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arXiv:1001.4816

The Morse potential and phase-space quantum mechanics

B. Belchev, Mark A. Walton

Published Jan 26, 2010• 5 citations
arXivHTMLPDF

Abstract

We consider the time-independent Wigner functions of phase-space quantum mechanics (a.k.a. deformation quantization) for a Morse potential. First, we find them by solving the $\ast$-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the $\ast$-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols of both diagonal and off-diagonal density operator elements in the energy basis are found this way. The Wigner transforms of the density matrices built from the known wave functions are then shown to confirm the solutions.

DOI
10.48550/arxiv.1001.4816

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We consider the time-independent Wigner functions of phase-space quantum mechanics (a.k.a. deformation quantization) for a Morse potential. First, we find them by solving the ∗\ast∗-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the ∗\ast∗-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols of both diagonal and off-diagonal density operator elements in the energy basis are found this way. The Wigner transforms of the density matrices built from the known wave functions are then shown to confirm the solutions.
We consider the time-independent Wigner functions of phase-space quantum mechanics (also known as deformation quantization) for a Morse potential. First, we find them by solving the *-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the *-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols found this way are of both diagonal and off-diagonal density operator elements in the energy basis. The Wigner transforms of the density matrices built from the known wavefunctions are then shown to confirm the solutions.
Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.
We generalize our modified spectral method for the solution of the coupled real partial differential equations in phase space for the stationary Wigner function of an energy eigenstate. This generalization allows us to apply our algorithm to arbitrary high-order partial derivatives without increasing the numerical costs. This is possible since we can derive a sum factorization formula converting a multiple sum into a simple product. We apply our method to evaluate the Wigner function of the Morse oscillator and an asymmetric double-well potential, and compare our results with the exact solution when it is known.
Abstract When considering quantum systems in phase space, the Wigner function is used as a function of quasi-probability density. Finding the Wigner function is related to the calculation of a Fourier transform of a certain composition of wave functions of the corresponding quantum system. As a rule, knowledge of the Wigner function is not the final goal, and the calculation of mean values of different quantum characteristics of the system is required. An explicit solution of the Schrödinger equation can be obtained only for a narrow class of potentials, therefore, in the majority of cases, numerical methods must be used to find wave functions. As a result, finding the Wigner function is related to the numerical integration of grid wave functions. When considering a one-dimensional system, the calculation of N2 Fourier integrals of the grid wave function is needed. To provide the necessary accuracy for wave functions, corresponding to the higher states of a quantum system, a larger number of grid nodes is needed. The goal of this work was to construct the numerical-analytical method for finding the Wigner function, which allows the number of computational operations to be considerably reduced. The quantum systems with polynomial potentials were considered, for which the Wigner function is represented as a series in some known functions.