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experiments:aharonov-bohm [2018/04/14 13:32] theodorekorovin [Abstract] |
experiments:aharonov-bohm [2018/04/15 12:00] ida [Concrete] |
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<tabbox Concrete> | <tabbox Concrete> | ||
+ | The magnetic field is only nonzero in the interior of the solenoid. However, the associated vector potential can be nonvanishing | ||
+ | also outside. Since the magnetic fields is zero outside, the value of $\vec{A}$ outside the solenoid has to be | ||
+ | pure gauge, i.e. a gauge transformation of $\vec{A}=0$: $\vec{\nabla}\times\vec{A}=\vec{0}$. | ||
+ | |||
+ | This is important because the region outside the solenoid is not simply connected | ||
+ | the vector potential cannot be gauged to zero everywhere, only patchwise. | ||
+ | |||
+ | We denote | ||
+ | by $\Psi_{1}^{(0)}$ and $\Psi_{2}^{(0)}$ the wave functions for the two electron beams without the solenoid. The total | ||
+ | wave function when we switch the magnetic field on is | ||
+ | \begin{eqnarray} | ||
+ | \Psi&=&e^{ie\int_{\Gamma_{1}}\vec{A}\cdot d\vec{x}}\Psi_{1}^{(0)}+ | ||
+ | e^{ie\int_{\Gamma_{2}}\vec{A}\cdot d\vec{x}}\Psi_{2}^{(0)} \nonumber \\ | ||
+ | &=&e^{ie\int_{\Gamma_{1}}\vec{A}\cdot d\vec{x}}\left[\Psi_{1}^{(0)} | ||
+ | +e^{ie\oint_{\Gamma}\vec{A}\cdot d\vec{x}}\Psi_{2}^{(0)}\right] | ||
+ | \label{eq:extra_phase} . | ||
+ | \end{eqnarray} | ||
+ | Here $\Gamma_{1}$ and $\Gamma_{2}$ denote two curves surrounding the solenoid | ||
+ | from different sides. In addition, $\Gamma$ is any closed loop surrounding it. | ||
+ | |||
+ | We can see here that the relative phase between the two beams going different paths, gets an additional contribution that depends on the value of the vector potential | ||
+ | \begin{eqnarray} | ||
+ | U=\exp\left[ie\oint_{\Gamma}\vec{A}\cdot d\vec{x}\right]. | ||
+ | \label{eq:wilson} | ||
+ | \end{eqnarray} | ||
+ | |||
+ | This way the presence of the magnetic field becomes visible through a changed interference pattern even though it is zero outside of the solenoid. | ||
+ | |||
+ | Take note that the quantity $U$ is independent of the [[advanced_tools:gauge_symmetry:gauge_fixing|gauge]] we are working in. Moreover, take note that the value of $U$ does not change when we continuously deform our curve $\Gamma$ around the solenoid, as long as both path stay on opposite sides of the solenoid. | ||
+ | |||
+ | ---- | ||
+ | |||
+ | |||
+ | * The best explanation can be found here: http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf at page 18ff | ||
+ | * See also https://www.dartmouth.edu/~dbr/topdefects.pdf | ||
+ | |||
+ | ---- | ||
<blockquote>For quite some time it was felt that such phase changes in the wavefunction | <blockquote>For quite some time it was felt that such phase changes in the wavefunction | ||
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R. G. Chambers in 1960) with results that confirmed the expectations of | R. G. Chambers in 1960) with results that confirmed the expectations of | ||
Aharonov and Bohm.<cite>page 6 in Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> | Aharonov and Bohm.<cite>page 6 in Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> | ||
- | |||
- | |||
- | ---- | ||
- | |||
- | |||
- | * The best explanation can be found here: http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf at page 18ff | ||
- | * See also https://www.dartmouth.edu/~dbr/topdefects.pdf | ||
<tabbox Abstract> | <tabbox Abstract> | ||