Modified WKB Description of Bound States for the Gaussian Confining Potential

The bound states energies and eigenfunctions are obtained within the framework of the variational modified WKB approach for the Gaussian confining potential. The accuracy of the results is shown to be fairly good.


Introduction
The Gaussian confining potential was used originally for the description of elastic scattering of nuclei by nuclei [1]. Later this attractive potential was applied in semiconductor physics in order to simulate the electron behavior in quantum dots [2]. The bound state energies for the Gaussian potential have been calculated in various papers [3][4][5][6][7] in the framework of different approximation techniques. However, the approximate eigenfunctions were studied insufficiently.
In the present paper, we use a new modified WKB method [8,9] to solve the radial Schrödinger equation One of the earliest and simplest methods of obtaining the approximate eigenvalues and eigenfunctions is the WKB method which was described in many books (see, e.g., [10,11]). In the context of WKB approach it is well known [11] that a suitable transformation of the initial equation improves the results of an approximation. We examine the power-law substitutions r = q s , s > 0, ψ(r) = r (s−1)/2s Ψ(q).
The transformed equation is where R. E. Langer [12] has used the particular cases s = 2 and s = ∞ (exponential substitution) when he applied the WKB approximation to the Kepler problem. In [9], the values s = 1 for l = 0 and s = 2 for l ̸ = 0 were chosen in the framework of the new modified WKB method [8] in the case of the power-law potentials. In the present paper, s is a variational parameter. The optimal value of this parameter will be found by means of minimizing the integral discrepancy in accordance with a new improved variational approach verified in [13].

Modified WKB method
The WKB approach deals with the logarithmic derivative Y (q) = d ln Ψ(q)/dq . The WKB series are the asymptotic expansions in powers of Plank's constanth of two particular logarithmic derivatives Y ± (q). The nontrivial properties of the WKB series were investigated in [14,15]. The usual WKB approximations contain a finite number of leading terms Y ± n (q) of the complete expansions. These approximations are not valid at the turning points, where Q(q) = 0.
The analysis of the well-known structure of the leading Y ± n (q) and recursion relations allows us to reconstruct the asymptotic WKB series as the infinite sums of new partial asymptotic series [8]. The complete series are approximated by a finite number of leading partial series. Explicit summations of two leading partial series [8] and their generalizations [9] give the new approximate logarithmic derivative Y app (q) ≡ Y (q; t) = b 1 (q)y 1 (a; t) + b 2 (q)y 2 (a; t) (6) with a mixture parameter t. The notations introduced are a(q) = 1 The functions (9) are expressed in terms of the well-studied Airy functions Ai(a) and Bi(a) [16].
It is not surprising that the asymptotics of our approximation coincide with the WKB asymptotics far away from the turning points. At the same time our approximation reproduces the known [17] satisfactory approximation near the turning points. Now we can construct the approximate radial wave functions for the bound states.
At the origin (r → 0, q → 0) there exist the exact expressions First, we must reproduce the correct limiting behavior. This requirement determines the value of t as where c(l, s) Two real turning points q − and q + (Q(q ± ) = 0) separate three regions.
In the first region, where 0 < q < q − , we select the unique approximate particular logarithmic derivative Y (q; t 0 ). In the second region, where q − < q < q + , we must describe the oscillatory solution of the original Schrödinger equation (2). Therefore in this case we select two approximate particular logarithmic derivatives Y (q; +i) and Y (q; −i). In the third region, where q > q + , we must describe only the decreasing solution of the original Schrödinger equation. Therefore in this case we select the approximate particular logarithmic derivative Y (q; 0) or Y (q; ∞) in accordance with the sign of dQ(q)/dq. Note that in the case where l = 0 and s = 1 we put q − = 0.
The turning points are ordinary nonsingular points in our approach in contrast with the conventional WKB method. Matching particular solutions at the turning points we obtain the continuous approximate radial wave function ψ app (r, s) = ⟨r|ψ app (s)⟩ = N app r (s−1)/2s Ψ app (q), (12) where Ψ app (q) is represented as if 0 < q < q − , if q − < q < q + , if q > q + .

We have the new quantization condition
Note that up to now the value of s is not fixed. We consider s as a variational parameter and ψ app (r, s) as a trial function in the variational approach.

Application to the Gaussian potential
It is convenient to apply our approximation using the dimensionless quantities Then the Schrödinger equation is rendered to bê with the Hamiltonian where the Gaussian potential quickly tends to zero from −A. This potential results in a finite number of energy levels.
In order to choose the optimal value of a variational parameter within the framework of the variational approach it is worthwhile to recast the Schrödinger equation in the form for the normalized eigenfunctions ψ(x) = ⟨x|ψ⟩. In accordance with (21) a discrepancy vector is introduced: for a normalized trial function ψ app (x, s) = ⟨x|ψ app (s)⟩ which is not an exact solution of the Schrödinger equation. Following [13], an integral discrepancy can be determined: (23) This discrepancy characterizes goodness of the approximation and is equal to zero for an exact solution of the Schrödinger equation.
We propose to determine the optimal value s o of a variational parameter from the minimality condition for the integral discrepancy. Note that we look for the absolute minimum of d(s).
In addition to the integral discrepancy we use a relative virial error Following previous papers [3][4][5][6][7], we give the numerical results for the particular value A = 400. In this case the bound states exist for l = 0 − 10. Full number of bound states is 51. We show our results for small l in Table 1 and for large l in Table 2.
In Tables 1,2 we estimate our approximation when the value of a variational parameter is optimal (s = s o ) and hence the integral discrepancy is minimal. We see that s o = 1 for l = 0 and various n as in [9] but the value of s o depends on n if l ̸ = 0. Here we use the notations d = d(s o ), vir = vir(s o ) and e app = e app (s o ). Our approximate energy levels e app are compared with the results e num of the precise numerical calculations [5] for small l. Note that the accuracy of our approximation is fairly good for l = 3 − 7 and values of e app are close to precise values of e num [4,5]. There are only unpublished numerical results of Buck quoted in [3] in the case of large l . One can see that two states l = 8, n = 3 and l = 10, n = 0 were omitted in [3]. We perform a more detail description of these states.

Conclusion
The modified WKB method yields the satisfactory qualitative description of the wave functions in the case of the radial Schrödinger equation. The application to the Gaussian potential demonstrates the sufficiently well quantitative properties of the proposed approach. In addition, the use of new method allows us to predict the existence of bound states which have been missed earlier.