Aharonov–Bohm Effect

The Aharonov–Bohm effect is a quantum interference effect in which a charged particle acquires a measurable phase from electromagnetic potentials around an excluded flux region, even when the magnetic field vanishes everywhere along the particle’s accessible paths.

Formula Hook

For a magnetic flux $\Phi_B$ enclosed by a closed path $C$, the relative phase is

$$\Delta\varphi_{\mathrm{AB}} = \frac{q}{\hbar} \oint_C \mathbf A\cdot d\mathbf r = \frac{q\Phi_B}{\hbar},$$

with orientation and charge-sign conventions understood. The magnetic flux period is

$$\Phi_0=\frac{h}{\lvert q\rvert}.$$

For superconducting condensate phases, where the relevant charge is $2e$, the common flux quantum is $h/(2e)$.

Canonical Home

See Aharonov–Bohm Effect for the geometric gauge-phase explanation. The wave-mechanics introduction is Aharonov–Bohm Effect: First Encounter, and the nearby mathematical language is Holonomy and Homotopy and Winding.

Common Confusions

• The effect is not a local magnetic-force effect along the field-free paths.
• The vector potential is gauge-dependent, but the closed-loop phase modulo $2\pi$ is gauge-invariant.
• The magnetic field is not zero everywhere; the idealized flux is excluded from the accessible region.
• A ring spectrum shifted by enclosed flux is the same phase mechanism in a bound geometry.

Related Entries

• Amplitude
• Commutator
• Quantum Hall Effect

References

• Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Physical Review 115, 485-491, 1959.
• A. Tonomura et al., “Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave,” Physical Review Letters 56, 792-795, 1986.
• M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect, Springer, 1989.