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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Inequivalence of the classes of quantum and classical harmonic potentials: Proof by example

Gautam Kumar Ghosh, Rainer W. Hasse

Published Aug 15, 1981• 18 citations
Publisher

Abstract

It has been proven by explicit construction that there exists a quantum harmonic potential (equally spaced energy levels) which does not belong to the class of classical harmonic potentials (frequency independent of energy). In this paper we prove the converse, also by example, that all classical harmonic potentials are not necessarily quantum harmonic potentials. Thus all classical harmonic potentials are not quantum harmonic potentials and vice versa.

DOI
10.1103/physrevd.24.1027

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It has been proven by explicit construction that there exists a quantum harmonic potential (equally spaced energy levels) which does not belong to the class of classical harmonic potentials (frequency independent of energy). In this paper we prove the converse, also by example, that all classical harmonic potentials are not necessarily quantum harmonic potentials. Thus all classical harmonic potentials are not quantum harmonic potentials and vice versa.
There are an infinite number of potentials whose classical angular velocity ${\ensuremath{\omega}}_{c}$ is independent of energy. There are also an infinite number of potentials whose quantum eigenenergies are equally spaced, $\ensuremath{\hbar}{\ensuremath{\omega}}_{q}$. If ${\ensuremath{\omega}}_{c}={\ensuremath{\omega}}_{q}$, then in the WKB approximation these classes are the same. (Certain specific potentials belong to both of these classes exactly.) However, we show by a different specific example that in general these classes are not equivalent.
It is well known that to quantize any dynamical system it is necessary for a generator of the classical motion to exist in the form of a Hamiltonian function. It is shown, by using the example of a damped harmonic oscillator, that a particular class of inequivalent classical Hamiltonians exist which make quantization of the system ambiguous. Hence it is conclued that it is not sufficient for the Hamiltonian to merely generate the motion, but it must also be necessarily related via a canonical transformation to the total energy of the system.
The first application of quantum ideas to a system with a nonuniform potential energy is to the harmonic oscillator: a pendulum, or a mass on a spring. The result is a quantization of energy with equally spaced energy levels. The wave function is localized in the potential well but can do quantum tunneling beyond the turning points into classically forbidden regions. A wave function can be a negative quantity, but the related position probability is always positive. The wave function shows the particle is not in any single place; in a sense it is in many places simultaneously. The fact that the lowest energy quantum state does not vanish is an example of zero-point energy. States of very high energy show a strong analogy to classical oscillator behavior, an example of the Bohr correspondence principle.
Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.