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One can show that for the phase difference $\Delta$ between the two wave functions (slit 1 and slit 2) it holds the first equality on the LHS

$$\Delta=\oint_{\partial\Omega} \vec{A}\overset{!}{=}\int_\Omega \nabla\times \vec{A}=\int_\Omega \vec{B}=\Phi_m$$

This is then used to proof a physically important result, namely that the phase difference is non-vanishing, because there is a flux through $\Omega$ due to the solenoid. There are issues with this.

I'm not a mathematician, but I would never use Stokes here. There is a huge singularity inside $\Omega$ $$\vec{A}_{solenoid}\propto\frac{\hat{\phi}}{r}$$

I'm sure there is a more formal argument than using stokes here?

Qmechanic
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  • Where did you get that expression for A? – knzhou May 19 '20 at 07:24
  • @knzhou Which one exactly, the expression for the vector potential of the solenoid or the relation to the phase difference? –  May 19 '20 at 09:23
  • @ChiralAnomaly Yes, this would be helpful I think. Thank you. But isn't it true, that neither $A$ nor $B$ should have any singularities in $\Omega$, in order to use stokes? You make it sound like it only matters that $B$ doesn't diverge. –  May 20 '20 at 05:27
  • Related: https://physics.stackexchange.com/questions/123423/basic-question-on-the-aharonov-bohm-effect?rq=1 –  May 20 '20 at 05:31
  • @ChiralAnomaly Don't rush. Looking forward to see this. Thanks. –  May 20 '20 at 21:38

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