Garrett approximation revisited

TL;DR

This paper revisits three variants of the Garrett approximation for quantum wells, analyzing their accuracy for symmetric and asymmetric cases, and finds that simpler variants are surprisingly more precise, with applications discussed.

Contribution

The study provides a comparative analysis of Garrett approximation variants, highlighting the accuracy of simpler forms for quantum well problems.

Findings

Simplest variants of Garrett approximation are most accurate.

Accuracy varies between symmetric and asymmetric wells.

Applications to quantum dots and neutron guides are discussed.

Abstract

Three variants of the Garrett approximation are studied and their accuracy is analyzed, for symmetric and asymmetric square wells. Quite surprisingly, the simplest variants are also the most accurate. The applications to quantum wells, quantum dots and capillary neutron guides are briefly discussed.

Equations

$$E_n^{\left(0\right)}=n^2\frac{{\pi }^2{\hslash }^2}{2m{L}^2}$$

$$\delta =\frac{\hslash }{\sqrt{2m\left(V-E\right)}}$$

$${\delta }^{\left(1\right)}=\frac{\hslash }{{\left[2m\left(V-E_n^{\left(0\right)}\right)\right]}^{1/2}}$$

$$E_n^{\left(1\right)}=n^2\frac{{\pi }^2{\hslash }^2}{2m{\left(L+2{\delta }^{\left(1\right)}\right)}^2}$$

$${\delta }^{\left(2\right)}=\frac{\hslash }{{\left[2m\left(V-E_n^{\left(1\right)}\right)\right]}^{1/2}}=\frac{\hslash \left(L+2{\delta }^{\left(1\right)}\right)}{\left[2mV{\left(L+2{\delta }^{\left(1\right)}\right)}^2-{\pi }^2{\hslash }^2{n}^2\right]}$$

$$E_n^{\left(2\right)}=n^2\frac{{\pi }^2{\hslash }^2}{2m{\left(L+2{\delta }^{\left(2\right)}\right)}^2}$$

$$P=\sqrt{2mV}\frac{L}{2\hslash }=\frac{1}{p}$$

$$\frac{E_n^{\left(0\right)}}{V}=\frac{{\pi }^2{n}^2}{4}{p}^2$$

$$\frac{2{\delta }^{\left(1\right)}}{L}=\frac{p}{\sqrt{1-\frac{{\pi }^2{n}^2}{4}{p}^2}}$$

$$\frac{2{\delta }_n^{\left(2\right)}}{L}=\frac{p}{{\left(1-\frac{{\pi }^2{n}^2}{4}\frac{p^2}{{\left(1+\frac{p}{{\left(1-\frac{{\pi }^2{n}^2}{4}{p}^2\right)}^{1/2}}\right)}^2}\right)}^{1/2}}$$

$${\delta }^{\left(l+1\right)}=\frac{\hslash \left(L+2{\delta }^{\left(l\right)}\right)}{\left[2mV{\left(L+2{\delta }^{\left(l\right)}\right)}^2-{\pi }^2{\hslash }^2{n}^2\right]}$$

$$\underset{q\to \infty }{\lim }{\delta }^{\left(q\right)}=\Delta ,y=\frac{2\Delta }{L}$$

$$4P^2{y}^4+8P^2{y}^3+\left(4P^2-{\pi }^2{n}^2-4\right)y^2-8y-4=0$$

$$y^4+2y^3+\left(1-\left(\frac{{\pi }^2{n}^2}{4}+1\right)p^2\right)y^2-2p^{2}y-p^2=0$$

$$y^2-2p^{2}y-p^2=0$$

$$y=p\sqrt{p^2+1}+p^2\simeq p+p^2,p\ll 1,n\sim 1$$

$$E_n=\frac{{\hslash }^2{k}_n^2}{2m}=n^2\frac{{\pi }^2{\hslash }^2}{2m{\left(L+2{\delta }_n\right)}^2}=\frac{{\pi }^2{\hslash }^2{n}^2}{2m{L}^2}\frac{1}{{\left(1+y_n\right)}^2},y_n=\frac{2{\delta }_n}{L}$$

