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Journal of Physics B Atomic and Molecular Physics

A correspondence principle for strongly coupled states

Ian Colin Percival, D Richards

Published Aug 1, 1970• 92 citations
Publisher

Abstract

Heisenberg's correspondence principle for non-relativistic matrix elements has been generalized: quantal transition amplitudes between strongly coupled states are expressed as Fourier components of integrals over classical trajectories. The new theory reduces to Heisenberg's correspondence principle in particular cases. The theory is applicable whenever the change in quantum number is small compared with the initial quantum number. In these situations it is more comprehensive than both quantal first order perturbation theory and the sudden approximation. Furthermore, compared with standard quantal methods, the evaluation of amplitudes is particularly simple. The theory is worked out for one- and three-dimensional separable systems and the generalization to a system of arbitrary dimension is indicated.

DOI
10.1088/0022-3700/3/8/005

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Heisenberg's correspondence principle for non-relativistic matrix elements has been generalized: quantal transition amplitudes between strongly coupled states are expressed as Fourier components of integrals over classical trajectories. The new theory reduces to Heisenberg's correspondence principle in particular cases. The theory is applicable whenever the change in quantum number is small compared with the initial quantum number. In these situations it is more comprehensive than both quantal first order perturbation theory and the sudden approximation. Furthermore, compared with standard quantal methods, the evaluation of amplitudes is particularly simple. The theory is worked out for one- and three-dimensional separable systems and the generalization to a system of arbitrary dimension is indicated.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
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After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.
A semiclassical expression for matrix elements of an arbitrary operator with respect to the eigenstates of an integrable Hamiltonian is derived. This is essentially the Heisenberg correspondence principle, and it is shown via the Weyl correspondence that the approximation is valid through the lowest two orders in \ensuremath{\Elzxh}. The result is used to prove that an asymptotic form of the Clebsch-Gordan coefficients for two large and one small angular momenta is valid through two orders. \textcopyright{} 1996 The American Physical Society.
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.
Abstract We develop a non-perturbative method for calculating partition functions of strongly coupled quantum mechanical systems with interactions between subsystems described by a path integral of a dual system. The dual path integral is derived starting from non-interacting subsystems at zeroth order and then by introducing couplings of increasing complexity at each order of an iterative procedure. These orders of interactions play the role of a dual time and the full quantum partition function is expressed as a transition amplitude in the dual system. More precisely, it is expressed as a path integral from a deformation-operators dependent initial state at zero time/order to the inverse-temperature dependent final state at later time/order. We provide examples of strongly coupled systems with up to first-order interactions (e.g. Ising model) and arbitrary high-order interactions (e.g. 1+1D1+1\hbox {D}1+1D 1 + 1 D QFT). We also discuss a possible emergence of space-time, quantum field theories and general relativity in context of the dual path integral.