Quantum particles in a suddenly accelerating potential

TL;DR

This paper develops a formalism to analyze quantum particles in potentials undergoing sudden changes in velocity or acceleration, revealing phenomena like quantum revivals, coherent states, and agreement with classical expectations in specific scenarios.

Contribution

It introduces a new formalism for quantum particles in rapidly changing potentials and characterizes their behavior, including revivals and coherent states, under sudden accelerations.

Findings

Quantum revivals occur at specific times regardless of parameters.

Sudden changes produce coherent states in harmonic oscillators.

Transition probabilities match classical expectations and previous formulas under certain conditions.

Abstract

We study the behavior of a quantum particle trapped in a confining potential in one dimension under multiple sudden changes of velocity and/or acceleration. We develop the appropriate formalism to deal with such situation and we use it to calculate the probability of transition for simple problems such as the particle in an infinite box and the simple harmonic oscillator. For the infinite box of length $L$ under two and three sudden changes of velocity, where the initial and final velocity vanish, we find that the system undergoes quantum revivals for $\Delta t = \tau_0 \equiv \frac{4mL^2}{\pi\hbar}$, regardless of other parameters ($\Delta t$ is the time elapsed between the first and last change of velocity). For the simple harmonic oscillator we find that the states obtained by suddenly changing (one change) the velocity and/or the acceleration of the potential, for a particle initially in an eigenstate of the static potential, are {\sl coherent} states. For multiple changes of acceleration or velocity we find that the quantum expectation value of the Hamiltonian is remarkably close (possibly identical) to the corresponding classical expectation values. Finally, the probability of transition for a particle in an accelerating harmonic oscillator (no sudden changes) calculated with our formalism agrees with the formula derived long time ago by Ludwig and recently modified by Dodonov~\cite{Dodonov21}, but with a different expression for the dimensionless parameter $\gamma$. Our probability agrees with the one of ref.~\cite{Dodonov21} for $\gamma \ll 1$ but is not periodic in time (it decays monotonously), contrary to the result derived in ref.~\cite{Dodonov21}.