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Physical Review

A Note on Local Momentum in Wave Mechanics

Lloyd A. Young

Published Nov 1, 1931• 46 citations
Publisher

Abstract

The present paper shows the exact relations between dynamical quantities of classical theory and their wave mechanical analogues. A new quantity local momentum is defined and discussed and the Bohr quantum condition is revived in a form compatible with modern theory.
DOI
10.1103/physrev.38.1612

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Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger equation. The following are proved: The “quantum-mechanical momentum” which is nearly equal to the corresponding classical momentum in “classical region” can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrödinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to “quantum-mechanical quantization condition”, equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the “classical region” of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrödinger equation always exists the corresponding classical system.
Abstract The present work is a discussion of a number of simple problems of the free motion of electrons and atoms from the point of view of the wave mechanics. The author has had the advantage of many conversations with Prof. N. Bohr on the subject, and it was composed under this inspiration. Bohr has recently published his views on the foundations of the theory, and it would be out of place to enter here deeply into the matter, but some review of principles is unavoidable if the technical processes are to be understood. Perhaps the chief point of the work is to show how a simple, even old-fashioned, technique is entirely adequate to deal with these very new problems. 1. General Principles. The matrix and the wave mechanics have both been already developed to great lengths as a calculus of stationary states, but they have not yet got so far in what we may call dynamics, a description of the progress of events. More and more complicated phenomena have been fitted into the same scheme, but not much has been done in making this scheme intuitively understandable. One of the most important contributions is a recent work by Heisenberg, who develops an “uncertainty relation” by showing how each observation of a system, say an electron, always itself introduces some disturbance, so that the exact state at the beginning of each new experiment is essentially unknowable. Heisenberg works from the matrix point of view, but Bohr has pointed out that the “uncertainty relation,” exhibited by Heisenberg simply as an experi­mental result, becomes quite natural on wave principles, and is indeed nothing more than an expression of those principles.
Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by Trev=9mua2/4hbarpiT_{rev} = 9 mu a^2/4hbar piTrev​=9mua2/4hbarpi where aaa and mumumu are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral (60circ−60circ−60circ60^{circ}-60^{circ}-60^{circ}60circ−60circ−60circ) triangle, such as the half equilateral (30circ−60circ−90circ30^{circ}-60^{circ}-90^{circ}30circ−60circ−90circ) triangle and other `foldings', which have related energy spectra and revival structures.
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.