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Journal of Statistical Mechanics Theory and Experiment

Explicit form for the kernel operator matrix elements in eigenfunction basis of harmonic oscillator

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, Evgenii V. Burlakov

Published Feb 1, 2020• 10 citations
Publisher

Abstract

Abstract In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems. Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.

DOI
10.1088/1742-5468/ab6f60

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Abstract In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems. Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.
Abstract The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.
In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form W=Sp⁡[ρW]W=\operatorname{Sp}\left[ ρ\mathcal{W} \right]W=Sp[ρW], where ρρρ is the density matrix, and W\mathcal{W}W is the universal density matrix. The density matrix ρρρ for each quantum system is different, and the universal matrix W\mathcal{W}W is the same for any quantum system. Thus, the matrix W\mathcal{W}W has a fundamental character. In the work, the elements of the universal matrix W\mathcal{W}W were found explicitly and their properties were investigated. The diagonal elements of the matrix W\mathcal{W}W are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix W\mathcal{W}W contain frequency oscillations responsible for dissipations in the quantum systems.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
We propose a phase-space representation concept in terms of the Wigner function for a quantum harmonic oscillator model that exhibits the semiconfinement effect through its mass varying with the position. The new method is used to compute the Wigner distribution function exactly for such a semiconfinement quantum system. This method suppresses the divergence of the integrand in the definition of the quantum distribution function and leads to the computation of its analytical expressions for the stationary states of the semiconfined oscillator model. For this quantum system, both the presence and absence of the applied external homogenous field are studied. Obtained exact expressions of the Wigner distribution function are expressed through the Bessel function of the first kind and Laguerre polynomials. Furthermore, some of the special cases and limits are discussed in detail.
Abstract It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J+, J0, J−} of LASU(2). The Hamiltonian of the system is proportional to J0, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.
Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of qqq and the eigenstates of ppp. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.