Abstract
We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box. We introduce the “blurring” potential in the form of the modified WoodsSaxon potential and solve the corresponding Schrödinger equation. It is found that the increase of the degree of blurring δ of the QW leads to the increase of number of the energy levels inside it and to increase of deviations from the quadratic dependence ε (n) (ε is the particle energy, n is the energy level number) typical for the infinite square QW, especially, for the energy levels close to the QW “tops”. It is most surprising that for relatively “large” values of δ the difference between the levels energies of such well and the appropriate (with the same n) levels energies of the square QW with the same depth changes sign (from positive to negative) as number n increases. We also conclude that the asymmetry of the QW and nonequality
Keywords:
infinite square quantum well; the “blurring” potential; the asymmetry of QW; the positiondependent effective mass
1. Introduction
As it is known, the study of model systems, for which there are simple analytic solutions of the time independent Schrödinger equation, makes it possible to understand the methods of quantum mechanics more comprehensively. In addition, the results obtained are of independent interest, since they reflect, in some approximation, the properties of the corresponding real systems. One of such idealized system is the particle in a onedimensional box model [1][1] D.J. Griffths, Introduction to Quantum Mechanics (Prentice Hall, Nova Jersey, 1995), p. 24., [2][2] D.F. Styer, Am. J. Phys. 69, 56 (2001). (also known as the infinite square quantum well (QW)) that describes a particle which can only move freely along a linear segment of finite length. Inside this segment the potential is considered equal to zero. At all other points of the straight line the potential goes to infinity. The particle in a box model is mainly used as an approximation for the description of quite complicated quantum systems. For example, the behaviour of electrons in some chemical compounds can be precisely modelled using particle in a box [3][3] T. Wimpfheimer, J. Lab. Chem. Educ. 3, 19 (2015)..
The energy spectrum of such QW is very simple [1][1] D.J. Griffths, Introduction to Quantum Mechanics (Prentice Hall, Nova Jersey, 1995), p. 24.:
where
There are many articles on square quantum wells (finite or infinite) in the literature. The educational papers in this area can be divided into the groups of works devoted to: improving the procedures for approximate finding energy spectrum in the finite square QW [^{4}[4] P.H. Pitkanen, Am. J. Phys. 23, 111 (1955).–^{8}[8] D.W.L. Sprung, H. Wu and J. Martorell, Eur. J. Phys. 13, 21 (1992).];^{}[5] B.I. Barker, G.H. Rayborn, J.W. Ioup and G.E. Ioup, Am. J. Phys. 59, 1038 (1991). ^{}[6] B.O.F. Alcantara and D.J. Griffths, Am. J. Phys. 74, 43 (2006). ^{}[7] P.G. Guest, Am. J. Phys. 40, 1175 (1972). solving the timedependent Schrödinger equation in the case of a onedimensional square QW with moving walls [^{9}[9] D.N. Pinder, Am. J. Phys. 58, 54 (1990)., ^{10}[10] J.G. Cordes, D. Kiang and H. Nakajima, Am. J. Phys. 52, 155 (1984).]; the study of symmetry properties of the quantum systems with squarewell potential [^{11}[11] F. Leyvraz, A. Frank, R. Lemus and M.V. Andrés, Am. J. Phys. 65, 1087 (1997), ^{12}[12] D. ShiHai and M. ZhongQi, Am. J. Phys. 70, 520 (2002).]; the investigations of the relativistic particle in a box [^{13}[13] C.G. Adler, Am. J. Phys. 39, 305(1971)., ^{14}[14] P. Alberto, S. Das and E.C. Vagenas, Eur. J. Phys. 39, 025401 (2017).]; the considerations of boundary conditions for an infinite squarewell potential in quantum mechanics [^{15}[15] M. Bowen and J. Coster, Am. J. Phys. 49, 80 (1981)., ^{16}[16] R. Seki, Am. J. Phys. 39, 929 (1971).] et al. However, from the theoretical point of view two issues are to be clarified. What are the possible reasons for the deviations from the model of a onedimensional infinite square QW in reality? How do these factors transform the energy spectrum (1)?
There are several factors that cause the deviations from the infinite square QW model for real microscopic systems. These are:

The finite QW depth.

The small “blurring” of the QW walls.

The asymmetry of QW.

The positiondependent effective mass in solids.

