Solutions of the equations of motion in classical and quantum theories

Journal Article · Wed Sep 01 00:00:00 EDT 1971 · Annals of Physics

DOI:https://doi.org/10.1016/0003-4916(71)90012-1· OSTI ID:4141098

Bialynicki-Birula, I. ^[1]

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1. Univ. of Pittsburgh, PA (United States)

The purpose of the present paper is to elucidate the relationship between the time dependence of quantum operators in the Heisenberg picture and the time dependence of the corresponding dynamical variables in the underlying classical theory. This problem is studied in the nonrelativistic particle mechanics and in the field theory. Further, it is shown how the operator solutions of the quantum equations of motion are related to the corresponding solutions of the classical equations of motion. An explicit formula is given, which expresses quantum operators in the Heisenberg picture in terms of their classical counterparts. This formula is particularly useful in the study of the classical limit of the quantum theory. The dependence on $$\hbar$$ of the matrix elements of the coordinate operators and the field operators is explicitly given, which enables one to study the quantum corrections to the classical theory in all orders. Coherent states of the quantum system play an essential role in the formalism.

• Research Organization: Univ. of Pittsburgh, PA (United States)
• Sponsoring Organization: US Atomic Energy Commission (AEC)
• NSA Number: NSA-24-017624
• OSTI ID: 4141098
• Alternate ID(s): OSTI ID: 4721983
• Report Number(s): NYO--3829-45
• Journal Information: Annals of Physics, Journal Name: Annals of Physics Journal Issue: 1 Vol. 67; ISSN 0003-4916
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

Atomic & Molecular Physics
CHARGED PARTICLES
DIFFERENTIAL EQUATIONS
FIELD THEORY
HEISENBERG PICTURE
MOTION
PARTICLES
QUANTUM MECHANICS
Quantum Mechanics