TY - JOUR
T1 - Algebraically Self-Consistent Quasiclassical Approximation on Phase Space
AU - Poirier, Bill
N1 - Funding Information:
I would like to thank Professor John C. Light for his support and guidance, and Profs. John C. Light and Robert. G. Littlejohn for many useful discussions. This work was supported, in part, under NSF Grant CHE-9634440.
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 -