A Coherent Bi-Directional Virtual Detector for the 1-D Schr\"odinger Equation

TL;DR

This paper introduces a bi-directional virtual detector for the 1-D Schrödinger equation that accurately measures energy spectra, resolves wave propagation directions, and reduces interference, improving upon existing virtual detector methods.

Contribution

It proposes a novel bi-directional virtual detector that maintains coherence, resolves propagation direction, and accurately reproduces spectra in the continuum limit.

Findings

The detector can reproduce the exact spectrum assuming a constant potential.

It resolves the direction of wave propagation.

It mitigates nonphysical interference effects.

Abstract

The virtual detector is a commonly utilized technique to measure the properties of a wavefunction in simulation. One type of virtual detector measures the probability density and current at a set position over time, permitting an instantaneous measurement of momentum at a boundary. This may be used as the boundary condition between a quantum and a classical simulation. However, as a tool for measuring spectra, it possesses several problems stemming from its incoherent nature. Another form of virtual detector measures the wavefunction's complex value at a set position in real space over time and Fourier analyzes it to produce an energy spectrum. The spectra it produces are exact provided that the wavefunction propagated through the detector in one direction. Otherwise it will produce a spectrum that includes interference between forward and backward propagating wavepackets. Here we propose a virtual detector which maintains all the benefits of this coherent virtual detector while also being able to resolve the direction of propagation and mitigate nonphysical interference by use of a second measurement point. We show that, in the continuum limit, this bi-directional virtual detector can reproduce an equivalent wavefunction assuming a globally constant potential. It is therefore equivalent to the exact spectrum.