Problems with variable Hilbert space in quantum mechanics

TL;DR

This paper investigates the issues arising from variable Hilbert spaces in quantum mechanics, especially in the context of a particle in a changing potential well, revealing non-intuitive transition probabilities and their implications for fundamental physics.

Contribution

It highlights the problem of inapplicability of standard transition probability calculations when the Hilbert space changes, and introduces a regularization approach to understand transition phenomena.

Findings

Transition probabilities less than 1 at zero time change

Regularization reveals transitions into continuous spectrum

Potential implications for early universe models

Abstract

The general problem is studied on a simple example. A quantum particle in an infinite one-dimensional well potential is considered. Let the boundaries of well changes in a finite time $T$. The standard methods for calculating probability of transition from an initial to the final state are in general inapplicable since the states of different wells belong to different Hilbert spaces. If the final well covers only a part of the initial well (and, possibly, some outer part of the configuration space), the total probability of the transition from any stationary state of the initial well into {\bf all} possible states of the final well is less than 1 at $T\to 0$. If the problem is regularized with a finite-height potential well, this missing probability can be understood as a non-zero probability of transitions into the continuous spectrum, despite the fact that this spectrum disappears at the removal of regularization. This phenomenon ("transition to nowhere'') can result new phenomena in some fundamental problems, in particular at description of earlier Universe. We discuss also how to calculate the probabilities of discussed transitions at final $T$ for some ranges of parameters.