Editing Aharonov–Bohm effect (section)

==Electric effect==

Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the [[Schrödinger equation]], the phase of an eigenfunction with energy <math>E</math> goes as <math>e^{-iEt/\hbar}</math>. The energy, however, will depend upon the electrostatic potential <math>V</math> for a particle with charge <math>q</math>. In particular, for a region with constant potential <math>V</math> (zero field), the electric potential energy <math>qV</math> is simply added to <math>E</math>, resulting in a phase shift:

:<math>\Delta\varphi = -\frac{qVt}{\hbar} ,</math>

where ''t'' is the time spent in the potential.

For example, we may have a pair of large flat conductors, connected to a battery of voltage <math>\Delta V</math>. Then, we can run a single electron double-slit experiment, with the two slits on the two sides of the pair of conductors. If the electron takes time <math>t</math> to hit the screen, then we should observe a phase shift <math>e\Delta V t/\hbar</math>. By adjusting the battery voltage, we can horizontally shift the interference pattern on the screen.

The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a time dependent potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a constant bias voltage ''V'' relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phase shift as above, and was observed experimentally in 1998, albeit in a setup where the charges do traverse the electric field generated by the bias voltage. The original time dependent electric Aharonov–Bohm effect has not yet found experimental verification.<ref>
{{cite journal
 |author=van Oudenaarden, A
 |date=1998
 |last2=Devoret
 |first2=Michel H.
 |last3=Nazarov
 |first3=Yu. V.
 |last4=Mooij
 |first4=J. E.
 |title=Magneto-electric Aharonov–Bohm effect in metal rings
 |journal=[[Nature (journal)|Nature]]
 |volume=391 |issue= 6669|pages=768
 |doi=10.1038/35808
|bibcode = 1998Natur.391..768V |s2cid=4426127
 }}</ref>