$$K_n=L{k}_n=\frac{\pi n}{1+y_n}$$

$$K^{\left(2\right)}\left(P,n\right)=\frac{\pi n}{1+y^{\left(2\right)}\left(P,n\right)},y^{\left(2\right)}\left(P,n\right)=\frac{2{\delta }_n^{\left(2\right)}}{L}$$

$$K_4\left(P,n\right)=\frac{\pi n}{1+y_4\left(P,n\right)}$$

$$K_0\left(P,n\right)=\frac{\pi n}{1+y_0\left(P\right)}$$

$$K_B\left(P,n\right)=2{\alpha }_n=\frac{2P}{1+P}\left(\frac{n\pi }{2}-\frac{1}{6{\left(1+P\right)}^3}{\left(\frac{n\pi }{2}\right)}^3\right)$$

$$\frac{K}{2}=\frac{n\pi}{2}-\arcsin\frac{K}{2P}\ \$$

$${\epsilon }^{\left(2\right)}\left(P,n\right)=\frac{K_{ex}\left(P,n\right)-K^{\left(2\right)}\left(P,n\right)}{K_{ex}\left(P,n\right)}$$

$${\epsilon }_a\left(P,n\right)=\frac{K_{ex}\left(P,n\right)-K_a\left(P,n\right)}{K_{ex}\left(P,n\right)},witha=4,0,B$$

$$\left(-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V\left(x\right)\right)\psi =E\psi$$

$${\psi }^{\prime \prime }+\left[k^2-U\left(x\right)\right]\psi =0$$

$$V\left(x\right)=\frac{{\hslash }^2}{2m}U\left(x\right).$$

$$U\left(x\right)=U_3\theta \left(b-x\right)+U_2\theta \left(x-b\right)\theta \left(a-x\right)+U_1\theta \left(x-2\right)$$

$$\psi\left(x\right)=\left\{\begin{array}[]{c}A_{1}e^{-K_{1}x},\ x>a\\ A_{2}\sin\left(kx+\varphi\right),\ b<a<a\\ A_{3}e^{K_{3}x},\ x<b\end{array}\right.$$

Full text

1 Introduction

Surprisingly or not, one of the most elementary problems of quantum mechanics - a particle in a symmetric square well - is still under debate: if its wave function can be easily expressed in terms of elementary functions, the bound states energy eigenvalues are given by transcendental equations, which defy exact solutions. A large number of approximations was proposed - based on graphical constructions [1], [2], [3], on mathematical tricks [4] [5], [6], [7] or physical ideas [8]. In this paper, we shall pay attention to an approach based on a physical idea, due to Garrett [8]: as the main difference between the infinite and finite square well is the fact that, in the first case, the wall is impenetrable, but in the second one, the wave function penetrates the wall on a certain distance $\delta$, the energy of a bound state $E_{n}$ in a finite well of length $L$ should be satisfactorily approximated by the energy of the corresponding bound state, $E_{n}^{\left(0\right)}$, in an infinite well of length $L+2\delta$.

Garrett’s idea is interesting from educational point of view, as it provides a way of understanding quantum phenomena without solving Schrödinger equation [9]; it was recently discussed in textbooks [10]. More than this, it has several applications in the theoretical description of quantum wells [11], quantum dots [12], capillary neutron guides [13] and infrared photodetectors [14]. A more attentive analysis pointed out that, besides the original approach proposed by Garrett, there are two more variants of this approximation [6], [9] which, in some cases, may provide more accurate results. The goal of this paper is to make a detailed (mainly numerical) investigation of the accuracy of each of these three variants, in the calculation of the energy levels of several symmetric and asymmetric square wells. The application of Garrett approach to asymmetric wells is appealing, as it could be used for a simple and quite precise evaluation of energy levels in stepped wells, so important for semiconductor heterostructures (see for instance [15]).

