Lecture Notes#

Setup: solenoid in a multiply-connected region#

A long solenoid carries a uniform magnetic field \(\boldsymbol{B}\) confined to its interior; outside the solenoid \(\boldsymbol{B} = 0\). An electron beam is split into two paths \(\mathcal{C}_\text{upper}\) and \(\mathcal{C}_\text{lower}\) that pass around opposite sides of the solenoid and recombine on a screen.

In the field-free region outside the solenoid the vector potential is not zero. Its curl vanishes locally, but the region is multiply connected — any loop enclosing the solenoid cannot be contracted to a point without crossing the field region — so \(\boldsymbol{A}\) cannot be written as the gradient of a single-valued function on the whole field-free region. By Stokes’ theorem, the circulation of \(\boldsymbol{A}\) around any loop encircling the solenoid equals the magnetic flux through any surface that spans the loop,

\[ \oint_\mathcal{C}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l} = \int_\mathcal{S}(\nabla\times\boldsymbol{A})\cdot\mathrm{d}\boldsymbol{S} = \int_\mathcal{S}\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S} = \Phi, \]

regardless of the choice of \(\mathcal{S}\) — the flux is the same through any cap, because \(\nabla\cdot\boldsymbol{B} = 0\) guarantees that no field lines are lost in between.

Classically, the electron experiences no Lorentz force along either path (\(\boldsymbol{F} = q\boldsymbol{v}\times\boldsymbol{B} = 0\)), so the two paths should produce identical interference fringes. The quantum-mechanical result is that the fringe pattern shifts when the flux through the solenoid is changed.

Homework#

1. Invisible solenoid. An electron double-slit experiment is performed around an extremely thin solenoid carrying flux \(\Phi\). The electron never enters the solenoid, so \(\boldsymbol{B} = 0\) everywhere on its path.

(a) Show that the fringe pattern shifts by \(\Delta\Phi_{\mathrm{AB}}/(2\pi)\) fringes, where \(\Delta\Phi_{\mathrm{AB}} = e\Phi/\hbar\). For \(\Phi = h/(2e)\), how many fringes does the pattern shift?

(b) A skeptic says: “The solenoid has fringing fields that leak out and deflect the electron classically.” Design an idealised experiment that rules out this objection. (Hint: what if the solenoid is enclosed in a superconducting shield?)

(c) If \(\Phi\) is increased continuously from \(0\) to \(\Phi_0 = h/e\), the fringe pattern returns to its original position. Explain why the interference intensity is periodic in flux with period \(\Phi_0\), while \(\Phi\) itself remains a continuous parameter.

2. Which path matters. Two paths \(\mathcal{C}_1\) and \(\mathcal{C}_2\) connect points \(A\) and \(B\) in a region where \(\boldsymbol{B} = 0\).

(a) Show that the phase difference \(\Delta\Phi_{\mathrm{AB}} = \Phi_{\mathrm{path}}[\mathcal{C}_1] - \Phi_{\mathrm{path}}[\mathcal{C}_2]\) depends only on whether the combined loop \(\mathcal{C}_1 - \mathcal{C}_2\) encloses flux, not on the shapes of the individual paths.

(b) A third path \(\mathcal{C}_3\) also connects \(A\) to \(B\) but winds twice around the solenoid. What is \(\Phi_{\mathrm{path}}[\mathcal{C}_3] - \Phi_{\mathrm{path}}[\mathcal{C}_1]\) in terms of \(\Phi\)?

(c) Classify all paths from \(A\) to \(B\) by their winding number around the solenoid. How does this classification reflect the multiply-connected topology of the field-free region?

3. Gauge dependence of single paths. Under a gauge transformation \(\boldsymbol{A} \to \boldsymbol{A} + \nabla\alpha\), the phase accumulated along a single path \(\mathcal{C}\) from \(P\) to \(Q\) changes.

(a) Show that \(\Phi_{\mathrm{path}}'[\mathcal{C}] = \Phi_{\mathrm{path}}[\mathcal{C}] + (q/\hbar)[\alpha(Q) - \alpha(P)]\).

(b) Explain why this makes the single-path phase unobservable, but the relative phase between two paths sharing endpoints remains gauge-invariant.

