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A NEW APPROACH TO THE JWKB TREATMENT.

Nasr Youssef. Saad

Published Jan 1, 1980• 0 citations
OpenAlex

Abstract

The JWKB approximation is widely used in many problems in quantum mechanics for its simplicity and also because a closed-form solution often provides more insight than a numerical one. Unfortunately, like all approximations, the JWKB method has its limitations: Firstly, it may fail near or at a classical turning point, and secondly, it employs solutions for a potential which deviates from the actual potential W(x). In this work, a new method is proposed to avoid the limitations of the JWKB method while keeping its advantages. First, the singularity around the classical turning point is resolved not only for a linear potential as proposed by Langer, but also for any potential with a linear leading term through the turning point, and second, a solution is constructed in closed form which is an exact solution for the actual potential W(x). The new method consists in finding a substitute potential U(x) in such a way that, together with its (DELTA){U,x} it is equal to the actual W(x). The solution is then based on U(x) but actually is the solution for W(x) because W(x) = U(x) + (DELTA){U,x}. The new method is first applied to potentials with only two turning points. The eigenvalue problem is reduced to the solution of a linear equation and the energy eigenvalues are exactly given. Then, potentials with more than two turning points are studied and the eigenvalue problem is again reduced to the solution of a polynomial equation whose order depends upon the number of turning points. Then the method is applied to the hindered or restricted rotation. The energy levels are found to be associated with the N-th roots of 1 in the complex plane, i.e. on the unit circle. The method described here is of rather general nature and can be used to solve a number of problems in quantum mechanics, where a solution for the Schrodinger equation in closed form is needed. The applications provided in this work are only given as examples of the applicability and the relative simplicity of the method.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1980 .S223. Source: Dissertation Abstracts International, Volume: 41-03, Section: B, page: 1002. Thesis (Ph.D.)--University of Windsor (Canada), 1980.