The paper has the following structure. The second section is merely a reformulation of previous results [8], [6], [9], i.e. we introduce the three variants of Garrett approximation and obtain convenient formulas for the calculation of dimensionless wave vectors of the bound states in the square well. It is the starting point for the evaluation of errors of each variant, implicitly of determining its adequacy for a certain bound state. The same scheme is applied to the Barker approximation. The third section is a comparative analysis of Garrett and Barker approximations for finite square wells: we find out which variant is the most appropriate (i. e. the most precise) for a specific case; the most relevant results are conveniently presented as plots - and tables, included as auxiliary material. In the fourth section, the same treatment is applied to the simple asymmetric well. In the fifth one, we discuss the applications of Garrett approach to quantum dots and capillary neutron guides. The last one is devoted to conclusions.

2 Garrett’s approximation for the bound states energy of finite

square wells

In order to expose Garrett’s approach, let us mention that the energy levels of a particle of mass $m$ in an infinite rectangular well of length $L$ is given by the well-known formula:

[TABLE]

The same particle, moving in a finite square well of depth $V$ and same length, can propagate in the classically forbidden region, where its wave function decays exponentially with a characteristic length $\delta:$

[TABLE]

whith $E$ - the energy of its bound state. Garrett notices that ”the use of this length to modify the effective width of the infinite well will lead to a simple iterative approximation for the energy states of the finite well”.

In the first iteration, the energy $E_{n}^{\left(0\right)}$ of the $n-$th level of the infinite well (1) can be introduced into (2), to provide the first approximation for the penetration of the $n-$th wave function of the finite well into the classically forbidden region:

[TABLE]

So, making in (1) the substitution $L\rightarrow L+2\delta^{\left(1\right)},$ we shall obtain a first approximation of the $n-$th state of the finite well:

[TABLE]

In the second iteration, we can substitute $E_{n}^{\left(1\right)}$ into (3)

[TABLE]

and get a second order correction of the penetration length, to be used to the substitution $L\rightarrow L+2\delta^{\left(2\right)}$ in (1), providing a second order approximation of the $n-$th state of the finite well:

[TABLE]

Defining $P,$ a dimensionless quantity, characterizing both the potential $\left(L,\ V\right)$ and the particle $\left(m\right),$ and its inverse $1/p$

[TABLE]

and noticing that

[TABLE]

we can write the penetration lengths in a dimensionless form:

[TABLE]

[TABLE]

Clearly, higher order iterations will produce too cumbersome expressions, instead of (9) and (10), but, by the substitutions $\delta^{\left(1\right)}\rightarrow\delta^{\left(l\right)}$ and $\delta^{\left(2\right)}\rightarrow\delta^{\left(l+1\right)}$, the relation (5) will be replaced by an equally simple one:

[TABLE]

where the index $n$ of $\delta_{n}^{\left(l\right)}$ was dropped, in order to avoid too complicated notations. Taking the limit $l\rightarrow\infty$ in (11) and defining:

[TABLE]

we get for the dimensionless penetration depth $y$:

[TABLE]

or, equivalently, with $P\rightarrow\frac{1}{p}$:

[TABLE]

It is easy to check that, for deep wells, i.e. for small values of $p,$ in the first order approximation, $\ y\left(p\right)\simeq p;$ so, the quartic and the cubic terms in (14) can be neglected. Also, for deep levels, $\left(\frac{\pi^{2}n^{2}}{4}+1\right)p^{2}\ll 1,$ and (14) becomes:

[TABLE]

with the positive root:

[TABLE]

Let us comment now on the Garrett’s iterative approximation. In his original paper, he uses only two iterations. The result obtained in this way, Eq. (10) (which was not explicitly written by Garrett) discourages the attempt of going to higher orders. However, it is easy to apply consistently Garrett’s idea, i.e. to consider an infinite number of iterations, according to Eqs. (11-14). In this situation, it would be interesting to investigate the following aspects:

(*1) The consistent application of Garrett’s idea (considering an infinite number of iterations, which generates a quartic equation, (14)) gives better results than Garrett’s original two-iteration approach, Eq. (10) ?

