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Moyal star product approach to the Bohr-Sommerfeld approximation. (arXiv:math-ph/0409039v1 CROSS LISTED)

Matthew Cargo, Alfonso Gracia-Saz, Robert G. Littlejohn, Matthias Reinsch, Pedro de M. Rios

Published Aug 17, 2020• 0 citations
OpenAlex

Abstract

The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order $\hbar^2$ (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in $\hbar$, starting with $\hbar^0$, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.

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Quantum normal forms, Moyal star product and Bohr–Sommerfeld approximation

Matthew Cargo, Alfonso Gracia-Saz, Robert G. Littlejohn +2 more

A normal form transformation is carried out on one-dimensional quantum Hamiltonians that transforms them into functions of the quantum harmonic oscillator. The method works with the Weyl transform (or 'symbol') of the Hamiltonian. The Moyal star product is used to carry out the normal form transformation at the level of symbols. Diagrammatic techniques are developed for handling the expressions that result from higher order terms in the Moyal series. Once the normal form is achieved, the Bohr–Sommerfeld formula for the eigenvalues, including higher order corrections, follows easily.

arXiv:math-ph/0409039•Journal of Physics A Mathematical and General•Feb 17, 2005•31 citations

The Bohr-Sommerfeld quantization rule and the Weyl correspondence

Petros N. Argyres

With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.

Physics Physique Fizika•Nov 1, 1965•30 citations

Semiclassical Approach to Quantum Oscillator Ground State Energy with Moyal Product Part II

Francesco R. Ruggeri

In (1), it is argued that if the classical energy of an oscillator E = pp/2 + wwxx/2 (m=1) is written as E = (p+iwx)(p-iwx) and this in turn written using the Moyal star product { d/dp left d/dx right - d/dx left d/dp right } in terms of a parameter ( i hbar /2), one obtains: E = pp/2 + wwxx/2 + hbar w/2 i.e. a semiclassical expression for the quantum result with the ground state energy hbar w/2. In Part I, we showed that this result follows from combining a time-independent Schrodinger equation in x-space with one in p (momentum) - space. The two second order equations yield second order derivatives composed of two parts. One part returns the form pp/2m and k/2 xx and the other returns a constant which is the ground state energy if one considers the ground state p and x wavefunctions which are of the form exp(- (pp/2 + wwxx/2) /b) where b is the ground state energy.. The first part cancels with kinetic energy from the p-space equation and .5kxx from the x-space equation. Thus, this combination of two Schrodinger equations really involves three pieces: ((A)) an energy term in p,x, ((B)) a cancellation term of this energy and ((C)) a constant ground state energy term. In (2) it is stated that the Moyal star product ** has the property: Integral dx dp f(p,x) g(p,x) = Integral dx dp f(p,x) ** g(p,x) This is then used to argue that: H(p,x) ** f(p,x) = E f(p,x) ((1)). For the ground state oscillator case, only the first and second order operator terms contribute. The first order operator term is zero, while the second order term behaves in exactly the same manner as the combined x-space and p-space time-independent Schrodinger equations used in Part I. Thus the combined p and x space time-independent Schrodinger equations for the oscillator Hamiltonian pp/2 +wwxx/2 is identical to the Moyal approach.

•May 3, 2023•0 citations

Semiclassical Moyal quantum mechanics for atomic systems

B. R. McQuarrie, T. A. Osborn, George C. Tabisz

The Moyal formalism utilizes the Wigner transform and associated Weyl calculus to define a phase-space representation of quantum mechanics. In this context, the Weyl symbol image of the Heisenberg evolution operator admits a generic semiclassical expansion that is based on classical transport and related $O({\ensuremath{\Elzxh}}^{2})$ quantum corrections. For two atom systems with a mutual pair interaction described by a spherically symmetric potential, the predictive power and convergence properties of this semiclassical expansion are investigated via numerical calculation. The rotational invariance and tensor structure present are used to simplify the semiclassical dynamics to the point where numerical computation in the six-dimensional phase space is feasible. For a variety of initial Gaussian wave functions and a selection of different observables, the $O({\ensuremath{\Elzxh}}^{0})$ and $O({\ensuremath{\Elzxh}}^{2})$ approximations for time dependent expectation values are determined. The interactions used are the Lennard-Jones potentials, which model helium, neon, and argon. The numerical results obtained provide a first demonstration of the practicality and usefulness of Moyal quantum mechanics in the analysis of realistic atomic systems.