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The JWKB approximation is widely used in many problems in quantum mechanics for its simplicity and also because a closed-form solution often provides more insight than a numerical one. Unfortunately, like all approximations, the JWKB method has its limitations: Firstly, it may fail near or at a classical turning point, and secondly, it employs solutions for a potential which deviates from the actual potential W(x). In this work, a new method is proposed to avoid the limitations of the JWKB method while keeping its advantages. First, the singularity around the classical turning point is resolved not only for a linear potential as proposed by Langer, but also for any potential with a linear leading term through the turning point, and second, a solution is constructed in closed form which is an exact solution for the actual potential W(x). The new method consists in finding a substitute potential U(x) in such a way that, together with its (DELTA){U,x} it is equal to the actual W(x). The solution is then based on U(x) but actually is the solution for W(x) because W(x) = U(x) + (DELTA){U,x}. The new method is first applied to potentials with only two turning points. The eigenvalue problem is reduced to the solution of a linear equation and the energy eigenvalues are exactly given. Then, potentials with more than two turning points are studied and the eigenvalue problem is again reduced to the solution of a polynomial equation whose order depends upon the number of turning points. Then the method is applied to the hindered or restricted rotation. The energy levels are found to be associated with the N-th roots of 1 in the complex plane, i.e. on the unit circle. The method described here is of rather general nature and can be used to solve a number of problems in quantum mechanics, where a solution for the Schrodinger equation in closed form is needed. The applications provided in this work are only given as examples of the applicability and the relative simplicity of the method.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1980 .S223. Source: Dissertation Abstracts International, Volume: 41-03, Section: B, page: 1002. Thesis (Ph.D.)--University of Windsor (Canada), 1980.
Abstract Questions of quantization turn on a proper treatment of normally isolated regions where the JWKB description, expressed by eqn (2.13), breaks down. The most obvious of such special regions are those around the classical turning points, at each of which the analysis in Sections 2.2 and 2.3 yields an associated so-called Maslov (1972) phase correction of π/4; Delos (1986) and Littlejohn and Robbins (1987) discuss this Maslov correction from a more fundamental standpoint and Appendix A and parts of Section 11.2 bear on its choice. Other special regions around potential singularities are most conveniently handled by introducing a Langer (1932) correction term to the physical potential function (see Section 2.3). It is shown in Section 3.1 that systems with no other complications may be quantized by the Bohr-Sommerfeld rule (3.1)∮p(q)dq=(n+δ)h, n=0,1,2,3⋯ where n is the quantum number and b is a Maslov index, which takes the values ½, 0, ½ for vibrational, plane rotational, and orbital motion respectively.
Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB0_00​ method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``insoluble'' potentials are obtained. The eigenfunctions have the form of a standing wave, Ancos⁡(knx+δn)A_n\cos(k_nx+δ_n)An​cos(kn​x+δn​), and are the asymptote of the exact solution.
Abstract. – An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom. In the development of quantum mechanics, the Bohr-Sommerfeld quantization rules of the old quantum theory [1] occupy a position intermediate between classical and quantum mechanics. The WKB approximation [2–4] is a method for the approximate treatment of the Schrödinger wave function with another quantization rule [1]. Both quantization rules are approximate. We are going to derive an exact quantization rule for the one-dimensional Schrödinger equation: d2 ψ(x) = −2M dx2 2 [E − V (x)] ψ(x), (1) ¯h where M is the mass of the particle, and the potential V (x) is a piecewise continuous real function of x satisfying for definiteness V (x)> E, − ∞ < x < xA or xB < x < ∞, V (x) = E, x = xA or x = xB, V (x) < E, xA < x < xB, (2) where xA and xB are two turning points. Between two turning points, the momentum is k(x) = √ 2M [E − V (x)]/¯h.
A method of approximating solutions of the one-dimensional Schr\"odinger equation is presented in this paper. The method closely resembles the usual WKB approximation. Whereas in the ordinary WKB method the exponential function is used as the basis of the approximation, in this paper the solutions of an arbitrary Schr\"odinger equation are used. The general advantage is that by proper choice of the arbitrary equation an improved approximation can be obtained. The method is illustrated by treating the potential well and potential barrier problems when there are two turning points. The approximations to the wave functions are continuous even across the turning points. The barrier transmission problem is treated uniformly for energies above and below the peak of the barrier.
Abstract It is a common practice to approach quantum mechanics through the study of a few, exactly soluble examples using the Schrödinger equation in position-space. The number of potential energy functions for which such closed-form solutions are available is, however, quite small. Luckily, many of them actually correspond reasonably well to actual physical systems; examples include the infinite well as a model of a free particle ina “box “ (Chapters 5 and 7), the harmonic oscillator (Chapter 9), the rigid rotator (Chapter 16), and the Coulomb potential for the hydrogen atom (Chapter
It is well-known that applications of the JWKB method to central-field problems in three dimensions require half-integral quantization of the angular momentum for their success. Thus, the square of the angular momentum must be represented by the term (𝓁 + 1 / 2 ) 2 ℏ 2 rather than 𝓁 (𝓁 + 1)ℏ 2 . This was first shown by Kramers in 1926 and has subsequently been discussed by several authors including, in particular, Langer (1937). While Kramers based his discussion on the ordinary radial variable r, Langer switched to the variable x defined by r = e x . In this new representation of the central-field problem, the expression (𝓁 + 1 / 2 ) 2 ℏ 2 emerges naturally. The ad hoc character of the Langer transformation has, however, often been emphasized. In the present communication, we choose a different entry to the problem. We keep the variable r and focus on physically equivalent forms of the radial Schrödinger equation in this variable. This leads to a smoother emergence of the (𝓁 + 1 / 2 ) 2 ℏ 2 term. Our analysis is carried out for a general dimension D . For a given D, there are D physically equivalent radial equations, corresponding to the subdimensions d = 1, 2, ..., D . We show that it is only the d = 2 equation that can be satisfactorily treated by the JWKB approximation. In the past, the focus was always on the d = 1 equation, and this was the reason behind the problems encountered by Kramers and Langer. As to the d = 2 equation, we finally show that this equation also is the most convenient starting point for determining the exact solutions of a central-field problem for general values of D and angular-momentum quantum number L .