TY - JOUR

T1 - Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

AU - Poirier, Bill

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2000/8

Y1 - 2000/8

N2 - The Wigner-Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary "*" operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas-Fermi theory, and a new quantization rule which - unlike semiclassical quantization - is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.

AB - The Wigner-Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary "*" operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas-Fermi theory, and a new quantization rule which - unlike semiclassical quantization - is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.

UR - http://www.scopus.com/inward/record.url?scp=0034359313&partnerID=8YFLogxK

U2 - 10.1023/A:1003632404712

DO - 10.1023/A:1003632404712

M3 - Article

AN - SCOPUS:0034359313

VL - 30

SP - 1191

EP - 1226

JO - Foundations of Physics

JF - Foundations of Physics

SN - 0015-9018

IS - 8

ER -