The Aharonov–Bohm effect and gauge invariance
Goal of the project: Understand why in quantum mechanics the gauge potential \(\boldsymbol{A}\) is physically significant even though the classical force depends only on the magnetic field \(\boldsymbol{B}\). Show that the Aharonov–Bohm effect is topological and is observed as a shift of the interference pattern.
We consider the interference of electrons in the presence of an infinitely long thin solenoid with magnetic flux \(\Phi\), while \[\boldsymbol{B} = 0 \quad \text{in the region where the electrons move}.\]
1. Theory: the phase and the gauge potential
In classical electrodynamics a particle feels only the local field \(\boldsymbol{B}=\nabla\times\boldsymbol{A}\). In quantum mechanics the phase structure of the wave function makes the vector potential \(\boldsymbol{A}\) observable.
The Schrödinger equation with a vector potential
Consider the Schrödinger equation for a particle of charge \(q\): \[i\hbar\frac{\partial\psi}{\partial t} = \frac{1}{2m}\left( - i\hbar\nabla - q\boldsymbol{A} \right)^2\psi .\]
Explain why the potential \(\boldsymbol{A}\) can affect the phase of the wave function even when \(\boldsymbol{B}=0\).
The phase factor
Let \(\psi_0(\boldsymbol{r},t)\) be a solution of the Schrödinger equation with \(\boldsymbol{A}=0\). Show that for nonzero \(\boldsymbol{A}\) the solution can be written as \[\psi(\boldsymbol{r},t) = \exp\!\left( \frac{i q}{\hbar} \int_{\mathcal C} \boldsymbol{A}\cdot d\boldsymbol{l} \right) \psi_0(\boldsymbol{r},t),\] where the integral is taken along the path of the particle.
The phase difference in an interferometer
Consider a double-slit experiment. Let the electron travel along two paths \(\mathcal C_1\) and \(\mathcal C_2\) that enclose the solenoid.
Show that the phase difference is \[\Delta\varphi = \frac{q}{\hbar} \oint \boldsymbol{A}\cdot d\boldsymbol{l} = \frac{q\Phi}{\hbar}.\]
Explain why the result depends only on the magnetic flux \(\Phi\) and not on the shape of the paths.
2. Practical part: simulating the interference
The aim is to show numerically that a change of the magnetic flux shifts the interference fringes.
The interference pattern
Model the intensity on the screen, \[I(x) = I_0\left[ 1+\cos\!\left( \frac{2\pi d}{\lambda L}x + \frac{q\Phi}{\hbar} \right) \right],\] where \(d\) is the distance between the slits, \(L\) is the distance to the screen and \(\lambda\) is the de Broglie wavelength of the electrons.
Plot \(I(x)\) for several values of \(\Phi\).
Derive this formula.
Animation
Make an animation showing how the interference pattern shifts as the flux changes, \[\Phi \in [0,\,\Phi_0], \qquad \Phi_0=\frac{h}{q}.\]
Monte Carlo
Simulate the arrival of individual electrons on the screen:
draw the coordinates of the hits from the distribution \(I(x)\);
plot histograms for different values of \(\Phi\);
show how the interference pattern builds up statistically.
3. Discussion and physical meaning
Gauge invariance
Explain why the phase of the wave function depends on the choice of gauge, while the observable phase difference \(\Delta\varphi\) does not.
Topological character
Explain why the shift of the pattern is periodic in the flux with period \(\Phi_0\) (an integer number of flux quanta is unobservable; only the fractional part of \(\Phi/\Phi_0\) matters) and is insensitive to local deformations of the paths.
4. What to submit
The theoretical derivation.
An animation of the interference pattern.
A minimal
gitrepository with the code (python) and the report.
References
- Y. Aharonov, D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115, 485 (1959).