Units#

Review & Summary#

Gauge transformation#

\(\psi \to e^{i\chi(x)}\psi\), \(A_\mu \to A_\mu - \partial_\mu\chi\). Leaves all observables unchanged.

Covariant derivative#

\(D_\mu = \partial_\mu + iA_\mu\). Ensures the Schrödinger equation is gauge-covariant.

Minimal coupling#

\(H = \frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 + q\Phi\).

Aharonov-Bohm effect#

Phase shift \(\Delta\phi = \frac{q}{\hbar}\oint \mathbf{A}\cdot d\mathbf{l} = \frac{q\Phi_B}{\hbar}\) from enclosed magnetic flux, even where \(\mathbf{B}=0\).

Landau levels#

\(E_n = \hbar\omega_c(n+\tfrac{1}{2})\), with cyclotron frequency \(\omega_c = eB/m\) and magnetic length \(\ell_B = \sqrt{\hbar/eB}\).

Dirac quantization#

\(eg = n\hbar c/2\). Magnetic monopoles imply electric charge quantization.

Berry phase#

\(\gamma = i\oint \langle n(\mathbf{R})|\nabla_{\mathbf{R}} n(\mathbf{R})\rangle \cdot d\mathbf{R}\). Geometric phase from adiabatic cyclic evolution.