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Q deformations of the Heisenberg equations of motion

Daniel G. Caldi

Published Jan 1, 1991• 3 citations
OpenAlex

Abstract

Two formulations of q-quantum mechanics based on quantum deformations of the Heisenberg equations of motion, are discussed. In one, the commutator is replaced by the “quommutator”: [A,E], = qAB - (l/q)BA. The other involves using the quantum bracket of the time derivative of an operator in the equation of motion. Both have advantages and difficulties, which are discussed, along with several simple examples, the conclusion being that these q-quantum mechanics appear to be sensible.

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Two formulations of q-quantum mechanics based on quantum deformations of the Heisenberg equations of motion, are discussed. In one, the commutator is replaced by the “quommutator”: [A,E], = qAB - (l/q)BA. The other involves using the quantum bracket of the time derivative of an operator in the equation of motion. Both have advantages and difficulties, which are discussed, along with several simple examples, the conclusion being that these q-quantum mechanics appear to be sensible.
The authors consider various formulations of what may be called q-quantum mechanics. One involves replacing the commutator that appears in the Heisenberg equations of motion by the 'quommutator': (A,B) q =qAB-(1/q)BA. They formally integrate the equations of motion, study some simple examples and discuss various apparent difficulties such as lack of conservation of energy, lack of unitarity and the unboundedness of the norm. They also consider another formulation in which the quantum bracket of the time derivative of an operator is used instead of just the time derivative, but the commutator is unchanged, in the Heisenberg equations of motion. This nonlinear treatment while preserving conservation of energy, has its own set of difficulties due to the nonlinearities. They also suggest that the quantum parameter may function as a kind of regulator for the time evolution of the system.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.