A simpler graphical solution and an approximate formula for energy eigenvalues in finite square quantum wells

Lima, F. M. S.

doi:10.1119/10.0001694

The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.

In modern physics or introductory quantum mechanics textbooks, we find only a few quantum systems whose bound state energy levels are easy to be determined. One of the first examples of exactly solvable quantum states is that of a particle of mass m confined in an infinitely deep square quantum well (SQW). On choosing x = 0 at the center of the well potential, whose width is L, the potential energy is

V (x) = 0

inside it (i.e., for

- L / 2 < x < + L / 2

⁠) and

\infty

otherwise. For this potential, the wave functions for stationary states are

ψ_{n} (x) = A cos (k_{n} x), n odd

⁠, and

ψ_{n} (x) = B sin (k_{n} x), n even

⁠.¹ From the boundary condition

ψ_{n} (\pm L / 2) = 0

— i.e., the vanishing of the de Broglie wave function at the impenetrable walls, forming standing waves—it follows that

k_{n} = n π / L

⁠, so the allowed energy levels are¹^,

ϵ_{n} = \frac{ℏ^{2} k_{n}^{2}}{2 m} = n^{2} \frac{π^{2} ℏ^{2}}{2 m L^{2}} = n^{2} ϵ_{1} .

(1)

However, this naive model is an oversimplification, which is never realized in practice. A more realistic model is the finite SQW, in which a particle moves between walls with a finite height V₀, penetrating into the walls (i.e., the classically forbidden regions), which leads to longer de Broglie wavelengths and thus lower energy eigenvalues. This problem is treated in all modern physics and quantum mechanics textbooks, but, in spite of its simplicity, the transcendental equations arising from the matching conditions for the wave functions at the potential walls, whose roots correspond to the energy eigenvalues, do not admit exact closed-form solutions.^2,3 Though we can appeal to numerical root-finding (iterative) routines, some alternatives are found in the literature (approximate analytical or graphical solutions or series expansions), but they are of limited practical value because they are either of poor accuracy or fail in the limit of infinitely deep or very shallow well potentials (see, e.g., Refs. 5–10).

Since the finite SQW model has important applications in many branches of physics, from atomic and nuclear to solid-state and nanophysics (e.g., in semiconductor heterojunctions, ultra-thin metallic films, and atomic force microscopy), it is pedagogically valuable to search for a simple approximation scheme that yields correct results for both infinitely deep and very shallow SQWs.

For a better comparison to infinite SQWs, let us choose the potential of the finite SQW as

V (x) = 0

for

- ℓ < x < + ℓ

⁠, where

ℓ \equiv L / 2

⁠, and

V (x) = V_{0} > 0

otherwise. For this potential, the one-dimensional, time-independent Schrödinger equation can be written as

ψ ″ + k^{2} ψ = 0, - ℓ < x < + ℓ,

(2a)

ψ ″ - q^{2} ψ = 0, | x | > ℓ,

(2b)

where

k \equiv \sqrt{2 m E} / ℏ

and

q \equiv \sqrt{2 m (V_{0} - E)} / ℏ

⁠. Since

0 < E < V_{0}

for bound states, both k and q are positive real numbers, which makes these second-order differential equations with constant coefficients solvable in terms of trigonometric and exponential functions. The boundary condition

ψ (\pm \infty) = 0

reduces the general solution to

ψ (x) = {\begin{matrix} \pm C e^{qx}, x < - ℓ \\ {\begin{matrix} A cos (k x) \\ B sin (k x), - ℓ < x < + ℓ \end{matrix} \\ C e^{- q x}, x > + ℓ . \end{matrix}

(3)

Here, the upper (lower) sign and trigonometric function are for symmetric (antisymmetric) states. The matching conditions for

ψ (x)

and

ψ' (x)

at

x = \pm ℓ

then yield

A cos (k ℓ) = C e^{- q ℓ},

(4a)

A k sin (k ℓ) = C q e^{- q ℓ}

(4b)

for symmetric wave functions and

B sin (k ℓ) = C e^{- q ℓ},

(5a)