(c) One might claim: “I can always choose a gauge in which \(\boldsymbol{A} = 0\) outside the solenoid, so there is no AB effect.” Find the flaw. Can \(\boldsymbol{A}\) be set to zero everywhere in a multiply-connected region with nonzero enclosed flux by a single-valued gauge function?

4. Surface independence. Stokes’ theorem turns the AB loop integral into a magnetic flux through any surface \(\mathcal{S}\) that spans the loop \(\mathcal{C}\). The result should not depend on which surface one chooses.

(a) Let \(\mathcal{S}_1\) and \(\mathcal{S}_2\) be two oriented surfaces that both span the same loop \(\mathcal{C}\), together enclosing a volume \(V\) with \(\mathcal{S}_1 - \mathcal{S}_2 = \partial V\). Use the divergence theorem to show

(b) Maxwell’s equation \(\nabla\cdot\boldsymbol{B} = 0\) forces the two flux integrals to agree. Conclude that the AB phase is well-defined independently of the surface used in Stokes’ theorem.

(c) Suppose a magnetic monopole of charge \(g\) sat between \(\mathcal{S}_1\) and \(\mathcal{S}_2\), so that \(\nabla\cdot\boldsymbol{B} = \mu_0\rho_m\) with a delta-function source. Compute the discrepancy in terms of \(g\) and explain why the AB phase would be ambiguous absent the Dirac quantization condition introduced in 4.4.2 Dirac Monopole.

5. AB phase as Berry phase. The AB phase is the closed-loop Berry phase of a charged particle whose parameter happens to be its own position.

(a) Identify the parameter space: as the electron’s position \(\boldsymbol{r}\) winds around the solenoid, what plays the role of the parameter \(\boldsymbol{R}\) in the Berry-phase formalism of 4.2.1 Berry Phase?

(b) Show that, for a wavefunction \(\psi(\boldsymbol{r}) = \mathrm{e}^{\mathrm{i}q\int^{\boldsymbol{r}}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l}'/\hbar}\tilde\psi(\boldsymbol{r})\) with \(\tilde\psi\) the gauge-rotated background state, the Berry connection \(\boldsymbol{A}_\text{Berry} = \mathrm{i}\langle\psi(\boldsymbol{r})\vert\nabla_{\boldsymbol{r}}\vert\psi(\boldsymbol{r})\rangle\) reduces to \((q/\hbar)\boldsymbol{A}(\boldsymbol{r})\).

(c) Verify that the closed-loop Berry phase is \(\oint\boldsymbol{A}_\text{Berry}\cdot\mathrm{d}\boldsymbol{r} = q\Phi/\hbar\) and that this answer is independent of the gauge chosen for \(\boldsymbol{A}\).

6. Electric Aharonov-Bohm. A charged particle passes through a region of zero electric field, but the scalar potential \(\phi\) is nonzero and time-dependent.

(a) Two electron wavepackets travel through two tubes at different electrostatic potentials \(V_1\) and \(V_2\) for a time \(T\), with \(\boldsymbol{E} = 0\) everywhere the electrons travel. Show that the relative phase is \(\Delta\Phi = eT(V_1 - V_2)/\hbar\).

(b) Explain why this is analogous to the magnetic AB effect: the potential produces a measurable phase even though no force acts on the particle.

(c) Is the electric AB effect topological in the same sense as the magnetic one? What role does the multiply-connected geometry play in each case?

7. Toroidal magnet. Tonomura’s electron-holography test of the AB effect used a toroidal magnet (donut), not an infinite solenoid: the magnetic field is fully confined to the toroid’s interior, with zero field everywhere outside.

(a) An electron beam threads the central hole of the toroid; a reference beam bypasses it externally; the two recombine on a detector. Sketch the geometry, identify the loop \(\mathcal{C}\) that contributes to the AB phase, and show that \(\Delta\Phi_{\mathrm{AB}} = q\Phi_\text{inside}/\hbar\), where \(\Phi_\text{inside}\) is the flux through the toroid’s interior.

(b) Why is the toroidal geometry — with no field anywhere outside — a cleaner experimental test than the original infinite-solenoid geometry?

(c) Tonomura further coated the toroid with a superconducting shell to flux-trap the field inside. Using the Meissner effect (field expulsion by a superconductor), explain how this rules out the residual objection of stray fringing fields and makes the AB phase fully isolated from any local Lorentz force.