(*2) The simple approximation of the roots of the quartic equation, so restrictive, independent of the index of energy level $n$, obtained for large wells and deep levels, (15), (16), can provide useful results?

(*3) For practical applications, which one is more convenient: the ”consistent” approximation (14), the two-iteration approximation (10) or the $n-$independent approximation (16)?

Also, it is interesting to compare these variants of Garrett’s approximation with another simple result for the energy of the bound state in a finite rectangular well - Barker’s formula. Let us remind that these two approximations are obtained from two different perspectives: Garrett proposes a physical idea (the existence of a penetration depth); Barker et al. use a mathematical approximation (transforming the transcendental eigenvalue equations into easily solvable, low order algebraic equations).

In order to analyze these issues, we shall calculate the errors generated by each variant. Let us firstly introduce convenient notations. The energy of a bound state, in any variant of Garrett’s approximation is, according to (4) or (6):

[TABLE]

and it can be expressed in terms of the dimensionless wave vector

[TABLE]

We have to distinguish among three different formulas for $K$, corresponding to each of the three variants of Garrett’s approximation:

(1) the two iterations Garrett approximation, used in his original paper,

[TABLE]

where $\delta_{n}^{\left(2\right)}$ is defined in (10);

(2) the ”consistent” Garrett approximation, obtained after infinitely many iterations:

[TABLE]

where $y_{4}\left(P,n\right)$ is the root of the quartic equation (14), and

(3) the lowest order Garrett approximation, given by the root (16) of eq. (15):

[TABLE]

In order to compare the various Garrett approximations with Barker formula, we shall also define:

[TABLE]

where $\alpha_{n}$ refers to Barker’s notation, eq. (16) of [4].

The exact value of the dimensionless wave vector, i.e. the solution of the equation

[TABLE]

will be denoted $K_{ex}\left(P,n\right).$ The errors of the aforementioned approximations are defined as:

[TABLE]

[TABLE]

[TABLE]

3 Comparative analysis of Garrett and Barker approximations for

finite square wells

The numerical values of the errors of the three variants of Garrett approximation and of the dimensionless characteristic (penetration) lengths $y^{\left(2\right)}\left(P,n\right),\ \ y_{4}\left(P,n\right)$ for $P=1,...,10$ and for any $n$ characterizing each bound state, are given as auxiliary material. For a well with $P=10$, the plots of the absolute values of errors $\varepsilon_{4},\varepsilon_{0}$ for $n=1,2,...7$ and of $\varepsilon^{\left(2\right)}$ for $n=1,2,...6$ (this approximation is unphysical for $n=7$) are given in Fig. 1. Any other similar plot can be easily done, using the auxiliary material or the formulas (19-21).

The conclusions of this analysis are quite surprising. The consistent Garrett approximation is really useful only for shallow wells $\left(P=1\right),$ where it is much better even than Barker’s one, and the two iteration approximation is unphysical (complex). Otherwise, it is less precise then (or comparable to) the two-iteration approximation; actually, the main inconvenient of the two-iteration approach is that it is unphysical (complex) for the highest level of any of the wells examined here. Even more surprising is that the lowest order approximation is the most precise one (among the Garrett approximations), for highest levels; let us remind that it was obtained using approximations valid for deep wells (large $P$) and deep levels (small $n$). Typically, the lowest levels are better described by the two iteration approximation; the few exceptions, when the ”consistent” approximation is more precise, are numerically irrelevant, for instance: $\varepsilon_{4}\left(P=7,n=1\right)=1.6488\times 10^{-4}<\varepsilon^{\left(2\right)}\left(P=7,n=1\right)=1.6747\times 10^{-4}.$ Actually, excepting the case of shallow wells $\left(P=1\right),$ the only benefit of the consistent Garrett approximation is that it generates the lowest order approximation, which is surprisingly accurate!

