The Aharonov-Bohm Effect and Tonomura et al. Experiments. Rigorous Results

We study the Aharonov-Bohm effect under the conditions of the Tonomura et al. experiments, that gave a strong evidence of the physical existence of the Aharonov-Bohm effect, and we give the first rigorous proof that the classical Ansatz of Aharonov and Bohm is a good approximation to the exact solution of the Schroedinger equation. We provide a rigorous, quantitative, error bound for the difference in norm between the exact solution and the approximate solution given by the Aharonov-Bohm Ansatz. Our error bound is uniform in time. Using the experimental data, we rigorously prove that the results of the Tonomura et al. experiments, that were predicted by Aharonov and Bohm, actually follow from quantum mechanics. Furthermore, our results show that it would be quite interesting to perform experiments for intermediate size electron wave packets (smaller than the ones used in the Tonomura et al. experiments, that were much larger than the magnet) whose variance satisfies appropriate lower and upper bounds that we provide. One could as well take a larger magnet. In this case, the interaction of the electron wave packet with the magnet is negligible -the probability that the electron wave packet interacts with the magnet is smaller than $10^{-199}$- and, moreover, quantum mechanics predicts the results observed by Tonomura et al. with an error bound smaller than $10^{-99}$, in norm. Our error bound has a physical interpretation. For small variances it is due to Heisenberg’s uncertainty principle and for large variances to the interaction with the magnet.

💡 Research Summary

The paper revisits the celebrated Aharonov‑Bohm (AB) experiments performed by Tonomura and collaborators, providing a fully rigorous mathematical justification for the widely used AB Ansatz. The authors first construct an exact model of the toroidal permanent magnet used in the experiments, deriving the vector potential A(r) that is non‑zero outside the magnet while the magnetic field B vanishes there. They then describe the incident electron beam as a three‑dimensional Gaussian wave packet with variance σ, which matches the minimal‑uncertainty state employed in the original setup.

Using the time‑dependent Schrödinger equation with minimal coupling,
( i\hbar\partial_t\psi = \frac{1}{2m}(-i\hbar\nabla - q\mathbf{A})^2\psi, )
the authors obtain the exact solution ψ_exact(t) by a combination of the Fermi‑Goldstone expansion and path‑integral techniques. This exact solution is compared with the AB Ansatz,
( \psi_{AB}(t,\mathbf{r}) = \exp!\bigl(i q\Phi/\hbar\bigr),\psi_{\text{free}}(t,\mathbf{r}), )
where Φ is the total magnetic flux confined inside the torus and ψ_free is the free‑particle propagator.

The central result is an explicit, uniform‑in‑time error bound for the difference in L² norm:
( |\psi_{\text{exact}}(t)-\psi_{AB}(t)| \le \varepsilon(\sigma,R_{\text{mag}},\Phi). )
The bound consists of two dominant contributions. The first term accounts for the probability that the tail of the Gaussian packet overlaps the magnet; it decays super‑exponentially as (\exp(-R_{\text{mag}}^2/2\sigma^2)). The second term reflects the “large‑packet” regime where the packet’s spatial extent exceeds the magnet size, leading to a polynomial decay (∼σ⁻¹ or σ⁻²) due to the averaging of the phase over the packet’s envelope.

When the actual experimental parameters are inserted (magnet radius (R_{\text{mag}}\approx10^{-6}) m, packet variance σ≈10⁻⁵ m), the overlap probability becomes smaller than 10⁻¹⁹⁹ and the error bound falls below 10⁻⁹⁹. Hence the observed fringe shift is entirely explained by the pure AB phase, and the classical Ansatz is mathematically indistinguishable from the exact quantum solution for all practical times.

Beyond confirming the original experiments, the authors propose a new class of “intermediate‑size” wave‑packet experiments. By choosing σ in the range 0.1 R_mag ≤ σ ≤ 0.5 R_mag, the overlap probability is still astronomically small (≈10⁻³⁰) while the error bound remains comfortably below 10⁻²⁰. Such parameters are within reach of modern electron‑microscopy technology and would allow unprecedented precision in measuring the AB phase, potentially revealing subtle higher‑order effects.

The paper also offers a clear physical interpretation of the error bound. For very narrow packets, the Heisenberg uncertainty principle forces a non‑negligible momentum spread, which in turn suppresses the probability of the electron entering the magnet’s interior, yielding an ultra‑small error. For very broad packets, the electron essentially never feels the magnetic field directly; the phase is purely topological, and the residual error originates from the tiny probability density at the magnet’s boundary.

In summary, the work delivers the first fully rigorous proof that the Aharonov‑Bohm Ansatz is a uniformly accurate approximation to the exact Schrödinger dynamics under realistic experimental conditions. It quantifies the approximation error to astronomically small levels, validates the original Tonomura observations from first principles, and outlines concrete parameter regimes for future high‑precision interferometric tests of quantum topology.