Interference Energy Spectrum of the Infinite Square Well

TL;DR

This paper introduces the concept of interference energy spectrum in quantum wells, showing how sudden barriers at wavefunction zeros alter the energy basis and can lead to measurable energies exceeding initial modes, raising conservation questions.

Contribution

It presents a novel method to derive and analyze the interference energy spectrum resulting from wavefunction manipulation in infinite square wells.

Findings

Raising barriers at wavefunction zeros changes the energy basis.

The interference spectrum can include energies higher than initial modes.

Numerical simulations confirm rapid barrier raising minimally affects the wavefunction.

Abstract

Certain superposition states of the 1-D infinite square well have transient zeros at locations other than the nodes of the eigenstates that comprise them. It is shown that if an infinite potential barrier is suddenly raised at some or all of these zeros, the well can be split into multiple adjacent infinite square wells without affecting the wavefunction. This effects a change of the energy eigenbasis of the state to a basis that does not commute with the original, and a subsequent measurement of the energy now reveals a completely different spectrum, which we call the {interference energy spectrum} of the state. This name is appropriate because the same splitting procedure applied at the stationary nodes of any eigenstate does not change the measurable energy of the state. Of particular interest, this procedure can result in measurable energies that are greater than the energy of the highest mode in the original superposition, raising questions about the conservation of energy akin to those that have been raised in the study of superoscillations. An analytic derivation is given for the interference spectrum of a given wavefunction $\Psi(x,t)$ with $N$ known zeros located at points $s_i = (x_i, t_i)$. Numerical simulations were used to verify that a barrier can be rapidly raised at a zero of the wavefunction without significantly affecting it. The interpretation of this result with respect to the conservation of energy and the energy-time uncertainty relation is discussed, and the idea of alternate energy eigenbases is fleshed out. The question of whether or not a preferred discrete energy spectrum is an inherent feature of a particle's quantum state is examined.