Langer Modification, Quantization condition and Barrier Penetration in Quantum Mechanics

The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. However, this method is valid only in the region where the WKB condition holds, and the corresponding errors are not known. In this paper, we present an analytical approximation method to calculate the wave functions of the Schr\"odinger equation, which is applicable to a much wider range of the problems, and in each case the upper bounds of the errors are given explicitly. By properly choosing the freedom introduced in the method, the errors can be minimized, which significantly improves the accuracy of the calculations. A byproduct of the method is to provide a very clear explanation of the Langer modification encountered in the studies of the hydrogen atom and harmonic oscillator. To further test our method, we calculate (analytically) the wave functions for several exactly solvable potentials of the Schr\"odinger equation, and then obtain the transmission coefficients of particles over potential barriers, as well as the quantization conditions for bound states. We find that such obtained results agree with the exact ones extremely well, and represent significant improvement over the results obtained by the WKB approximation. Applications of our method to other fields are also discussed.


I. INTRODUCTION
A fundamental interest in quantum mechanics (QM) is to derive various physical quantities from the wave function of the Schrödinger equation. Due to the complexity of the equation, often various approximations have to be used. Among them, the WKB approximation has played an essential role in the development of QM and has been widely used in many fields of physics and chemistry [1][2][3][4][5][6]. In general, for a one dimensional Schrödinger equation which describes a particle of mass m moving with total energy E in a potential V (x), the WKB wave function Ψ(x) can be approximately written in the form where p(x) = 2m(E − V ) is the local momentum. The validity of the approximation is restricted to the regions where the WKB condition is fulfilled, In this way, one can treat the reduced Planck constant as a small parameter and extend the leading-order solution (2) to high-orders. However, it is well known that the above condition can be violated or not fulfilled completely in many cases

Potentials
V (x) q(x) Harmonic oscillator 1 2 mω 2 x 2 + 2 l(l+1) [1]. For example, the WKB condition is always violated around turning points, at which both Q and the WKB wave function (2) diverge. In addition, the WKB condition is also violated around singular points (poles) of p 2 (x). For instance, for the radial Schrödinger equation, the effective potential V (r) contains a centrifugal term which has a second-order pole at the origin r = 0. Other typical potentials encountered in QM are presented in Table 1. Note that it is exactly because of this second-order pole that the WKB approximation fails to give correct results for hydrogen atoms and harmonic oscillators [7]. This problem was studied by Langer several decades ago, and shown that it can be cured if one replaces l(l + 1) by (l + 1/2) 2 in V C (r) [8]. This modification now is considered as a standard ingredient of the WKB method in QM [1]. However, a rigorous and logically consistent derivation of this modification is still lacking.
Another situation that could violate the WKB condition is around the extreme point of p 2 (x) if Q arXiv:1902.09675v1 [quant-ph] 26 Feb 2019 |p /2p 3 | ∼ O(1). This can arise in the bound states with a potential well or in particle scattering with a potential barrier. In both cases the results from the WKB approximation becomes invalid if Q is not small enough at the extreme points.
The purpose of this paper is to present an analytical approximation method to calculate accurately the wave function of the Schrödinger equation (1) with singular potentials such as those given in Table 1, a method that has been shown to be powerful and robust for calculating the mode functions and primordial spectra of slow-roll infflationary models [9][10][11][12][13], inflation with nonlinear dispersion relations [14][15][16][17][18][19], and loop quantum cosmology [20][21][22][23], as well as studying the parametric resonance during inflation and reheating [24]. The major advantage of the method is that the errors in each order of approximations can be estimated and the upper bounds of the errors are always known. In particular, for certain models, it was found that the errors are no larger than 0.15% up to the third-order of approximations [18].
In the application of this method to hydrogen atoms and harmonic oscillators, we provide a rigorous derivation of the Langer modification and expressions for the quantization conditions and the barrier transmission cofficients. Applications of our method to some well-known examples are also presented, in order to further test it. More details will be reported elsewhere [25].

