Interactive simulation of the Aharonov-Bohm effect: adjust magnetic flux through a solenoid to observe quantum phase shift and interference pattern changes in a double-slit experiment
In classical electrodynamics, the electromagnetic force on a charged particle depends only on the local E and B fields. In 1959, Yakir Aharonov and David Bohm predicted a purely quantum mechanical effect: a charged particle acquires a phase shift even when traveling through a region where the magnetic field B = 0, provided the region is not simply connected (i.e., encloses magnetic flux). The phase difference between two paths is δφ = (e/ℏ) ∮ A·dl = (e/ℏ)Φ, where Φ is the enclosed magnetic flux and Φ₀ = h/e ≈ 4.14 × 10⁻¹⁵ Wb is the magnetic flux quantum for electrons. This is a topological phase — it depends only on the topology of the paths relative to the flux, not on the details of the trajectory. The effect was experimentally confirmed by Tonomura et al. (1986) using electron holography with a toroidal magnet shielded by a superconductor, definitively proving that the vector potential A (not just B) has physical significance in quantum mechanics.
In a double-slit experiment, the probability amplitude at a point on the screen is the sum of contributions from both slits: ψ = ψ₁ + ψ₂. The intensity is I = |ψ₁ + ψ₂|² = I₁ + I₂ + 2√(I₁I₂)cos(φ₁ - φ₂ + δφ_AB), where δφ_AB is the Aharonov-Bohm phase shift. When δφ_AB = 0: the central maximum appears at θ = 0 (symmetric). When δφ_AB = π: the central maximum shifts to where a minimum was — constructive and destructive interference swap. When δφ_AB = π/2: the pattern shifts by half a fringe width. The key insight is that shifting δφ by 2π restores the original pattern (gauge invariance), so only the fractional part of Φ/Φ₀ matters physically. The effect demonstrates that in quantum mechanics, the electromagnetic potentials (A, φ) — not just the fields (E, B) — are the fundamental physical quantities.
Fundamental physics: The AB effect proves that the vector potential A is physically real in quantum mechanics, not merely a mathematical convenience. This places gauge theory at the heart of modern physics — the Standard Model is a gauge theory. Berry phase: The AB effect inspired Michael Berry's discovery (1984) of geometric phases, generalizing the concept to any cyclic adiabatic quantum evolution. Topological insulators: Modern condensed matter physics uses AB-type phases to characterize topological materials (quantum Hall effect, topological insulators, Majorana fermions). SQUID magnetometers: Superconducting quantum interference devices use AB-type interference in superconducting loops to measure magnetic fields as weak as 10⁻¹⁵ T — the most sensitive magnetometers ever built. Quantum computing: Topological quantum computers (e.g., using anyons in fractional quantum Hall systems) exploit topological phases for inherently fault-tolerant quantum gates. Electron microscopy: Electron holography uses AB interference to image magnetic domain structures at nanoscale resolution. Molecular electronics: AB oscillations in molecular ring structures demonstrate quantum coherence in nanoscale circuits.
The top canvas shows a double-slit experiment viewed from above. A solenoid (represented by a circle) sits between the two paths from slits to screen. Charged particles travel from the source on the left through the two slits and along curved paths to the detection screen on the right. Use the Magnetic Flux slider (Φ/Φ₀) to adjust the flux through the solenoid. As you increase flux, observe how the interference pattern on the bottom canvas shifts — both the fringe spacing and the phase shift now respond visibly to λ and d. The Phase Shift display shows δφ = 2π(Φ/Φ₀). Start with Zero Flux: the standard double-slit pattern with central maximum. Try Quarter Quantum (Φ/Φ₀ = 0.25): δφ = π/2 and the pattern shifts partway. Half Quantum (Φ/Φ₀ = 0.5): δφ = π and the bright/dark pattern is inverted. Full Quantum (Φ/Φ₀ = 1.0): the pattern looks identical to zero flux due to 2π periodicity. Use Play/Reset to control the particle animation, and toggle Vector Potential to see the curling A field lines around the solenoid — notice that A is non-zero even where B = 0, illustrating the physical reality of the vector potential.