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Physica Scripta

Some Properties of Commutators and the Equations of Motion

A. D. Jannussis, A. Streclas, D. Sourlas, Kostas Vlachos

Published Mar 1, 1977• 11 citations
Publisher

Abstract

In this paper we calculate the commutators of operator functions for non-commuting operators on a Heisenberg ring. The commutator of two operators can be expressed with the help of the fundamental operators and their differentials. Using the theorem of the derivative of a function of operators with respect to any parameter, we can find tne equation of motion of a system, in canonical and non-canonical quantum mechanics.

DOI
10.1088/0031-8949/15/3/001

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In this paper we calculate the commutators of operator functions for non-commuting operators on a Heisenberg ring. The commutator of two operators can be expressed with the help of the fundamental operators and their differentials. Using the theorem of the derivative of a function of operators with respect to any parameter, we can find tne equation of motion of a system, in canonical and non-canonical quantum mechanics.
In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum mechanics, first term in the sum is formally equivalent to Poisson bracket in classical mechanics, which is a well-known result. The novelty respect other papers is the type of functions of operators considered: equations mentioned are proved valid when each function is an infinite series of positive and negative powers of these operators, as long as every series converges. Therefore, this formalism can be applied to functions such as the Coulomb potential, where the inverse of radial distance cannot be expressed as a Taylor series.
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 problem of how to obtain a quantum equation of motion with higher derivatives is discussed. First the basic ideas of generalized classical mechanics with higher derivatives are reviewed. The generalized canonical conjugate momenta and a function N, corresponding to the Hamiltonian function, are constructed. Then certain reduced momenta are introduced, which in absence of higher derivatives are identical to the canonical conjugate momenta. The function N is expressed in terms of the reduced momenta. In order to obtain a quantum equation of motion, the reduced momenta in N are replaced by operators fulfilling some given conditions; thus the corresponding operator N̂ can be identified with the Hamiltonian operator. With its help, and introducing a wave function, the quantum equation of motion with higher derivatives is obtained.
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.
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.