1940s of the twentieth century (for the English version of a more recent edition, see [20]) but remained unknown to Western physicists—a minor but significant consequence of the poor circulation of scientific information during the

A completely different approach was proposed by Siewert [21], who obtained an exact solution in an integral form; unfortunately, it is very complicated and of limited practical use. Recently, Siewert's solutions were discussed in the context of generalized Lambert functions

Among the papers which provide approximate analytical solutions of the eigenvalue Eqs. (18)–(20) and (22)–(24), the most popular one, authored by Barker et al. [23], is essentially a low-order algebraic approximation of sin x, cos x. Another interesting contribution is that of Garrett [24], who introduced an intuitive physical concept, the characteristic depth δ of a finite

ka

U xð Þ¼ < 0 ∞, Uð Þ¼� 0 < x < a U0, Uxð Þ¼ > a 0 (30)

<sup>P</sup> <sup>¼</sup> <sup>n</sup><sup>π</sup>

<sup>2</sup> (31)

perspective

90 Heterojunctions and Nanostructures

tion as

Cold War.

[22], a subject under intense investigation.

well, for a bound electron with energy E:

subject, in the last 60 years.

$$s = 1/2, \ c = 0.212 \text{ or } s = 1, \ c = 0.101 \tag{34}$$

#### 4. The parabolic approximation

To solve the eigenvalue equations, or—more generally—Eq. (16), with y≷0, means, as already mentioned, to obtain the inverse of the function y xð Þ defined by (16), i.e., to obtain the function x yð Þ: Geometrically, the inverse of the function y xð Þ, plotted as a curve whose generic point is ð Þ x; y , is its symmetric with respect of the first bisectrix. A generic point of the inverse function has the coordinates ð Þ y; x :

Clearly, only the monotonic functions can be inverted; for instance, in our case, the function sin x=x must be replaced with its restriction on their intervals of monotony, and this restriction will be actually inverted. We shall consider rsn <sup>≃</sup> <sup>n</sup> � <sup>1</sup> 2 � �π and approximate the bump of the function sin x=x on the interval 2ð Þ nπ;ð Þ 2n þ 1 π with a segment of parabola. It is easy to see that the ascendant part of this parabola is given by the equation:

$$y = \frac{4}{\left(2n + \frac{1}{2}\right)\pi^3} \left\{ \frac{\pi^2}{4} - \left[ \mathbf{x} - \left(2n + \frac{1}{2}\right)\pi \right]^2 \right\},\tag{35}$$

$$2n\pi < \mathbf{x} < \left(2n + \frac{1}{2}\right)\pi$$

Solving this equation for y

$$\mathbf{x} = \left(2n + \frac{1}{2}\right)\pi - \sqrt{\frac{\pi^2}{4} - \frac{y}{4}\left(2n + \frac{1}{2}\right)\pi^3} \tag{36}$$

semi-infinite well (when one of the walls is infinite), or more realistic cases, when the walls are rounded (see [11, 12]). These potentials can model a semiconductor heterojunction (a thin semiconductor slice sandwiched between two different, larger semiconductors), a metallic film

We shall indicate now an approach for solving the eigenvalue Eqs. (18)–(20) and (22)–(24) providing an exact solution, written as a series expansion. We shall first illustrate this method

dp <sup>¼</sup> <sup>ζ</sup>1ð Þ<sup>p</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � p<sup>2</sup>ζ1ð Þp

2

, p ∈½ � 0, 1 , ζ1ð Þp ∈

X2nð Þ¼ p ζnð Þp , X2n�<sup>1</sup>ð Þ¼ p ξnð Þp , n ¼ 1, 2, …, (45)

2

þ x

pζ1ð Þ¼ p sin ζ1ð Þp (40)

cos <sup>ζ</sup>1ð Þ� <sup>p</sup> <sup>p</sup> (41)

Quantum Wells and Ultrathin Metallic Films http://dx.doi.org/10.5772/intechopen.74150 93

π <sup>2</sup> ; <sup>π</sup>

ζ1ð Þ¼ 0 π (44)

� � (43)

(42)

(46)

deposited on a semiconductor (in vacuum), and so on.

with the function ζ1ð Þp :

we get

5. The differential form of transcendental equations

Taking the derivative with respect to p in both sides of the equation

dζ1ð Þp

cos ζ1ð Þ¼� p

Using Eq. (40) and taking into account that we are in the second quadrant

1 � p<sup>2</sup>ζ1ð Þp

dXnð Þx

2

þ p

replacing p by x and relaxing the restriction p > 0, the equations for the eigenvalues of the

dx ¼ � Xnð Þ<sup>x</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

1 � x<sup>2</sup>Xnð Þx

we obtain the differential form of the equation for ζ1ð Þp :

dp ¼ � <sup>ζ</sup>1ð Þ<sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

dζ1ð Þp

wave vectors can be written in a unitary form:

with the initial condition:

Putting

and making the change x \$ y, we get for the root ζ2<sup>n</sup> [27]:

$$\zeta\_{2n}^{(pr)}(\mathbf{x}) = \left(2n + \frac{1}{2}\right)\pi - \sqrt{\frac{\pi^2}{4} - \frac{\mathbf{x}}{4}\left(2n + \frac{1}{2}\right)\pi^3},\tag{37}$$

$$0 < \mathbf{x} < \frac{1}{\left(2n + \frac{1}{2}\right)\pi}$$

Following exactly the same steps, similar expressions can be obtained for ζ2nð Þ x < 0 and for all the functions ζq, ξ<sup>q</sup> their parabolic approximations can be obtained. A special case is ζ<sup>1</sup> :

$$
\zeta\_{2n}^{(par)}(\mathbf{x}) = \pi \sqrt{1-\mathbf{x}} \tag{38}
$$

The method cannot be applied, evidently, for ξ1, as the function to be inverted has no bump.

The explicit expressions of the parabolic approximation for the functions ξ<sup>n</sup> ð Þ n > 1 and ζn, obtained in [27] are simple, but cumbersome, and will not be given here.

It is possible to improve the parabolic approximation in two ways:

(1) To express the numerical coefficients in formulas similar to Eq. (36) using analytic approximations for the roots of the equations tan x ¼ x and tan x ¼ �1=x: Actually, these transcendental equations can be transformed in approximate, tractable, algebraic equations, using the algebraic approximations of the tan function, proposed by de Alcantara Bonfim and Griffiths [26] and generalized by other authors [28]. This approach is sometimes called "improved parabolic approximation."

(2) To approximate the bumps of the functions sin x=x and cos x=x with a cubic curve (polynomial); this approach is sometimes called cubic approximation. The calculations are elementary, but cumbersome, and will not be given here [29].

For an algebraic approximation of ζ1, we can use a formula similar to the cos approximation in Eq. (34), namely,

$$\frac{\sin x}{x} \simeq \frac{1 - \left(\frac{x}{x}\right)^2}{\sqrt{1 + 0.2x^2}}\tag{39}$$

proposed in [30].

The finite square well is a good starting point for similar quantum mechanical problems, i.e., the asymmetric well (when the walls of the well, see Figure 1, have different heights), the semi-infinite well (when one of the walls is infinite), or more realistic cases, when the walls are rounded (see [11, 12]). These potentials can model a semiconductor heterojunction (a thin semiconductor slice sandwiched between two different, larger semiconductors), a metallic film deposited on a semiconductor (in vacuum), and so on.
