Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: minimal coupling, gauge transformations, the Aharonov–Bohm effect, Landau levels, magnetic moments and the Pauli Hamiltonian, the Zeeman effect, and the magnetism of atoms. No worked example sits above the answers.

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p^\mapsto p^- q A

H^= \frac{( p ^ - q A ) ^{2}}{2 m} + q ϕ

B = \nabla \times A

Δ φ = \frac{q Φ}{ℏ}

Chapter navigation

Intuition

Bring the chapter together: minimal coupling, gauge transformations, the Aharonov–Bohm effect, Landau levels, magnetic moments and the Pauli Hamiltonian, the Zeeman effect, and the magnetism of atoms. No worked example sits above the answers.

A sailor reads the wind from the waves it raises, without ever seeing the wind itself. Classical physics reads the fields; quantum mechanics feels the potentials underneath them, through the phases they leave.

Before you practise

$\hat{H}=\frac{(\hat{\mathbf{p}}-q\mathbf{A})^{2}}{2m}+q\phi$ ; $\psi\to e^{iq\chi/\hbar}\psi$ with $\mathbf{A}\to\mathbf{A}+\nabla\chi$ ; $E_{n}=\hbar\omega_{c}(n+\tfrac12)$ with $\omega_{c}=|q|B/m$ .

g_{J}=1+\frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}

Reminders

The Aharonov–Bohm phase is $q\Phi/\hbar$ ; the flux quantum is $h/e$ .
The spin’s $g$ is 2: $\vec{\mu}_{S}=-2\mu_{B}\hat{\mathbf{S}}/\hbar$ .
Closed shells are diamagnetic; atoms with a moment are paramagnetic, with $\chi\propto1/T$ .

Applications

Practice

Try it

An electron in gallium arsenide moves with the effective mass $0.067m_{e}$ . How many times larger is its Landau-level spacing than a free electron’s in the same field? Give one decimal place.