The Wentzel–Kramers–Brillouin approximation method applied to the Wigner function

Abstract

An adaptation of the WKB method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in

Application of WKB Method in Approximating Wave Functions in Square Potential Wells

“…The WKB Method is an approximation of the solutions for linear differential equations, including the Schrodinger Equation. Its amplitude is taken to vary slowly in respect to the de Broglie wavelength, thus giving the approximation a semiclassical nature [5].…”

The Phase Space Model of Nonrelativistic Quantum Mechanics

“…This analogy is, however, misleading, because the function , although real, usually take both positive and negative values. Even at a spatial point from the interval , at which the density of the probability of detection of the system is 0, the Wigner function for may be different from 0 [ 68 ].…”

“…As we remember from Section 4, a star product on an arbitrary symplectic manifold can be calculated in frames of the formal series calculus. Thus, the expressions ( 67), (68), and (69) need to be adapted to that calculus (compare [54]).…”

Formal series of generalized functions and their application to deformation quantization

“…For the phase space (R 2 , dq ∧ dp) equipped with the Moyal product (4.1) a relationship between a wave function and its respective state called a Wigner function, is known [16]. And as it was proved in [17], if the wave function is of a compact support then its Wigner function is not. One can extend this observation on mixed states.…”