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experiments:aharonov-bohm [2018/04/14 13:29]
theodorekorovin [Abstract]
experiments:aharonov-bohm [2018/05/06 11:56] (current)
jakobadmin [Abstract]
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-====== ​The Aharonov-Bohm Experiment ======+====== Aharonov-Bohm Experiment ======
  
  
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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   * 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   * 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>​
-  
 <tabbox Abstract> ​ <tabbox Abstract> ​
  
  
  
-The Aharonov-Bohm experiment can be understood nicely using [[advanced_tools:​fiber_bundles|fiber bundles]]:+The Aharonov-Bohm experiment can be understood nicely using [[advanced_tools:​fiber_bundles|fiber bundles]]. A different path through around the solenoid leads to a different path through the fiber bundle and thus to a netto phase difference¬†
  
 +{{ :​experiments:​aharnovbohm.png?​nolink&​600 |}}
 +The gauge field $A$ is responsible that electrons moving on opposite sides around the solenoid also need to take different paths around the fiber bundle. In the picture above, the gauge field corresponds to the ramps that tell us how the phase factor of an electron changes as it moves through space.
  
-[{{ :​advanced_tools:​fiberramps-aharonovbohm.png?​nolink |[[http://​gregnaber.com/​wp-content/​uploads/​GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] 
  
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experiments/aharonov-bohm.1523705383.txt.gz · Last modified: 2018/04/14 11:29 (external edit)