# What was the theoretical basis for the Ehrenberg–Siday–Aharonov–Bohm effect?

Wikipedia about the prediction of the Aharonov–Bohm effect:

Werner Ehrenberg (1901–1975) and Raymond E. Siday first predicted the effect in 1949.

What was the theoretical basis for this prediction?

And BTW, an older question about Has The Aharonov-Bohm Effect Been Experimentally Proven? is not answered clearly. It seems that the claimed Aharonov–Bohm effect in 1959 was a prediction too. So which different theoretical basis was used by Aharonov and Bohm?

It's well described in the wikipedia page, but it just falls out from the Hamiltonian for a particle in a region with non-zero $\vec{A}$ but not necessarily non-zero $\vec{B}$ or $\vec{E}$:

Using cylindrical coordinates for a particle confined to a ring of radius $R$ with a solenoid of radius $r < R$ inside,

$H = \frac{1}{2m} \left( -\hbar^2\nabla^2 + \frac{2i\hbar q}{c}\vec{A}\cdot \vec{\nabla} + \frac{q^2}{c^2}|\vec{A}|^2 \right)$

This can be expressed

$H = -\frac{\hbar^2}{2mR^2} \left( \frac{\partial}{\partial \theta} - K \right)^2; K = i\frac{q \Phi}{2c\hbar}$

Where $\Phi$ is the flux. When you work out solutions with boundary conditions $\psi(\theta) = \psi(\theta + 2\pi)$, you will see the energy only cares about the flux, but the wavefunction itself still depends on $\vec{A}$, lending itself to potential interference effects, hence the Abraham Bohm effect.

• Señor, please explain it more in detail. Thank you in advance. – HolgerFiedler May 29 '17 at 4:18
• @HolgerFiedler what detail would you like? – Señor O May 29 '17 at 4:43
• Why for a shielded field there is predicted a force? Connect your answer with this. – HolgerFiedler May 29 '17 at 5:26
• @HolgerFiedler "The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (V, A), despite being confined to a region in which both the magnetic field B and electric field E are zero" – Señor O May 29 '17 at 6:31
• this says nothing about a force – Señor O May 29 '17 at 6:31