II. UNIFORM ASYMPTOTIC APPROXIMATION METHOD
Let us first write the standard form of (1) in the form [26,27], where g(x) + q(x) = −p 2 (x)/ 2 . Note that, for any given p(x), here we introduce two functions g(x) and q(x), and to fix them uniquely, we require that the errors in each order of approximations be minimized. This is one of the major gradients of the method. Then, such defined g(x) in general can have zero points g(x i ) = 0, which are called turning points in the uniform asymptotic approximation. Except such points, g(x) may also have other types of transition points, such as poles and extreme points. According to the theory of the second-order ordinary differential equations [26,27], the wave function Ψ(x) sensitively depends on the number and nature of turning points, poles and extreme points. Analyzing the corresponding error control function around poles and extreme points provides the main guidance on how to determine g(x) and q(x) [26,27]. In addition, around each of the turning points x i , we require |q(x)| |g(x)/(x − x i )|, while away from them we require |q(x)| |g(x)| [26,27]. At the turning points, the WKB condition (3) is violated, and the wave function (2) becomes invalid. As mentioned above, the turning points can have different types. For a single turning point, say, denoted by x 0 , the wave function Ψ(x) can be written as [26] Ψ where Ai(ξ) and Bi(ξ) are the Airy functions, a 0 and b 0 are two integration constants, and ξ(x) is chosen to be a monotonic function of x, defined by |ξ|dξ = |g(x)|dx with ξ(x 0 ) = 0. The errors are controlled by the error control function, For a pair of turning points x 1 and x 2 , they could be: (1) both real and different x 1 = x 2 ; (2) both real but equal x 1 = x 2 ; or (3) complex conjugate x 1 = x * 2 . In each case, between these points, g(y) usually has one extreme. If this extreme is a minimal point of g(y), we can construct the wave function Ψ(x) in terms of the parabolic cylinder functions U (−ζ 2 0 /2, If the extreme is a maximal point, the wave function can be also constructed in terms of parabolic cylinder functions, but now in terms of W (ζ 2 0 /2, √ 2ζ) and In both cases, the variable ζ(x) is a monotonic function of x which defined by |ζ 2 − ζ 2 0 |dζ = |g(x)|dx with ζ(x 1 ) = −|ζ 0 |, ζ(x 2 ) = |ζ 0 |, and ζ 2 |g(x)|dx|, where " + " (" − ") corresponds to the case in which the two turning points x 1 and x 2 are real (complex conjugate). The associated error control function of the above two wave functions is, The wave functions given in (6), (8) and (9) are valid in the neighborhoods of the turning points x i . The extension of them beyond these points crucially depends on the behaviors of the corresponding error control functions in the extended regions. In the following let us consider it for the case with a second-order pole, as shown in (4).

A. Second-order pole and Langer modification
As mentioned previously, for the radial Schrödinger equation, the effective potential V C (r) given by (4) contains a second-order pole at the origin, at which we have (ξ, ζ) → ±∞, and Since near the second-order pole, we have g(x) ∼ a/x 2 , which leads to In order for the wave functions (6), (8) and (9) valid near the origin, we must require the convergence of the error control functions, which leads to which is nothing but exactly the Langer modification. Therefore, the latter is simply the result of the condition that the error control function be finite near the pole.

B. Extreme point and the elimination of the error term
The extreme points of g(x) in general arise from quantum system with a potential well or barrier in the region between two turning points. These extreme points are the same as the bottom of the well or the top of the potential if one chooses q(x) = 0. As mentioned previously, the existence of the extreme points will make the WKB approximation invalid if Q is not small enough at the extreme. Such cases rise when the potential wells or barriers are sharply peaked, for which g(x) has two coalescent turning points x 1 and x 2 . To see this clearly, let us write g(x) in the form where f (x) is a finite and regular function with f (x i ) = 0. Then, we expect the dominant contribution to the integral of (10) arises from the lower limit. Therefore, we can formally expand the error control function I (ζ) about the turning points and find, Note that in deriving the above expression we had ignored the small corrections. We also note that when the turning points x 1 and x 2 are closed to each other, the dominant contributions to the error control function come from the ln |x 2 −x 1 | term, which becomes divergent when x 1 = x 2 . It is somehow surprising that such dominant contributions seemingly had never been noticed before, and later they will play an essential role in determining the extension of the wave functions to the regions near the extreme point.
With the knowledge of the error control function (15), we are now in a position to eliminate the dominant error term in (15) by properly choosing q(x) in the second term of (15). To achieve this, we expand q(x) at one of the turning points, Then the elimination of the first error term in (15) requires This represents one of the important conditions for the choice of the function q(x). Additional requirements include that |q(x)| be negligible in comparing to |g(x)| in the regions that is away from the extreme and turning points. Then, we expect that the right hand side of (17) does not contain q 0 and should be independent of the nature of the two turning points x 1 and x 2 . For this requirement, one can relate q 0 to the derivatives of the function g(x) at the extreme point x m via the relation,