B k cos (k ℓ) = - C q e^{- q ℓ}

(5b)

for antisymmetric ones. Dividing Eq. by Eq. and Eq. by Eq. , these pairs of equations are reduced to

(6)

θ tan θ = q ℓ,

(6a)

θ cot θ = - q ℓ,

(6b)

respectively, where

θ \equiv k ℓ = (\sqrt{2 m E} / ℏ) ℓ

is a suitable dimensionless variable. Though these equations are enough for a graphical solution, as done, e.g., in Refs. 2–4, on seeking for a simpler solution, let us rewrite them as

θ sin θ = q ℓ cos θ,

(7a)

θ cos θ = - q ℓ sin θ .

(7b)

On taking the square on both sides of each equation, we apply the trigonometric identity

{sin}^{2} θ + {cos}^{2} θ = 1

to further reduce them to

(8)

| cos θ | = \frac{θ}{θ_{m a x_{}}},

(8a)

| sin θ | = \frac{θ}{θ_{max}},

(8b)

where

θ_{max} \equiv (\sqrt{2 m V_{0}} / ℏ) ℓ

corresponds to the maximum value that θ can assume. Since

θ > 0

by virtue of Eq. , one must add the conditions

tan θ > 0

to Eq. and

cot θ = 1 / tan θ < 0

to Eq. , which is why only the right-hand “half” of the trigonometric curves in Eq. were taken into account in Fig. 1. In this figure, θ_n,

n = 1, 2, \dots, N

⁠, are the abscissae of the points where the straight-line

y = θ / θ_{max}

intersects those curves. The total number N of such points is given by the ratio

θ_{max} / (π / 2)

rounded up to the nearest positive integer, i.e.,

N = ⌊ \frac{θ_{max}}{(π / 2)} ⌋ + 1,

(9)

which is just the number of allowed energy levels. Here,

⌊ x ⌋

is the integer part of x. Note that the line

y = θ / θ_{max}

crosses the trigonometric curves at points with higher and higher ordinates, as indicated by vertical dashed lines in Fig. 1, which makes the abscissae

θ_{n}

of these points move toward the lower end of the n-th interval with the increase in n since each trigonometric curve is a decreasing function of θ. (Therefore,

(n - 1) π / 2 < θ_{n} < θ_{n - 1} + π / 2

and

(N - 1) π / 2 < θ_{N} < min {θ_{max}, θ_{N - 1} + π / 2}

⁠. These upper bounds should be taken into account in the optimization of numerical root-finding routines applied to this problem.) This simple graphical solution allows for a better understanding of the dependence of the energy eigenvalues on the SQW physical parameters. The increase in V₀ for a given L, or that of L for a given V₀, increases θ_max, thus reducing the slope of the line

y = θ / θ_{max}

⁠, which increases the number N of intersections with the trigonometric curves. In the limit of infinitely deep SQWs,

θ_{max} \to \infty

⁠, and then, the crossing-points will tend to the points on the axis of abscissae with

θ_{n} = n π / 2

⁠, which makes

E_{n} = ℏ^{2} θ_{n}^{2} / (2 m ℓ^{2})

tend to

n^{2} ϵ_{1}

⁠, which is just the well-known result for infinite SQWs [see Eq. ]. Inversely, the decrease in V₀ for a given L, or that of L for a given V₀, reduces θ_max, thus increasing the slope of the line

y = θ / θ_{max}

until the number N of intersections to be reduced to only one, which occurs when

θ_{max} < π / 2

⁠, or, equivalently,

V_{0} < π^{2} ℏ^{2} / (2 m L^{2}) = ϵ_{1}

⁠, the ground-state energy for the corresponding infinitely deep SQW. This shows that there is at least one stationary bound state in a finite SQW, no matter how shallow it is. For very shallow SQWs,

θ_{max} ≪ π / 2

⁠, so the Taylor approximation

cos θ \approx 1 - θ^{2} / 2!