Existence of two additional mechanical degrees of freedom of the particle.
The aim of this article is the consideration of the influence of such factors on the deviations from energy spectrum (1) in the real QW systems (heterostructures, quantum dots and atomic nuclei, etc.) that can be approximately modelled using particle in a box.
2. The “blurring” of the walls and finite quantum well depth
The assumption of the vertical walls is nonphysical because the force
Let us consider the finite “blurring” potential in the form of the modified WoodsSaxon potential [^{17}[17] R.D. Woods and D.S. Saxon, Phys. Rev. 95, 577 (1954)., ^{18}[18] V.K. Dolmatov, J.L. King and J.C. Oglesby, J. Phys. B: At. Mol. Opt. Phys. 45, 105102 (2012).] that represents a square well with rounded edges:
where V_{0} is the QW depth;
The solid line is the “blurring” quantum well (2). The dashed line is the infinite square quantum well.
We write the Schrödinger equation for (2) as
where
To satisfy the Dirichlet condition, we represent the wave function in the form:
then the Schrödinger Eq. (3) can be transformed to the Euler's hypergeometric differential equation [19][19] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972), p. 555.:
The solution of this equation is the Gauss's hypergeometric function
Therefore, the even parity solution of our problem is
where C is the normalizing factor and
The odd parity solution is given as
where
The conditions (9), (11) quantize energies ε_{n}, where quantum number n numbers the energy levels in order of increasing their energy.
Fig. 2, 3 show the results of numerical calculations (for the numerical solution of the transcendental equations hereinafter we apply the “fsolve” command of the Maple software [20][20] Maple User Manual, http://www.maplesoft.com/documentation center/maple2017/UserManual.pdf.
http://www.maplesoft.com/documentation c...
) of particle energy ε depending on continuous variable n, based on Eqs. (4
11), at different values of δ and u. It should be noted that in Eqs. (8) and (10)n is the integer variable. But we let it change continuously to get smooth curves ε_{n} (n).
Dimensionless energy ε as a function of continuous variable n. (1)
Dimensionless energy ε as a function of continuous variable n. (1)
From Fig. 2 it is seen that taking into account the nonzero value of δ will cause the deviations from quadratic dependence ε(n) typical for the infinite square QW (see Eq. (1)), especially, for the energy levels close to the QW “tops”. With an increase in the degree of blurring of QW δ, the number of the energy levels inside it increases too. It is most surprising is that for relatively “large” values of δ the difference between the levels energies of such well and the appropriate (with the same n) levels energies of square QW with the same depth changes sign (from positive to negative) as number n increases. The above results are not previously reported in literature and can help readers to probe the limits of applicability of the square (infinite or finite) QW.
As it expected the decrease of a well depth (that is the increase of the dimensionless quantity u; see Fig. 3) causes the decrease of number of such discrete states. Therefore, there always exist at least one discrete level located near the top of QW (2) as for the case of the symmetric finite square QW. This level corresponds to the even parity solution (8) with n = 0.
Our numerical calculations (which are done using Eq. (1) and Eq. (12) and putting α = 1) also indicate that the location of the levels within finite and infinite square QW approximations is practically the same (the maximum relative error in the determination of the levels position is less than 5%) if
3. The asymmetry of quantum well
The QW, where the inversion symmetry with respect to the quantum well center is broken by some means, is called the asymmetric QW. The symmetry breaking can be achieved in several different ways, for example, varying the material (alloy) composition in QW or applying an electric field along the growth direction [21][21] H.C. Liu and F. Capasso, Intersubband transitions in quantum wells: Physics and device applications I (Academic Press, London, 2000), p. 26. . The energy spectrum of such a QW differs from the infinite square QW not only due to the finite well depth but also due to the asymmetry of the potential.
For the simplest case of the asymmetric square QW the transcendental equation for finding particle energy
where
Dimensionless energy
We see that
4. The positiondependent effective mass
In solid state physics, a particle effective mass is the quasimass of a particle that takes into account its interaction with the internal electric field of a condensed matter (crystal). The effective mass of particles in lowdimensional semiconductors is positiondependent. Within the simplest approximation the effective mass inside the well,
where
We seek solution of the stationary Schrödinger equation in the form:
where
Using Eqs. (13) (15), we derive:
Solving system (17) and using Eq. (16), we finally get:
Eq. (16) obtained here has that one advantage that it describes the energy spectrum of both even and odd states sumultaneously, whereas usually [23][23] V. Barsan and M.C. Ciornei, Eur. J. Phys. 38, 015407 (2017). two equations are applied to find eigenvalues.
In Fig. 5 we plot dependences
Dimensionless energy ε as a function of continuous variable n for: (1)
At fixed value of n the particle energy is an increasing function of parameter β. We see that the deviation of parameter β from 1 can play a significant role only for the relatively “shallow” well near the QW top. These obtained features of the energy spectrum are of pedagogical value, since they can grasp the applicability limits of the model of an infinitely deep QW, for which difference between
5. Existence of two additional mechanical degrees of freedom of the particle
In reality the motion of a particle occurs not in one but in three directions. These two extra motions can be finite as in the case of rotational motion in a quantum dot or have a quasifree character (the socalled lateral motion) as for a twodimensional nanosystem. As an example let us consider the infinite spherical QW. The position of the energy levels in such a well is determined via the roots of equation [24][24] S. Flügge, Practical Quantum Mechanics I (SpringerVerlag, New York, 1974).
where
It is interesting that for l = 0 (that is in the absence of orbital motion of the particle) equation (17) can be solved analytically and we get the energy spectrum in form (1) with only even values of
Thus, the adding of the new mechanical degrees of freedom to the particle in box problem significantly complicates its solution and the interpretation of results. This circumstance should be explained to students on such and similar examples.
6. Conclusions
We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box. We introduce the “blurring” potential in the form of the modified WoodsSaxon potential and solve the corresponding Schrödinger equation. It is found that the increase of the degree of blurring δ of the QW leads to the increase of number of the energy levels inside it and to increase of deviations from the quadratic dependence
References

