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

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 o-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.

Dispersion chain of Vlasov equations

“…On the other hand, expression (6) admits the Vlasov-Moyal approximation [7,8] and may be represented as vμ = +∞ n=0 (−1) n+1 (h/2) 2n m 2n+1 (2n + 1)!…”

The Wigner function negative value domains and energy function poles of the harmonic oscillator

“…To transform expression (A.7) let us calculate the sum 19) can be rewritten in a compact form using the Heaviside function…”

Wigner function of a quantum system with polynomial potential

“…To consider classical dissipative systems instead of the Liouville equation we can use the modified Vlasov equation [16][17][18][19] which in contrast to the Liouville equation has the non-zero right-hand side responsible for dissipative processes.…”

“…Such a representation makes it possible to analyze the anharmonic corrections effect on the Wigner function behavior, to evaluate dissipative effects leading to a variability of the quasi-probability density along phase trajectories. Another advantage of the basis {Ψ k } is the ability to explicitly obtain expressions for the matrix elements w n,k of the kernel operator Ŵ [19]. Knowledge of explicit expressions for elements w n,k (x, p) can significantly reduce the computation time and increase the Wigner function accuracy in various applied studies [6,[20][21][22].…”

“…The matrix eigenvalues E s correspond to the energy spectrum of a quantum system with the potential U N (x). The matrix eigenvectors contain the expansion coefficients c k give a density matrix ρ (s) , using which one can represent the Wigner function in the form W s = Tr ρ (s) W , where the matrix W corresponds to the kernel operator [19].…”