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Physical Review A

Vector fields, line integrals, and the Hamilton-Jacobi equation: Semiclassical quantization of bound states

N. De Leon, Eric J. Heller

Published Jul 1, 1984• 41 citations
Publisher

Abstract

In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ associated with nonlinear dynamical systems with Hamiltonians of the form $H=T(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})+V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; (2) the WKB wave function ${\ensuremath{\Psi}}^{\mathrm{WKB}}$ is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; and (3) if the transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr\"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.

DOI
10.1103/physreva.30.5

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In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates \ensuremathγ\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremathγ\ensuremath→​ associated with nonlinear dynamical systems with Hamiltonians of the form H=T(p\ensuremath→)+V(q\ensuremath→)H=T(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})+V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})H=T(p\ensuremath→​)+V(q\ensuremath→​). For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to \ensuremathγ\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremathγ\ensuremath→​ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in \ensuremathγ\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremathγ\ensuremath→​ coordinates; (2) the WKB wave function \ensuremathΨWKB{\ensuremath{\Psi}}^{\mathrm{WKB}}\ensuremathΨWKB is separable in \ensuremathγ\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremathγ\ensuremath→​ coordinates; and (3) if the transformation to \ensuremathγ\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremathγ\ensuremath→​ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schrödinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −iℏ-i\hbar−iℏ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.
The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.
The quantization method based on the quantum Hamiltonian Jacobi equation, is extended to two-dimensional non-separable but integrable Hamiltonians. It is shown that each wave function for those systems corresponds to a well-defined family of classical trajectories, enveloped by a caustic. The energy eigenvalues and the values of the wave functions on the caustic are obtained by solving the 1-dim quantum Hamilton Jacobi equation of the caustic' arcs. Results are in good agreement with those obtained by usual methods.
Abstract The quantization of the dynamics of point systems is closely connected with the Hamilton-Jacobi theory of the calculus of variations for simple integrals, the latter being a suitable mathematical formalism for describing the classical laws of point dynamics. This connexion is evident from the fact that, in order to quantize a dynamical system, one has first to know what the pairs of canonically conjugate variables are, and also from the fact that the quantum laws, if expressed in terms of commutation brackets, have exactly the same form as the classical laws, if expressed in terms of Poisson brackets. In dealing with the quantization of the dynamics of continuous media, e.g. the electromagnetic and other fields, which has been developed along different lines, following Heisenberg and Pauli (1929), it seems tempting to try to base the method of quantization on an extended Hamilton-Jacobi theory of the calculus of variations for multiple integrals, the latter being the appropriate formalism for describing most relevant systems of continuum physics. In a previous paper (Weiss 1936), referred to as I, such an attem pt has been made, by extending the notion of pairs of conjugate variables. That attem pt led to quantum relations on an arbitrary space-like section in space-time, provided that one postulated that, if the space-like section becomes especially a space section, the quantum relations should go over into those of Heisenberg and Pauli.
The canonical functional action in the path integral in phase space is discretized by linking each pair of consecutive vertebral points - and or and - through the invariant complete solution of the Hamilton-Jacobi equation associated with the classical path defined by these extremes. When the measure is chosen to reflect the geometrical character of the propagator (it must behave as a density of weight in both of its arguments), the resulting infinitesimal propagator is cast in the form of an expansion in a basis of short-time solutions of the wave equation, associated with the eigenfunctions of the initial momenta canonically conjugated to a set of normal coordinates. The operator ordering induced by this prescription is a combination of a symmetrization rule coming from the phase, and a derivative term coming from the measure.