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Physical Review

Two Notes on Quantum Conditions

John Hasbrouck Van Vleck

Published Dec 1, 1923• 7 citations
Publisher

Abstract

Selection of coordinates for use in evaluating the Sommerfield quantum integrals.---It is well known that these coordinates need not necessarily satisfy Lagrange's equations, and a general criterion for the choice of such variables is given in the present paper, in which it is shown that all sets of $p$'s and $q$'s which (a) satisfy Hamilton's equations, (b) separate the variables, and (c) are of such a character that the average value of the kinetic energy is equal to that of $\frac{1}{2}\ensuremath{\Sigma}{1}^{n}{p}_{i}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}}_{i}$, will give the correct energy levels when employed in the quantum integrals $\ensuremath{\int}{p}_{i}d{q}_{i}={n}_{i}h$; ($i=1,2,\ensuremath{\cdots}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},n$). This general class of coordinate systems includes as special cases the familiar normalized Schwarzschild angle variables and Lagrangian generalized coordinates which separate the variables.Combination of Hamilton's principle with Trkal's method for quantizing conditionally periodic systems is proposed which, like Hamilton's principle, determines the character of the motion permitted by the Newtonian dynamics, and in addition quantizes the orbits by specializing the constants of integration in accordance with the conventional quantum conditions. This variation principle includes formally in a single equation the results of classical dynamics and the Sommerfield quantum conditions, which together determine the sizes and shapes of electronic orbits. The principle is illustrated by an application to a non-linear oscillator in which the potential involves the second and fourth powers of the displacement from the equilibrium position.

DOI
10.1103/physrev.22.547

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Selection of coordinates for use in evaluating the Sommerfield quantum integrals.---It is well known that these coordinates need not necessarily satisfy Lagrange's equations, and a general criterion for the choice of such variables is given in the present paper, in which it is shown that all sets of ppp's and qqq's which (a) satisfy Hamilton's equations, (b) separate the variables, and (c) are of such a character that the average value of the kinetic energy is equal to that of 12\ensuremathΣ1npiq\ifmmode˙\else˙\fii\frac{1}{2}\ensuremath{\Sigma}{1}^{n}{p}_{i}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}}_{i}21​\ensuremathΣ1npi​q\ifmmode˙\else˙\fi​i​, will give the correct energy levels when employed in the quantum integrals \ensuremath∫pidqi=nih\ensuremath{\int}{p}_{i}d{q}_{i}={n}_{i}h\ensuremath∫pi​dqi​=ni​h; (i=1,2,\ensuremath⋯\ifmmode⋅\else\textperiodcentered\fi\ifmmode⋅\else\textperiodcentered\fi,ni=1,2,\ensuremath{\cdots}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},ni=1,2,\ensuremath⋯\ifmmode⋅\else\textperiodcentered\fi\ifmmode⋅\else\textperiodcentered\fi,n). This general class of coordinate systems includes as special cases the familiar normalized Schwarzschild angle variables and Lagrangian generalized coordinates which separate the variables.Combination of Hamilton's principle with Trkal's method for quantizing conditionally periodic systems is proposed which, like Hamilton's principle, determines the character of the motion permitted by the Newtonian dynamics, and in addition quantizes the orbits by specializing the constants of integration in accordance with the conventional quantum conditions. This variation principle includes formally in a single equation the results of classical dynamics and the Sommerfield quantum conditions, which together determine the sizes and shapes of electronic orbits. The principle is illustrated by an application to a non-linear oscillator in which the potential involves the second and fourth powers of the displacement from the equilibrium position.
The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
The orbit equation for the motion of a quantum in a potential is proposed by defining uniquely the quantum mechanical momentum from the Schrodinger wave equation of a one-particle system and using the Hamilton-Jacobi formalism. On this occasion it is necessary to introduce a new dynamical postulate, “initial condition”, which is independent of the quantization condition as well as of the Schrodinger wave equation. The momentum is determined in such a way that it is approximately equal to the corresponding classical mechanical momentum in the classical region of validity of the coordinates. The momentum in the non-classical region is in general very small in comparison with that in the classical region. The quantum-mechanical motion well corresponds to the classical motion. If a system is macroscopic, the motion of the particle is described by Newtonian mechanics and the quantization condition is of no validity. The “quantization of magnetic flux” derives from the quantization condition. It is closely related to the energy level structure of the atomic system. The above argument is applied to the scalar electromagnetic and sound wave equations to give the dynamical motion of a photon and a phonon without recourse to the so-called second quantization method.
Ensembles and expectation values Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization. In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition. The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble. Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.
In a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle. It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian-operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long-standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
Abstract The transition from classical mechanics to quantum mechanics has been formulated in various ways. But all these ways have the common feature that they use as starting point mechanics not in its Newtonian form, but in the form which it was given by Lagrange and Hamilton. The essence of this form consists in the fact that the differential equations are represented as Euler equations of a variational principle for v functions, the generalized coordinates, of one variable, the time. In its purely mathematical aspect the quantization method may, therefore, be regarded as a mode of taking certain notions and certain relations arising from a variational principle for several dependent and one independent variable and of giving them a new meaning.