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Power-like potentials: From the Bohr—Sommerfeld energies to exact ones

J. C. del Valle, Alexander V. Turbiner

For one-dimensional power-like potentials [Formula: see text], [Formula: see text], the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for [Formula: see text] but below them for [Formula: see text]. The ground state BSE as function of [Formula: see text] are of the same order of magnitude as the exact energies for linear [Formula: see text], quartic [Formula: see text] and sextic [Formula: see text] oscillators but their relative deviation grows with [Formula: see text], reaching the value 4 at [Formula: see text]. For physically important cases [Formula: see text], for the 100th excited state BSE coincide with exact ones in 5–6 figures. It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction [Formula: see text] (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function [Formula: see text] for all [Formula: see text]. It grows slowly with increasing [Formula: see text] for fixed quantum number [Formula: see text], while it decays with quantum number growth at fixed [Formula: see text]. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of [Formula: see text] (and [Formula: see text]).

arXiv:2108.00327•International Journal of Modern Physics A•Oct 20, 2021•9 citations

Moyal quantum mechanics: the semiclassical Heisenberg dynamics

T. A. Osborn, Molzahn, F H

The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.

•Jan 1, 1994•46 citations
The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order ℏ2\hbar^2ℏ2 (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in ℏ\hbarℏ, starting with ℏ0\hbar^0ℏ0, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.
A normal form transformation is carried out on one-dimensional quantum Hamiltonians that transforms them into functions of the quantum harmonic oscillator. The method works with the Weyl transform (or 'symbol') of the Hamiltonian. The Moyal star product is used to carry out the normal form transformation at the level of symbols. Diagrammatic techniques are developed for handling the expressions that result from higher order terms in the Moyal series. Once the normal form is achieved, the Bohr–Sommerfeld formula for the eigenvalues, including higher order corrections, follows easily.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
In (1), it is argued that if the classical energy of an oscillator E = pp/2 + wwxx/2 (m=1) is written as E = (p+iwx)(p-iwx) and this in turn written using the Moyal star product { d/dp left d/dx right - d/dx left d/dp right } in terms of a parameter ( i hbar /2), one obtains: E = pp/2 + wwxx/2 + hbar w/2 i.e. a semiclassical expression for the quantum result with the ground state energy hbar w/2. In Part I, we showed that this result follows from combining a time-independent Schrodinger equation in x-space with one in p (momentum) - space. The two second order equations yield second order derivatives composed of two parts. One part returns the form pp/2m and k/2 xx and the other returns a constant which is the ground state energy if one considers the ground state p and x wavefunctions which are of the form exp(- (pp/2 + wwxx/2) /b) where b is the ground state energy.. The first part cancels with kinetic energy from the p-space equation and .5kxx from the x-space equation. Thus, this combination of two Schrodinger equations really involves three pieces: ((A)) an energy term in p,x, ((B)) a cancellation term of this energy and ((C)) a constant ground state energy term. In (2) it is stated that the Moyal star product ** has the property: Integral dx dp f(p,x) g(p,x) = Integral dx dp f(p,x) ** g(p,x) This is then used to argue that: H(p,x) ** f(p,x) = E f(p,x) ((1)). For the ground state oscillator case, only the first and second order operator terms contribute. The first order operator term is zero, while the second order term behaves in exactly the same manner as the combined x-space and p-space time-independent Schrodinger equations used in Part I. Thus the combined p and x space time-independent Schrodinger equations for the oscillator Hamiltonian pp/2 +wwxx/2 is identical to the Moyal approach.
The Moyal formalism utilizes the Wigner transform and associated Weyl calculus to define a phase-space representation of quantum mechanics. In this context, the Weyl symbol image of the Heisenberg evolution operator admits a generic semiclassical expansion that is based on classical transport and related $O({\ensuremath{\Elzxh}}^{2})$ quantum corrections. For two atom systems with a mutual pair interaction described by a spherically symmetric potential, the predictive power and convergence properties of this semiclassical expansion are investigated via numerical calculation. The rotational invariance and tensor structure present are used to simplify the semiclassical dynamics to the point where numerical computation in the six-dimensional phase space is feasible. For a variety of initial Gaussian wave functions and a selection of different observables, the $O({\ensuremath{\Elzxh}}^{0})$ and $O({\ensuremath{\Elzxh}}^{2})$ approximations for time dependent expectation values are determined. The interactions used are the Lennard-Jones potentials, which model helium, neon, and argon. The numerical results obtained provide a first demonstration of the practicality and usefulness of Moyal quantum mechanics in the analysis of realistic atomic systems.
For one-dimensional power-like potentials [Formula: see text], [Formula: see text], the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for [Formula: see text] but below them for [Formula: see text]. The ground state BSE as function of [Formula: see text] are of the same order of magnitude as the exact energies for linear [Formula: see text], quartic [Formula: see text] and sextic [Formula: see text] oscillators but their relative deviation grows with [Formula: see text], reaching the value 4 at [Formula: see text]. For physically important cases [Formula: see text], for the 100th excited state BSE coincide with exact ones in 5–6 figures. It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction [Formula: see text] (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function [Formula: see text] for all [Formula: see text]. It grows slowly with increasing [Formula: see text] for fixed quantum number [Formula: see text], while it decays with quantum number growth at fixed [Formula: see text]. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of [Formula: see text] (and [Formula: see text]).
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.