becomes accurate, and we are left with

1 - θ^{2} / 2 = θ / θ_{max}

whose only positive solution is

θ_{1} = \frac{1}{θ_{max}} (\sqrt{1 + 2 θ_{max}^{2}} - 1) \approx θ_{max} - \frac{θ_{max}^{3}}{2},

(10)

where

θ_{max}^{2} ≪ 1

justifies the binomial approximation. On writing this result in terms of E₁ and V₀ and neglecting high-order terms, one promptly finds

E_{1} \approx V_{0} - \frac{2 m ℓ^{2} V_{0}^{2}}{ℏ^{2}},

(11)

which is the correct result for very shallow SQWs, as established, e.g., in Eq. (4.19) of Ref. 11.

Since the trigonometric curves in Fig. 1 are identical, I realized that all the roots θ_n,

n = 1, \dots, N

⁠, can be computed by reducing the trigonometric functions in Eq. to a single cosine curve over

[0, π / 2]

containing all “roots”

{\tilde{θ}}_{n} = θ_{n} - (n - 1) π / 2

⁠, as indicated in Fig. 2. Our approximation scheme consists in interpolating the function

y = cos θ

by its three points (0, 1),

(π / 3, 1 / 2)

⁠, and

(π / 2, 0)

⁠, marked with black bullets in Fig. 2. This yields the parabola (see the red, dashed curve)

y = - \frac{3}{π^{2}} θ^{2} - \frac{1}{2 π} θ + 1 .

(12)

The exact

{\tilde{θ}}_{n}

values will be approximated by α_n, the abscissae of the points where the parallel straight-lines

y = \frac{θ}{θ_{max}} + (n - 1) \frac{π / 2}{θ_{max}}

(13)

intersect the interpolation curve, within the interval

(0, π / 2)

⁠. These points are marked with red bullets in Fig. 2. For each

n = 1, \dots, N

⁠, with N as given in Eq. , the only positive solution of

3 z^{2} + (\frac{1}{2} + \frac{1}{Z}) z + \frac{n - 1}{2 Z} - 1 = 0,

(14)

where

z \equiv θ / π

and

Z \equiv θ_{max} / π

⁠, is

2 z_{n} = - b + \sqrt{b^{2} - \frac{2 (n - 1)}{3 Z} + \frac{4}{3}},

(15)

where

b \equiv 1 / 6 + 1 / (3 Z)

⁠. Since

{\tilde{θ}}_{n} \approx α_{n} = π z_{n}

⁠,

\begin{matrix} E_{n} \approx \frac{ℏ^{2}}{2 m ℓ^{2}} {[π z_{n} + (n - 1) π / 2]}^{2}, \\ = \frac{π^{2} ℏ^{2}}{2 m L^{2}} {(2 z_{n} + n - 1)}^{2}, \\ = {(2 z_{n} + n - 1)}^{2} ϵ_{1} . \end{matrix}

(16)

Let us show how our approximation scheme works using a SQW whose physical parameters, in atomic units, (In this unit system, lengths are expressed in units of $a_{B} \equiv ℏ^{2} / (m_{0} k_{e} e^{2}) = 0.529$ Å, the Bohr radius, and energies are in units of $H t \equiv m k_{e}^{2} e^{4} / ℏ^{2} = 27.2114$ eV, the Hartree energy. The main advantage of this system is that $m_{0} = ℏ = k_{e} = e = 1$ ⁠.) are $L = 2$ and $V_{0} = 49 π^{2} / 32 = 15.11$ ⁠, which were chosen in a manner that $θ_{max} = \frac{7}{4} π$ ⁠, the value used in Figs. 1 and 2. For this value, Eq. (9) returns $N = ⌊3.5⌋ + 1 = 4$ ⁠. Then, since $Z = 7 / 4$ ⁠, one promptly finds $b = 1 / 6 + 4 / 21 = 5 / 14$ ⁠, with which our approximate formula in Eq. (15) returns $2 z_{1} = 0.8515, 2 z_{2} = 0.6821, 2 z_{3} = 0.4789$ ⁠, and $2 z_{4} = 0.2068$ ⁠. With these values in hands, being $ϵ_{1} = π^{2} ℏ^{2} / (2 m L^{2}) = π^{2} / 8$ a.u., Eq. (16) returns the following approximate values (in a.u.): $E_{1} = 0.8946 (0.8805), E_{2} = 3.4905 (3.4864), E_{3} = 7.5811 (7.6788)$ ⁠, and $E_{4} = 12.6868 (12.9883)$ ⁠. (The values in parenthesis are the exact energy eigenvalues, found by solving Eqs. (8a) and (8b) numerically.) Our approximate values are in good agreement with the corresponding exact ones.

