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Physical Review A

Some structural and numerical aspects of Heisenberg’s matrix mechanics with application to one-dimensional nonpolynomial potentials

Ching‐Teh Li, Gwo-Hong Suen, Abraham Klein

Published Apr 18, 2001• 2 citations
Publisher

Abstract

The equation-of-motion method based on Heisenberg's matrix mechanics, utilized in this paper, was originally developed within the context of the nuclear many-body problem. The method was later applied successfully to anharmonic oscillators and other problems in quantum mechanics with polynomial potentials in one and two dimensions. Here we apply the method to the one-dimensional quantum system with the nonpolynomial potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ as an illustration of its applicability to a wider class of problems than has been previously considered. We examine some of the structural and numerical aspects of this one-dimensional quantum system, and show that it is essential to work with $\mathrm{sinh}x$ instead of x as the dynamical variable for this nonpolynomial potential problem. This choice is suggested by a preliminary study of the exactly solvable Poschl-Teller potential, $\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,$ where our methods provide an exact algebraic solution. For the potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.

DOI
10.1103/physreva.63.052110

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The equation-of-motion method based on Heisenberg's matrix mechanics, utilized in this paper, was originally developed within the context of the nuclear many-body problem. The method was later applied successfully to anharmonic oscillators and other problems in quantum mechanics with polynomial potentials in one and two dimensions. Here we apply the method to the one-dimensional quantum system with the nonpolynomial potential \ensuremathλsinh2x,\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,\ensuremathλsinh2x, as an illustration of its applicability to a wider class of problems than has been previously considered. We examine some of the structural and numerical aspects of this one-dimensional quantum system, and show that it is essential to work with sinhx\mathrm{sinh}xsinhx instead of x as the dynamical variable for this nonpolynomial potential problem. This choice is suggested by a preliminary study of the exactly solvable Poschl-Teller potential, \ensuremath−\ensuremathλ/cosh2x,\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,\ensuremath−\ensuremathλ/cosh2x, where our methods provide an exact algebraic solution. For the potential \ensuremathλsinh2x,\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,\ensuremathλsinh2x, the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.
Based on an equations-of-motion method in the framework of Heisenberg's matrix mechanics, we investigate the conditions under which a one-dimensional quantum mechanical system becomes exactly solvable. By linearizing the equation of motion which is a double-commutation relation of some appropriately chosen function of the position operator with the Hamiltonian, we obtained a set of nontrivial exactly solvable potentials in one dimension. These potentials not only can be solved analytically in closed forms but also contain both the Morse potential and the Poschl-Teller potential as their limiting cases. They may thus be valuable for some potential model calculations as well as for testing various approximation schemes. We also examine these potentials in the framework of supersymmetric quantum mechanics, which is particularly useful for studying exactly solvable potentials.
For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number nnn that the matrix elements $〈n|x|{n}^{\ensuremath{'}}〉$ and $〈n|p|{n}^{\ensuremath{'}}〉$ should be rapidly decreasing functions of $|n\ensuremath{-}{n}^{\ensuremath{'}}|$. Matrix elements of polynomials in xxx and ppp should therefore be well approximated by a finite number of terms in their sum-rule decomposition. From the matrix elements of the equations of motion for xxx and for ppp and of the commutator ∣x,p∣|x,p|∣x,p∣, one thereby obtains closed sets of nonlinear algebraic equations to characterize subspaces of the Hilbert space of exact eigenfunctions. The approximations are also derived from a novel variational principle, and numerous variant approximation schemes are suggested. Essentially exact numerical results are obtained and compared with previous work. The broad applicability of the techniques is emphasized.
For a class of problems it is shown that a solution to Newton's equation of motion may be interpreted as the generating function for the matrix elements of the coordinate operators in the Heisenberg representation. For three one-dimensional problems, the harmonic oscillator, the particle in a box, and the x4{x}^{4}x4 anharmonic oscillator, solutions to the quantum-mechanical problem are given in terms of the classical solution. A brief examination is made of a two-dimensional problem in which the angular momentum is conserved.
Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm's hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton's principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities
This paper considers the Heisenberg equations of motion $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}=\ensuremath{-}i[q, H]$, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{p}=\ensuremath{-}i[p, H]$, for the quantum-mechanical Hamiltonian H(p,q)H(p, q)H(p,q) having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle-variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of the anharmonic oscillator whose Hamiltonian is H=p22+q44H=\frac{{p}^{2}}{2}+\frac{{q}^{4}}{4}H=2p2​+4q4​.