4.2.2 Aharonov-Bohm Effect#
Prompts
Describe the Aharonov-Bohm experimental setup. Why does a phase shift occur despite zero magnetic field outside the solenoid?
How does the Aharonov-Bohm effect demonstrate that the vector potential \(\boldsymbol{A}\) has physical meaning in quantum mechanics?
Derive the condition for flux quantization from the requirement that the wavefunction be single-valued around a closed loop.
What is a flux quantum \(\Phi_0 = h/(2e)\)? Why does the factor of 2 appear for superconductors?
How do superconducting devices like SQUIDs use flux quantization to measure magnetic fields with extraordinary sensitivity?
Lecture Notes#
Overview#
The Aharonov-Bohm (AB) effect demonstrates that a charged particle’s quantum phase depends on the electromagnetic vector potential \(\boldsymbol{A}\), even in regions where the magnetic field vanishes. This reveals that the gauge potential is physically real in quantum mechanics—not merely a mathematical convenience. The effect leads directly to flux quantization: in systems with a macroscopic wavefunction (superconductors), magnetic flux is restricted to discrete quantum units \(\Phi = n\Phi_0\).
The Aharonov-Bohm Setup#
A solenoid produces a uniform magnetic field \(\boldsymbol{B}\) inside but zero field outside. An electron beam splits around the solenoid—one path above, one below—then recombines at a screen.
Classical prediction: No effect. The electron experiences no Lorentz force (\(\boldsymbol{F} = -e\boldsymbol{v} \times \boldsymbol{B} = 0\)) in the field-free region, so both paths are identical.
Quantum reality: The interference pattern shifts. The two beams acquire different quantum phases, even though neither experiences a magnetic force.
For a solenoid of radius \(R\) with internal field \(B\) along \(\hat{\boldsymbol{z}}\), the vector potential outside (\(r > R\)) in cylindrical coordinates is
where \(\Phi_B = \pi R^2 B\) is the enclosed flux. The curl vanishes (\(\nabla \times \boldsymbol{A} = 0\)), yet \(\boldsymbol{A} \neq 0\). This is possible because the region is not simply connected—\(\boldsymbol{A}\) cannot be written as a gradient globally.
Aharonov-Bohm Phase Shift
A charged particle traveling around a solenoid acquires a phase shift:
where \(\Phi_B\) is the total magnetic flux enclosed by the path. This phase shift occurs despite \(\boldsymbol{B} = 0\) along the particle’s trajectory.
Two-Path Interference#
In quantum mechanics, a charged particle accumulates phase \(\phi[\mathcal{C}] = (q/\hbar)\int_{\mathcal{C}} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\) from minimal coupling (\(\hat{\boldsymbol{p}} \to \hat{\boldsymbol{p}} - q\boldsymbol{A}\)). Two paths around the solenoid acquire different phases:
where the combined paths form a closed loop encircling the solenoid. The interference pattern shifts by \(\Delta\phi/(2\pi)\) of a fringe spacing.
Topological Invariance
The circulation \(\oint_{\mathcal{C}} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l} = \Phi_B\) depends only on whether the loop encircles the solenoid (winding number), not on its shape or size. By Stokes’ theorem, the circulation equals the enclosed flux \(\int \boldsymbol{B} \cdot \mathrm{d}\boldsymbol{S}\).
Discussion
In classical mechanics, a particle’s trajectory depends only on \(\boldsymbol{E}\) and \(\boldsymbol{B}\) (the Lorentz force). The vector potential \(\boldsymbol{A}\) does not appear. In quantum mechanics, \(\boldsymbol{A}\) enters the Schrödinger equation via minimal coupling, making it observable in gauge-invariant combinations. Is the AB effect a sign that quantum mechanics is “deeper” than classical mechanics, or simply that we must describe particles by wavefunctions rather than trajectories?
Gauge Invariance and Physical Meaning#
Under a gauge transformation \(\boldsymbol{A} \to \boldsymbol{A} + \nabla\chi\), \(\psi \to \mathrm{e}^{\mathrm{i}q\chi/\hbar}\psi\), the phase on a single path changes by \((q/\hbar)[\chi(\text{end}) - \chi(\text{start})]\). But the relative phase between two interfering paths sharing the same endpoints is gauge-invariant:
The Observable is Relative Phase
The relative phase difference between two paths is gauge-invariant and physically measurable. The absolute phase on a single path is gauge-dependent and unobservable. The circulation \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l} = \Phi_B\) is gauge-invariant because it equals the enclosed flux \(\int \boldsymbol{B} \cdot \mathrm{d}\boldsymbol{S}\).
The Vector Potential is Physical in QM
Classical electromagnetism treats \(\boldsymbol{A}\) as a mathematical convenience—only \(\boldsymbol{E}\) and \(\boldsymbol{B}\) are “real.” The AB effect demonstrates that in quantum mechanics, gauge-invariant combinations of \(\boldsymbol{A}\) (like \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\)) have direct observable consequences. This does not contradict gauge invariance; it shows that \(\boldsymbol{A}\) encodes physical information that \(\boldsymbol{B}\) alone does not capture in multiply connected regions.
