Bound states of Newton's equivalent finite square well

TL;DR

This paper investigates how a family of Newton's equivalent Hamiltonians affects the bound state energies and wavefunctions in a finite square well, revealing parameter-dependent differences from standard quantum results.

Contribution

It introduces a method to analyze bound states for a family of NEHs in finite square wells, highlighting quantum differences from classical equivalence.

Findings

Bound state energies depend on the parameter β.

As β approaches zero, results recover standard quantum finite square well energies.

The wavefunctions are infinitely differentiable at well boundaries for all NEHs.

Abstract

In this paper, a 1-parameter family of Newton's equivalent Hamiltonians (NEH) for finite square well potential is analyzed in order to obtain bound state energy spectrum and wavefunctions. For a generic potential, each of the NEH is classically equivalent to one another and to the standard Hamiltonian yielding Newton's equations. Quantum mechanically, however, they are expected to be differed from each other. The Schr\"{o}dinger's equation coming from each of NEH with finite square well potential is an infinite order differential equation. The matching conditions therefore demand the wavefunctions to be infinitely differentiable at the well boundaries. To handle this, we provide a way to consistently truncate these conditions. It turns out as expected that bound state energy spectrum and wavefunctions are dependent on the parameter $\beta,$ which is used to characterize different NEH. As $\beta\to 0,$ the energy spectrum coincides with that from the standard quantum finite square well.