Abstract
A general approach to conserved physical quantities originating from dynamical symmetries is presented for quantum-mechanical systems. It is illustrated that a general ansatz for the Hamiltonian leads to a differential equation for the central potential that can be solved analytically. This indicates that additional integrals of motion are closely connected to the functional form of the potential. In nonrelativistic three-dimensional quantum mechanics, we show that, besides the trivial case of a constant potential, the Coulomb and harmonic potentials are the only two examples that give rise to additional integrals of motion known as the Runge-Lenz-Laplace vector and the Demkov-Fradkin second-rank tensor, which is in agreement with earlier results. Tensors of rank higher than 2 are only conserved quantities for a constant potential and basically denote a generalization of momentum conservation for higher ranks. We also consider the relativistic case by studying the Hamiltonian form of the Dirac equation. Here we show that only a constant and the Coulomb potential lead to conserved quantities. In the case of a Coulomb potential, this is the pseudoscalar Johnson-Lippmann-Biedenharn operator, which reduces to a spin projection of the Runge-Lenz-Laplace vector in the nonrelativistic limit.
- Received 30 November 2018
DOI:https://doi.org/10.1103/PhysRevA.99.032116
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