Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Physical Chemistry A

The JWKB Method in Central-Field Problems. Planar Radial Wave Equation and Resolution of Kramers' Dilemma

Jens Peder Dahl, Wolfgang P. Schleich

Published Apr 21, 2004• 8 citations
Publisher

Abstract

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 .

DOI
10.1021/jp049616r

Related papers

WKB expansion for the angular momentum and the Kepler problem: from the torus quantization to the exact one

Marko Robnik, Luca Salasnich

We calculate the WKB series for the angular momentum and the non-relativistic three-dimesional Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. In this way we derive Langer's rule, calculate the first correction to the leading Langer's term and conjecture the form of all higher terms.

arXiv:chao-dyn/9603014•Journal of Physics A Mathematical and General•Mar 7, 1997•25 citations

A NEW APPROACH TO THE JWKB TREATMENT.

Nasr Youssef. Saad

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.

•Jan 1, 1980•0 citations

Wave equations in momentum space

Maurice M. Lévy

Abstract The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Dec 7, 1950•56 citations

The “Radial Kinetic Energy” Term in the Schrödinger Equation for a Central Force

Frank S. Crawford

A straightforward derivation is given for the Hermitian operator pr ≡ 1r ħi δδrr corresponding to the radial component of linear momentum in the central force problem. The purpose is purely pedagogical; we believe that the radial term of the Schrödinger equation in spherical coordinates is more comprehensible when it appears in the form (pr2/2m)ψ = [(1r ħi δδrr)2/2m]ψ than when it appears, as it does in most textbooks, in the form −ħ22mr2 δδr[r2δδr]ψ. The latter expression is usually arrived at by a tedious transformation of variable done “in the appendix”; instead one may derive pr with a minimum of algebra and with the focus where it belongs, on the concepts of quantum mechanics.

American Journal of Physics•Aug 1, 1964•2 citations

On the Wentzel-Brillouin-Kramers Approximate Solution of the Wave Equation

Lloyd A. Young, G. E. Uhlenbeck

A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values $\stackrel{-}{{r}^{k}}$ (see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Land\'e formula for the doublet separations is derived. The value of ${\ensuremath{\psi}}^{2}(0)$ (0), which occurs in the calculation of the hyperfine separations in $^{2}S$ states, is also given.

Physical Review•Oct 1, 1930•42 citations

Wigner–Weyl correspondence and semiclassical quantization in spherical coordinates

Bill Poirier

The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time.

Journal of Mathematical Physics•Dec 1, 1999•6 citations
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 .
We calculate the WKB series for the angular momentum and the non-relativistic three-dimesional Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. In this way we derive Langer's rule, calculate the first correction to the leading Langer's term and conjecture the form of all higher terms.
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 The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.
A straightforward derivation is given for the Hermitian operator pr ≡ 1r ħi δδrr corresponding to the radial component of linear momentum in the central force problem. The purpose is purely pedagogical; we believe that the radial term of the Schrödinger equation in spherical coordinates is more comprehensible when it appears in the form (pr2/2m)ψ = [(1r ħi δδrr)2/2m]ψ than when it appears, as it does in most textbooks, in the form −ħ22mr2 δδr[r2δδr]ψ. The latter expression is usually arrived at by a tedious transformation of variable done “in the appendix”; instead one may derive pr with a minimum of algebra and with the focus where it belongs, on the concepts of quantum mechanics.
A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values rk−\stackrel{-}{{r}^{k}}rk− (see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Land\'e formula for the doublet separations is derived. The value of ${\ensuremath{\psi}}^{2}(0)$ (0), which occurs in the calculation of the hyperfine separations in 2S^{2}S2S states, is also given.
The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time.