To conclude, the responses to the questions put in the previous section are the following:

(*1) the consistent approximation is the only one to give good results for shallow wells $\left(P\sim 1\right),$ and Garrett’s original two-iteration approach is the most accurate for relatively low levels $\left(n\lesssim\frac{n_{\max}}{2}\right)$; actually, it is unphysical for the highest level $\left(n\sim n_{\max}\right)$

(*2) the $n-$independent approximation is the most accurate one for relatively high levels $\left(\frac{n_{\max}}{2}\lesssim n\lesssim n_{\max}\right)$

(*3) if we are interested in accuracy, we should made a case by case analysis, eventually guided by the additional materials, as there is no general rule; if we are interested in the simplest analytical approximation, we should choose the $n-$independent approximation.

One more remark: taking into account the validity of mathematical approximations done in order to obtain Barker approximation, it is supposed to work well for large $P$ and relatively small $n$; actually, as we can see from Tables 1-4, it gives excellent results for $P=2$ and for any larger $P$, if $n$ is relatively high.

4 The simple asymmetric well

Let as consider the simplest generalization of the symmetric rectangular well, called sometimes simple asymmetric square well. Its corresponding Schroedinger equation:

[TABLE]

can be written simpler, as

[TABLE]

if we introduce $U\left(x\right)$ instead of $V\left(x\right)$ by:

[TABLE]

We shall define, following Messiah [16], Ch. III, §6 (see also [17], §22, problem 2)

[TABLE]

where $\theta$ is the Heaveside function.

The bound state wave function has the form:

[TABLE]

We shall put:

[TABLE]

Without restricting the generality, we can choose $U_{2}=0$ and define:

[TABLE]

The eigenvalue equation associated to the solution (29) has the form:

[TABLE]

For a symmetric well, $\ P_{1}=P_{3}$ and (32) becomes:

[TABLE]

identical with (23).

As Garrett noticed, the approach used for the symmetric wells can be also applied here, for the $n-$th bound state, with $L$ replaced by $L+\delta_{l}+\delta_{r}=L\left(1+y_{l}+y_{r}\right)$ (where the indices $l$ and $r$ refer to the left, respectively right wall). The dimensionless penetration depths $y_{l},\ y_{r}$ are evaluated choosing that variant of Garrett approximation with smallest error, for a given pair $\left(P,n\right),$ characterizing the $n-$th bound state of a square well of strength $P.$

To see how the method works, let us consider the case $P_{3}=10,\ P_{1}=8,$ with $n_{\max}=6$ bound states. For $\left(P_{3}=10,\ n=1\right),$ the most accurate variant of Garrett approximation for a rectangular well gives $y_{4}\left(P_{3}=10,n=1\right)=0.10103,$ and for $\left(P_{1}=8,\ n=1\right)$, the best one is $y_{4}\left(P_{1}=8,n=1\right)=0.126942,$ so the Garrett approximation for the asymmetric well gives the dimensionless wave vector:

[TABLE]

[TABLE]

[TABLE]

to be compared to the ”exact” value of (32) with the same parameters,

$K_{ex}\left(P_{3}=10,P_{1}=8;\ n=1\right)=2.82264.$ For $P_{3}=10,P_{1}=8;\ n=2,$ the most accurate variant is again, for both walls, $y_{4},$ but for $P_{3}=10,P_{1}=8;\ n=3,$ for the both walls, the most accurate variant is the $n-$independent one, so $y_{0}(P_{3}=10,n=3)=y_{0}(P_{3}=10)$ and $y_{0}(P_{1}=8,n=3)=y_{0}(P_{1}=8).$ The approximate value is:

[TABLE]

[TABLE]

[TABLE]

to be compared to the ”exact” one, $K_{ex}\left(P_{3}=10,P_{1}=8;\ n=3\right)=8.4342.$ Actually, the errors are, in these two cases:

[TABLE]

