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Journal of Changsha University

Noether Theorem at the Quantal Level

Jiang Jin-Huan

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

Based on the configurationspace generating functional for a system with a finite number of degrees of freedom,the Noether theorem at the quantum level is derived.Appling to motion of the particles in potential,the conserved quantity at the classical level may not hold true at the quantum level.

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The time-dependent integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the nonstationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are also obtained from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.
For time-dependent systems the wavefunction depends explicitly on time and it is not a pure state of the Hamiltonian. We construct operators for which the above wavefunction is a pure state. The method is based on the introduction of conserved quantities QQQ and the pure states are defined by Q^ψ=qψ{\hat Q}ψ=qψQ^​ψ=qψ. The conserved quantities are constructed using parametrised mechanics and the Noether theorem.
In contrast with the case of symmetry groups, no unique definition for the dynamical group generators of a given quantum Hamiltonian has been adopted in the literature. Following some work by Malkin and Man'ko (1965), Lipkin (1969), Dothan (1970) proposed a general definition based on the symmetry group of the corresponding time-dependent quantum mechanical equation of motion for the system. The authors show that this definition arises naturally from the quantum version of time dependent Noether symmetry transformations and derive the dynamical algebras for some simple examples.