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American Journal of Physics

Forced Vibrations of a Harmonic Lattice in Quantum Mechanics

William Band

Published Sep 1, 1962• 0 citations
Publisher

Abstract

Schrödinger's equation is solved in an elementary fashion for a harmonic lattice subject to external forces that are functions explicitly only of time, and to damping forces proportional to velocity expectation values. The result shows that the response is a purely classical motion of the quantum mechanical probability distribution associated with the free lattice. The energy of the lattice is Eq + Ec, where Eq is any energy eigenvalue of the free undisturbed lattice existing before the external forces were applied, and Ec is the energy that would be obtained by classical mechanics for the response of the lattice to the disturbance. The energy propagates through the lattice as a purely classical wave motion superposed on any quantum mechanical background that may happen to exist. Anharmonic coupling terms are needed to cause quantum excitations and produce thermal equilibrium within the lattice. The conventional treatment of energy transport as due to unidirectional normal modes is criticized.

DOI
10.1119/1.1942151

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Schrödinger's equation is solved in an elementary fashion for a harmonic lattice subject to external forces that are functions explicitly only of time, and to damping forces proportional to velocity expectation values. The result shows that the response is a purely classical motion of the quantum mechanical probability distribution associated with the free lattice. The energy of the lattice is Eq + Ec, where Eq is any energy eigenvalue of the free undisturbed lattice existing before the external forces were applied, and Ec is the energy that would be obtained by classical mechanics for the response of the lattice to the disturbance. The energy propagates through the lattice as a purely classical wave motion superposed on any quantum mechanical background that may happen to exist. Anharmonic coupling terms are needed to cause quantum excitations and produce thermal equilibrium within the lattice. The conventional treatment of energy transport as due to unidirectional normal modes is criticized.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ \hat X X^ in x representation. The evolution of ⟨x^⟩ \left\langle {\hat x} \right\rangle ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.
The wave equations for the forced, damped, and forced and damped oscillators are solved in closed form for an arbitrary forcing function; the solutions produced being in one-to-one correspondence with the stationary states of the unforced, undamped Hamiltonian H 0 . The quantal motion is closely connected with the classical: for the forced oscillator the probability density is that of H 0 but moves as a whole with the classical motion; for the forced and damped oscillator this motion is accompanied by a contraction progressing eventually into a delta function at the classical position. Transition probabilities between states of H 0 are computed in the case of forced motion and depend solely on the classically acquired energy of the oscillator at any time. The transition probability vanishes strictly only when this energy has a value falling at the roots of a Laguerre polynomial associated with the transition. The classical dipole radiation emitted by a disturbed oscillator is, when the damping force is identified with the force of radiation damping, that of the classical oscillator: a shifted and broadened line.
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.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
Assuming a damping force proportional to the velocity, it is shown that the usual quantum condition between the physical variables, namely the displacement and the momentum, has to be modified by the damping. However, it is possible to change to new variables which are canonical. Then we can apply the usual quantum mechanics and use the usual physical interpretation through the statistical average values. The case of a one-dimensional damped harmonic oscillator under the action of an arbitrary external force has been worked out as an example. No difficulty arises and the results are reasonable.