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arXiv:quant-ph/9804076 · Journal of Nonlinear Mathematical Physics

Hamiltonian Formalism in Quantum Mechanics

Boris A. Kupershmidt

Published Jan 1, 1998• 2 citations
arXivHTMLPDF

Abstract

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures.

DOI
10.2991/jnmp.1998.5.2.8

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Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures.
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.
The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form ${p}_{k}{q}_{j}\ensuremath{-}{q}_{j}{p}_{k}=S{\ensuremath{\delta}}_{\mathrm{jk}}$, where SSS is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of $S(=\frac{\ensuremath{\hbar}}{i})$ leads to a theory identical with wave mechanics a part from a constant (unobservable) phase factor on the wave function. The fact that SSS may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.
We describe an ppp-mechanical (see funct-an/9405002 and quant-ph/9610016) brackets which generate quantum (commutator) and classic (Poisson) brackets in corresponding representations of the Heisenberg group. We \emph{do not} use any kind of semiclassic approximation or limiting procedures for ℏ→0\hbar \to 0ℏ→0. Harmonic oscillator considered within the approach. Keywords: Classic and quantum mechanics, Hamilton and Heisenberg equations, Poisson brackets, commutator, Heisenberg group.
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.