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Physical Review

Function in Quantum Mechanics Which Corresponds to a Given Function in Classical Mechanics

Hamilton Daughaday, B. P. Nigam

Published Sep 6, 1965• 12 citations
Publisher

Abstract

A generalization of the procedures of Weyl and McCoy for making transition from a classical function to a quantum-mechanical operator function corresponding to it has been carried out, in order to include terms involving more than one pair of conjugate variables. An application of the method is made to the case of two spinless interacting charges---the Darwin interaction. This result is verified by solving directly for the quantum-mechanical Hamiltonian operator, starting from equations which were derived from the classical equations by replacing the classical Poisson brackets by commutators and by requiring the equality of the energy and the Hamiltonian operator. An alternative expression for a quantized Hamiltonian, derived in the Appendix, reduces to the form given originally by Born and Jordan for a single set of conjugate variables.

DOI
10.1103/physrev.139.b1436

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A generalization of the procedures of Weyl and McCoy for making transition from a classical function to a quantum-mechanical operator function corresponding to it has been carried out, in order to include terms involving more than one pair of conjugate variables. An application of the method is made to the case of two spinless interacting charges---the Darwin interaction. This result is verified by solving directly for the quantum-mechanical Hamiltonian operator, starting from equations which were derived from the classical equations by replacing the classical Poisson brackets by commutators and by requiring the equality of the energy and the Hamiltonian operator. An alternative expression for a quantized Hamiltonian, derived in the Appendix, reduces to the form given originally by Born and Jordan for a single set of conjugate variables.
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.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
A quantum-mechanical counterpart to the classical mechanical variation of constants method is derived, with initial values of coordinates and momenta as “constants.” Use is made of a formal operator solution for nonautonomous or autonomous systems in classical mechanics, which we published earlier, and of the correspondence between Poisson brackets and commutators. An alternative unified Lie-algebraic derivation is also given. It is shown that the Schrödinger, Heisenberg, and interaction pictures in quantum mechanics do not correspond directly to the method of classical mechanical variation of these “constants.” A fourth picture, termed “mixed interaction,” is introduced and shown to so correspond. It complements the previous three in a symmetrical manner, bearing the same relation to the Heisenberg picture that the Schrödinger picture bears to the interaction one. The group-theoretic relationship to the interaction picture is noted, as is the relation to the usual variation-of-constants method in wave mechanics. For completeness, the classical counterparts of the Heisenberg and interaction pictures are also given. The present results arose from a comparison of quantum and classical treatments of collisions.
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.
We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to ccc-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where ccc-number and qqq-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.