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American Journal of Physics

Ehrenfest theorem and the classical trajectory of quantum motion

J. Nag, V Jayaram Menon, S. N. Mukherjee

Published Sep 1, 1987• 4 citations
Publisher

Abstract

Ehrenfest theorem asserts that the quantum mechanical motion of a particle when considered in the expectation value sense should agree with classical mechanics in the correspondence limit. An explicit verification of this result is presented in the one-dimensional case for motion in an infinite potential well (large quantum number limit) and a brief mention is made of the case of a smooth potential (ℏ→0 limit).

DOI
10.1119/1.14992

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Ehrenfest theorem asserts that the quantum mechanical motion of a particle when considered in the expectation value sense should agree with classical mechanics in the correspondence limit. An explicit verification of this result is presented in the one-dimensional case for motion in an infinite potential well (large quantum number limit) and a brief mention is made of the case of a smooth potential (ℏ→0 limit).
Ehrenfest’s theorem states that as a quantum state evolves in time, the rate of change of the expectation value of momentum is equal to the expectation value of the force. In the familiar ‘‘particle-in-a-box,’’ however, the probability of finding the particle at a wall—the only place where forces act—is zero, so at first glance it appears that the expectation value of the force should vanish for any state, which would violate Ehrenfest‘s theorem. This argument is flawed, however, since the forces at the walls of the box are infinite. We consider the box as a limit of a very deep square well and confirm that Ehrenfest’s theorem emerges safe and sound.
An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit ℏ→0. Several ways in which this limit may be performed are considered. It is shown that Schrödinger's equation for a single particle moving in an external potential V does not, except in special cases, lead, in this limit, to Newton's equation of motion for the particle. This shows that classical mechanics cannot be regarded as emerging from quantum mechanics—at least in this sense—upon straightforward application of the limit ℏ→0.
The classical limit of quantum mechanics is usually discussed in terms of Ehrenfest's theorem, which states that, for a sufficiently narrow wave packet, the mean position in the quantum state will follow a classical trajectory. We show, however, that that criterion is neither necessary nor sufficient to identify the classical regime. Generally speaking, the classical limit of a quantum state is not a single classical orbit, but an ensemble of orbits. The failure of the mean position in the quantum state to follow a classical orbit often merely reflects the fact that the centroid of a classical ensemble need not follow a classical orbit. A quantum state may behave essentially classically, even when Ehrenfest's theorem does not apply, if it yields agreement with the results calculated from the Liouville equation for a classical ensemble. We illustrate this fact with examples that include both regular and chaotic classical motions.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ħ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ~ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.