Flux Quantization#
A quantum wavefunction must be single-valued: traveling around a closed loop must return \(\psi\) to itself (up to a factor \(\mathrm{e}^{2\pi\mathrm{i}n}\)). For a superconductor, the macroscopic order parameter \(\Psi(\boldsymbol{r})\) describing the Cooper pair condensate (charge \(q = 2e\)) must satisfy:
Flux Quantization
Magnetic flux through a superconducting loop is quantized:
The factor of 2 in \(\Phi_0 = h/(2e)\) arises because the charge carriers are Cooper pairs (\(q = 2e\)), not individual electrons. For single-electron systems, \(\Phi_0^{(e)} = h/e\).
Derivation: The Factor of 2
In a normal metal, charge carriers are individual electrons (\(q = e\)), giving \(\Phi_0 = h/e\). In a superconductor below \(T_c\), electrons condense into Cooper pairs—bound pairs with total charge \(2e\) and zero spin. The macroscopic order parameter describes the pair amplitude, so it is the pair charge \(2e\) that enters the single-valuedness condition. This factor was confirmed experimentally by Doll & Näbauer (1961) and Deaver & Fairbank (1961), providing early evidence for Cooper pairing in BCS theory.
Discussion
Flux quantization in a superconductor requires two ingredients: (1) a macroscopic wavefunction with single-valuedness, and (2) Cooper pairing that sets the charge to \(2e\). Does knowing the flux quantum \(\Phi_0 = h/(2e)\) tell us that pairing must occur, or only that paired carriers exist? Could a system have flux quantization with \(\Phi_0 = h/e\)?
Experimental Confirmation#
Aharonov-Bohm effect: Tonomura et al. (1986) used electron holography on tiny solenoids (\(\sim\)500 nm diameter) to directly observe interference pattern shifts as solenoid current varied, in perfect agreement with theory.
Flux quantization: Doll & Näbauer (1961) and Deaver & Fairbank (1961) measured \(\Phi_0 = h/(2e)\) in superconducting tin cylinders. Modern SQUIDs routinely exploit flux quantization with extraordinary precision (see §4.2.3 Flux Ring).
See Also
4.2.1 Berry Phase: Geometric phase in parameter space (AB effect as Berry phase in real space)
4.2.3 Flux Ring: The concrete model of a particle on a ring, persistent currents, and SQUID applications
4.1.1 Gauge Principle: Minimal coupling and gauge freedom
Summary#
The Aharonov-Bohm phase \(\Delta\phi = q\Phi_B/\hbar\) arises from the vector potential in field-free regions.
The relative phase between two paths is gauge-invariant and observable through interference.
The vector potential has physical reality in quantum mechanics: gauge-invariant combinations of \(\boldsymbol{A}\) are directly measurable.
Flux quantization \(\Phi = n\Phi_0\) follows from single-valuedness of the macroscopic wavefunction; \(\Phi_0 = h/(2e)\) for Cooper pairs.
These phenomena are topological: they depend only on the winding number of the path, not microscopic details.
Homework#
1. An electron travels around a solenoid with enclosed flux \(\Phi_B = 10^{-15}\) Wb. (a) Compute the Aharonov-Bohm phase shift \(\Delta\phi = e\Phi_B/\hbar\) in radians. (b) If interference fringes are spaced by one wavelength \(\lambda = 0.1\) nm, by how many fringes does the pattern shift?
2. For a solenoid of radius \(R\) with uniform internal field \(B\) along \(\hat{\boldsymbol{z}}\): (a) Write the vector potential \(\boldsymbol{A}\) outside (\(r > R\)) in cylindrical coordinates. (b) Verify by direct integration that \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l} = \Phi_B\) for a circular loop of radius \(r = 2R\). (c) Use Stokes’ theorem to show this result holds for any closed path encircling the solenoid.
3. Under a gauge transformation \(\boldsymbol{A} \to \boldsymbol{A} + \nabla\chi\), show that: (a) the phase on a single path transforms as \(\phi'[\mathcal{C}] = \phi[\mathcal{C}] + (q/\hbar)[\chi(\text{end}) - \chi(\text{start})]\); (b) the relative phase \(\Delta\phi = \phi_1 - \phi_2\) between two paths sharing the same endpoints is gauge-invariant; (c) explain why this is essential for interpreting the AB effect.
4. Derive the flux quantization condition for a superconductor: (a) Write the single-valuedness constraint \(\oint \nabla\phi \cdot \mathrm{d}\boldsymbol{l} = 2\pi n\), where \(\phi\) is the phase of the macroscopic order parameter. (b) For Cooper pair carriers (\(q = 2e\)), show that \(\Phi = n\Phi_0\) with \(\Phi_0 = h/(2e)\). (c) Calculate \(\Phi_0\) numerically in Weber.
5. In a normal metal, charge carriers are individual electrons (\(q = e\)). (a) What would the flux quantum be? (b) How does the experimentally measured \(\Phi_0 = h/(2e)\) provide evidence for Cooper pairing?
6. The Aharonov-Bohm phase can be understood as a special case of Berry phase (§4.2.1). (a) Identify the “parameter space” for the AB effect—what parameters cycle as the particle traverses the loop? (b) How does the Berry phase perspective illuminate why the AB phase depends on topology (winding number) rather than path details?
7. Describe one experimental confirmation of the Aharonov-Bohm effect (Tonomura et al., 1986): (a) the physical setup, (b) the quantity measured and how it depends on flux, (c) the agreement between theory and experiment.