Entropy of very low energy localized states

TL;DR

This paper investigates the entropy of vacuum-bounded states at very low energies, demonstrating that their entropy can surpass that of traditional states with the same energy, and conjectures unbounded growth of certain length ratios as energy approaches zero.

Contribution

It introduces improved numerical techniques for analyzing vacuum-bounded states without finite outside regions and explores their entropy behavior at very low energies.

Findings

Entropy of vacuum-bounded states can be much higher than that of rigid box states at the same energy.

Numerical evidence supports the conjecture that the length ratio diverges as energy approaches zero.

Abstract

We expand on previous work involving "vacuum-bounded" states, i.e., states such that every measurement performed outside a specified interior region gives the same result as in the vacuum. We improve our previous techniques by removing the need for a finite outside region in numerical calculations. We apply these techniques to the limit of very low energies and show that the entropy of a vacuum-bounded state can be much higher than that of a rigid box state with the same energy. For a fixed $E$ we let $L_in'$ be the length of a rigid box which gives the same entropy as a vacuum-bounded state of length $L_in$. In the $E\to 0$ limit we conjecture that the ratio $L_in'/L_in$ grows without bound and support this conjecture with numerical computations.