III. IMPROVED QUANTIZATION CONDITIONS AND POTENTIAL BARRIER TRANSMISSION COFFICIENTS
With the above considerations, now we are at the position to generalize the wave functions near poles and extreme points. These wave functions then can be utilized to derive the quantization conditions for bound states or the quantum turning rate of a particle through a potential barrier. For bound states, if g(x) has two turning points x 1 and x 2 , and the wave function (8) leads to the quantization condition, When g(x) has only one turning point x 0 , the wave function (6) leads to the quantization condition, where x b is the boundary of the classically allowed region. For a particle passing through a potential barrier, the wave function (9) yields the transmission coefficients We note that ζ 2 0 is positive when x 1 and x 2 are real and negative when x 1 and x 2 are complex conjugate.

IV. EXAMPLES
For applications of the above quantization condition, we consider several representative exactly solvable systems, including hydrogen atoms, harmonic oscillators in three dimensions, Morse potential, Pöschl-Teller potential, and Eckart potential, as given in Table. I. For hydrogen atoms and harmonic oscillators, both potentials have a second-order pole at the origin and an extreme point in the region x ∈ (0, +∞). The choice of (13) not only eliminates the divergence in the error control function around the origin, but also satisfies the criterion (18) at the extreme point. For the Morse and Pöschl-Teller potentials, they both have an extreme point and the application of the criterion (18) leads to q 0 = 0 and q 0 = α 2 /4, respectively. Therefore, we can take q(x) = 0 for the Morse potential and q(x) = α 2 /(4 cosh 2 (αx)) for the Pöschl-Teller potential. Similar to the hydrogen atom and harmonic oscillator, the Eckart potential has a second-order pole at the origin and an extreme point. By requiring q(x) ∼ −1/(4x 2 ) near the origin and the criterion (18) to be satisfied, we find q(x) = − α 2 4 sinh 2 (αx) . Then, it can be seen that, while the WKB quantization condition fails to predict correct energy eigenvalues E n except for the Morse potential, the quantization condition (19) yields precisely the exact results for all these potentials.
To study the transmission coefficient (21), let us consider the Pöschl-Teller potential barrier with V 0 being positive 1 , for which we obtain while the WKB approximation gives [30] T = e −π( It is worth noting that the choice of q(x) is only possible when 8mv 0 / 2 > α 2 for positive v 0 . When 8mv 0 / 2 < α 2 , as mentioned previously, the choice of q(x) changes the properties of turning points significantly, which makes the approximation not applicable. In Fig. 1, we present the transmission coefficient (22), the WKB transmission coefficient (23) and the exact result, from which one can see that the transmission coefficient (22) fits the exact result extremely well.

V. SUMMARY AND OUTLOOK
In summary, we have presented a new analytical approximation method for the the Schrödinger equation, which overcomes various difficulties associated with the conventional WKB approximations. In particular, from such obtained wave functions we have derived the quantization conditions for bound states and the transmission coefficients for particle scattering over a potential barrier, which significantly improve the accuracy of the results obtained by the WKB approximations. Future investigations including the high-order approximations, extension to the nonlinear Schrödinger equation, and applications to various quantum systems with general potentials can be carried out based on the method presented in this paper. Our formulas are general and simple to use, and can be also applied to other physically interesting cases, for example, Hawking radiation and quasi-normal modes of black holes [31,32], primordial perturbations during inflation and reheating [33], and Schwinger vacuum pair productions due to laser pulses [34].   (23), and our "improved" transmission coefficient (22).