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American Journal of Physics

Ehrenfest’s theorem and the particle-in-a-box

Daniel S. Rokhsar

Published Nov 1, 1996• 17 citations
Publisher

Abstract

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.

DOI
10.1119/1.18367

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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.
"We study the issues of average forces and the Ehrenfest theorem for a particle restricted to a semi-infinite interval by an impenetrable wall. We consider and discuss two specific cases: (i) a free particle in an infinite step potential, and (ii) a free particle on a half-line. In each situation, we show that the mean values of the position, momentum and force, as functions of time, verify the Ehrenfest theorem (the state of the particle being a general wave packet that is a continuous superposition of the energy eigenstates for the Hamiltonian). However, the involved force is not the same in each case. In fact, we have the usual external classical force in the first case and a type of nonlocal boundary quantum force in the second case. In spite of these different forces, the corresponding mean values of these quantities give the same results. Accordingly, the Ehrenfest equations in the two situations are equivalent. We believe that a careful and clear consideration of how the two cases differ but, in the end, agree, is pertinent, and has not been included in the literature."
In this paper we have solved analytically the Schrodinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square of the linear momentum are equal to the classical time average of particle position and its square of the linear momentum, respectively.
In classical physics, if you want to calculate what’s going to happen in a given situation, you basically have two options open to you. You can carefully specify the positions and momenta of all the particles involved, list the forces acting, and then plug the lot into Newton’s laws of motion and calculate away. Alternatively, you can inspect all the kinetic energies, potential energies, and any other form of energy involved and work through a calculation based on energy conservation. As it happens, the second approach, using energy, is the more amenable to a straightforward quantum development. The first Ehrenfest relationship should be compared with the classical formulation px=mdxdt. Moving on from Ehrenfest’s theorem, let us revisit Heisenberg’s equation of motion and explore one of its significant consequences. The left-hand side of this relationship is the time rate of change of the expectation value of the operator. Let’s think carefully about what that means.
The Schrödinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though the particle is nowhere near the walls, it will be affected. It is shown that this force apart from a minus sign is equal to the expectation value of the gradient of the quantum potential for vanishing at the walls boundary condition. Variation of this force with time is studied. A selection of Bohmian trajectories of the confined particle is also computed.