(let us mention that \varepsilon_{0}\left(P_{3}=10,\ n=3\right)=6.02\times 10^{-3}\and $\varepsilon_{0}\left(P_{3}=8,\ n=3\right)=8.02\times 10^{-4}$) and

[TABLE]

(let us mention that \varepsilon_{4}\left(P_{3}=10,\ n=3\right)=6.27\times 10^{-4}\and $\varepsilon_{4}\left(P_{3}=8,\ n=3\right)=1.15\times 10^{-4}$). So, the error of the $n-$th bound state energy in the asymmetric well with strengths $\left(P_{3},P_{1}\right)$ is comparable to the error of the most precise variant of the Garrett approximation of the $n-$th bound state of the symmetric wells with strength $P_{3},\ P_{1}.$ The correctness of this empirical remark was verified in all cases we worked out (see the auxiliary material).

The errors for other values of $n$ are plotted in Fig.1, and can be easily obtained, for any pair $\left(P_{3},P_{1}\right)$, using the auxiliary material.

5 Applications

We shall shortly describe here some of the applications mentioned in the Introduction.

Garrett’s idea was used in replacing ”a stepped spherical potential” with an effective, impenetrable one, in order to calculate the thermodynamic properties of a system of non-interacting bosons confined in a quantum dot. The same approach was discussed in the context of semiconductor quantum dots [12].

In a study of interference effects in capillary neutron guides [13], Rohwedder examines both circular and rectangular cases. In the circular (cylindrical) case, the neutrons effectively ”see” a reflecting wall not at the radius $R,$ but a slightly larger ”effective” radius $R\,_{eff}\simeq R+d$. This can be interpreted as a waveguide-confined manifestation of the Goos-Hänchen effect [18]. For rectangular guides, with section $\left(a_{x},a_{y}\right),$ the variables can be easily separated, and the energy eigenvalues are approximately given by the corresponding spectrum of an infinite square well; its ”effective” width $a_{x,eff}\simeq a_{x}+d,\ a_{y,eff}\simeq a_{y}+d$ turns out to be slightly larger than the ”bare” width. The amount $d=\hbar/\sqrt{2MV}$ can once more be identified with the evanescent penetration depth of the lowest-lying eigenmodes, and - again - is an expression of the (waveguide-confined) Goos-Hänchen effect.

Finally, we can expect that, applying to a stepped rectangular well, an approach similar to that used in the previous section for asymmetric wells (i.e. associating to each wall a strength $P$ and a penetration depth $\delta$), we shall obtain similar accuracy in the evaluation of bound states energy.

6 Conclusions

Essentially, Garrett approximations consists in the following steps: (1) for the $n-$th bound state of a particle in a rectangular well, and for each wall of the well, we associate a penetration depth; (2) in this way, we define a larger, ”effective” well, with impenetrable walls; (3) the $n-$th level of this (infinite) well is a good approximation for the $n-$th level of the finite well.

We discussed in detail the three variants of this approximation and calculated its errors in a large number of cases; the error of Barker approximation, one of the most precise alternative approximations, is also obtained - it is typically smaller than Garrett’s. Quite surprisingly, the simplest variants give the most accurate results. The method works almost equally well for symmetric and asymmetric wells (slightly better, in the symmetric case). The applications for quantum wells, quantum dots and capillary neutron wave guides are shortly discussed.

Garrett approximation is an analytical one, based on a simple physical idea, and this is why it can be extended to more complicated rectangular potentials. One could object that it is unnecessary to use such an approximation, when a very precise result can be easily obtained numerically, but an analytic formula remains attractive, especially in this case, when its form - based on a result obtained for infinite wells - is so simple.

7 Auxiliary material

Table 1. $\left(P=1-5\right)$

[TABLE]

Table 2. $\left(P=6,7\right)$

[TABLE]

Table 3. $\left(P=8,9\right)$

[TABLE]

Table 4. $\left(P=10\right)$

[TABLE]

Acknowledgement 1

The author acknowledges the financial support of the ANCSI - IFIN-HH project PN 18 09 01 01/2018.