绰绰的物理实验室

Derivation of the Aharonov-Bohm Phase

The Aharonov-Bohm (AB) effect demonstrates that a charged particle is affected by the vector potential $\mathbf{A}$ even in regions where the magnetic field $\mathbf{B}$ is zero.

Consider a particle moving along a path $\gamma$ in a region where $\mathbf{B}=\mathrm{\nabla }\times \mathbf{A}=0$, but $\mathbf{A}\ne 0$. The Schrödinger equation can be solved by making the ansatz:

$$\psi (\mathbf{r},t)={\psi }_0(\mathbf{r},t)\exp (\frac{iq}{\hslash }{\int }_{{\mathbf{r}}_0}^{\mathbf{r}}\mathbf{A}({\mathbf{r}}^{\prime })\cdot d{\mathbf{l}}^{\prime })$$

where ${\psi }_0$ is the wavefunction in the absence of the vector potential ($\mathbf{A}=0$).

Substituting this ansatz into the Hamiltonian, the extra phase factor generates terms that exactly cancel the $\mathbf{A}$-dependent terms in the kinetic energy operator, verifying that this ansatz is indeed the solution.

The phase difference accumulated by the particle traveling along two different paths (Path 1 and Path 2) around a magnetic flux region is:

$$\mathrm{\Delta }{\phi }_{AB}=\frac{q}{\hslash }({\int }_{\text{Path }1}\mathbf{A}\cdot d\mathbf{l}-{\int }_{\text{Path }2}\mathbf{A}\cdot d\mathbf{l})=\frac{q}{\hslash }\oint \mathbf{A}\cdot d\mathbf{l}$$

Using Stokes' theorem, the closed loop integral of the vector potential is equal to the magnetic flux ${\mathrm{\Phi }}_B$ enclosed by the paths:

$$\mathrm{\Delta }{\phi }_{AB}=\frac{q}{\hslash }{\mathrm{\Phi }}_B=2\pi \frac{{\mathrm{\Phi }}_B}{{\mathrm{\Phi }}_0}$$