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The virtual detector is a commonly utilized technique to measure theproperties of a wavefunction in simulation. One type of virtual detectormeasures the probability density and current at a set position over time,permitting an instantaneous measurement of momentum at a boundary. This may beused as the boundary condition between a quantum and a classical simulation.However, as a tool for measuring spectra, it possesses several problemsstemming from its incoherent nature. Another form of virtual detector measuresthe wavefunction's complex value at a set position in real space over time andFourier analyzes it to produce an energy spectrum. The spectra it produces areexact provided that the wavefunction propagated through the detector in onedirection. Otherwise it will produce a spectrum that includes interferencebetween forward and backward propagating wavepackets. Here we propose a virtualdetector which maintains all the benefits of this coherent virtual detectorwhile also being able to resolve the direction of propagation and mitigatenonphysical interference by use of a second measurement point. We show that, inthe continuum limit, this bi-directional virtual detector can reproduce anequivalent wavefunction assuming a globally constant potential. It is thereforeequivalent to the exact spectrum.

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