^{[1]}D.J. Griffths, Introduction to Quantum Mechanics (Prentice Hall, Nova Jersey, 1995), p. 24.

^{[2]}D.F. Styer, Am. J. Phys. 69, 56 (2001).

^{[3]}T. Wimpfheimer, J. Lab. Chem. Educ. 3, 19 (2015).

^{[4]}P.H. Pitkanen, Am. J. Phys. 23, 111 (1955).

^{[5]}B.I. Barker, G.H. Rayborn, J.W. Ioup and G.E. Ioup, Am. J. Phys. 59, 1038 (1991).

^{[6]}B.O.F. Alcantara and D.J. Griffths, Am. J. Phys. 74, 43 (2006).

^{[7]}P.G. Guest, Am. J. Phys. 40, 1175 (1972).

^{[8]}D.W.L. Sprung, H. Wu and J. Martorell, Eur. J. Phys. 13, 21 (1992).

^{[9]}D.N. Pinder, Am. J. Phys. 58, 54 (1990).

^{[10]}J.G. Cordes, D. Kiang and H. Nakajima, Am. J. Phys. 52, 155 (1984).

^{[11]}F. Leyvraz, A. Frank, R. Lemus and M.V. Andrés, Am. J. Phys. 65, 1087 (1997)

^{[12]}D. ShiHai and M. ZhongQi, Am. J. Phys. 70, 520 (2002).

^{[13]}C.G. Adler, Am. J. Phys. 39, 305(1971).

^{[14]}P. Alberto, S. Das and E.C. Vagenas, Eur. J. Phys. 39, 025401 (2017).

^{[15]}M. Bowen and J. Coster, Am. J. Phys. 49, 80 (1981).

^{[16]}R. Seki, Am. J. Phys. 39, 929 (1971).

^{[17]}R.D. Woods and D.S. Saxon, Phys. Rev. 95, 577 (1954).

^{[18]}V.K. Dolmatov, J.L. King and J.C. Oglesby, J. Phys. B: At. Mol. Opt. Phys. 45, 105102 (2012).

^{[19]}M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972), p. 555.

^{[20]}Maple User Manual, http://www.maplesoft.com/documentation center/maple2017/UserManual.pdf
» http://www.maplesoft.com/documentation center/maple2017/UserManual.pdf 
^{[21]}H.C. Liu and F. Capasso, Intersubband transitions in quantum wells: Physics and device applications I (Academic Press, London, 2000), p. 26.

^{[22]}L.D. Landau and E.M. Lifshitz, Quantum Mechanics: NonRelativistic Theory (Pergamon Press, 1977), v. 3, 3ª ed., p. 63.

^{[23]}V. Barsan and M.C. Ciornei, Eur. J. Phys. 38, 015407 (2017).

^{[24]}S. Flügge, Practical Quantum Mechanics I (SpringerVerlag, New York, 1974).
Publication Dates

Publication in this collection
04 Nov 2019 
Date of issue
2020
History

Received
22 Aug 2019 
Reviewed
16 Sept 2019 
Accepted
17 Sept 2019