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arXiv:2206.07531

Bouncing Wave Packets, Ehrenfest Theorem, and Uncertainty Relation based upon a new Concept for the Momentum of a Particle in a Box

I. Albrecht, J. Herrmann, A. Mariani, U.‐J. Wiese, V. Wyss

Published Jun 15, 2022• 1 citation
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Abstract

For a particle in a box, the operator $-i\partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg-Robertson-Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.

DOI
10.48550/arxiv.2206.07531

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For a particle in a box, the operator −i∂x-i\partial_x−i∂x​ is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg-Robertson-Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.
For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.
Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.
Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.
This note shows that Heisenberg's choice for a wave function in his original paper on the uncertainty principle is simply a renormalized characteristic function of a stable distribution with certain restrictions on the parameters. Relaxing Heisenberg's restrictions leads to a more general formulation of the uncertainty principle. This reformulation shows quantum uncertainty can exist at a macroscopic level. These modifications also give rise to a new form of Schrodinger's wave equation as the equation of a vibrating string. Although a heat equation version can also be given, the latter shows the traditional formulation of Schrodinger's equation involves a hidden Cauchy amplitude assumption.
We prove the uncertainty relation σT σE≥ℏ/2σ_T \, σ_E \geq \hbar/2σT​σE​≥ℏ/2 between the time TTT of detection of a quantum particle on the surface ∂Ω\partial Ω∂Ω of a region Ω⊂R3Ω\subset \mathbb{R}^3Ω⊂R3 containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy EEE of the initial wave function. Here, σσσ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since TTT is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schrödinger. We also prove that if there is nonzero probability that the particle never reaches ∂Ω\partial Ω∂Ω (in which case we write T=∞T=\inftyT=∞), and if σTσ_TσT​ denotes the standard deviation conditional on the event T<∞T<\inftyT<∞, then σT σE≥(ℏ/2)Prob(T<∞)σ_T \, σ_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}σT​σE​≥(ℏ/2)Prob(T<∞)​.
The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.