In conclusion, in this note, I have presented a simpler graphical solution and an approximate analytical formula for the energy levels of stationary bound states of a particle in a finite SQW. Our graphical solution demands only one dimensionless parameter, namely, θ_max, which determines the straight-line $y = θ / θ_{max}$ plotted in Fig. 1, which crosses some identical trigonometric curves, allowing students to see that finite SQWs always have a finite number $N \geq 1$ of stationary bound states and, since this figure also shows that the decrease in V₀ increases the slope of the straight-line, they can understand why the energy eigenvalues $E_{n}$ deviate more and more from $ϵ_{n}$ ⁠, the corresponding eigenvalues for infinite SQWs. In Fig. 2, we succeeded in reducing those curves to a single cosine curve over the interval $[0, π / 2]$ ⁠. As this curve resembles a parabola, I have taken the endpoints, together the “midpoint” $(π / 3, 1 / 2)$ ⁠, for a quadratic interpolation. This has led to an accurate formula for the “roots” ${\tilde{θ}}_{n}$ ⁠, from which one promptly finds the corresponding energy eigenvalues and all this without artificial numerical factors or adjustable parameters. We also give a numerical example to illustrate how our approximation scheme works, and we think that it is so simple that the reader can easily repeat the computations for arbitrary values of $L$ and $V_{0}$ in a pocket calculator. As our method is not iterative, it is free of divergence issues, or slow convergence, as well as spurious roots, with the advantage of becoming more and more accurate in both limits of infinitely deep and very shallow SQWs, as given in Eqs. (1) and (11), respectively.

Of course, some authors have already developed graphical solutions for the finite SQW problem similar to the one proposed here. In Sec. 4.2 of Ref. 11, a more advanced textbook, Davies arrives at a pair of simple equations in his Eq. (4.14), which is the same as our Eq. (8), but, astonishingly, he turns back to the more complicated original result, our Eq. (6), to develop a graphical solution! In Ref. 12, the authors also arrive at that pair of equations, but their graphical analysis is somewhat obscured by some equations involving complex numbers, which follow from their choice of the reference level for the well potential (null in the barriers and negative into the well). Moreover, they adopt k for the axis of abscissae, which is not a dimensionless quantity. In Ref. 13, Mitin et al. employed a method very similar to ours, but they prefer to use the inverse trigonometric function $\arcsin x$ ⁠, which is not a single-valued function. In comparison to those methods found in textbooks, our method certainly is both simpler and more elegant.

On regarding previous attempts to approximate the allowed energy levels in finite SQWs, distinct methods are found in the literature, but they either present a poor accuracy or they do not remain accurate for both very shallow and very deep SQWs.^5–7,9 Reed's method,⁷ e.g., which is often mentioned in recent papers (see, e.g., Ref. 10), involves a rather complicated function, which certainly demands the use of a computer, which in turn could be used for solving Eq. (6) numerically or even for implementing a full numerical solution of the Schrödinger equation! As our interpolation of the cosine function includes the endpoints (0, 1) and $(π / 2, 0)$ ⁠, our approximate formula in Eq. (15) will also furnish the exact result for infinite SQWs and the correct result for very shallow SQWs. Finally, for those who insist in solving Eq. (6) numerically, our approximate energies could be taken as good starting points to improve the convergence of numerical routines applied to this problem.

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A simpler graphical solution and an approximate formula for energy eigenvalues in finite square quantum wells

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