Quantum Mechanics · Lesson 10

Chapter practice

Bring the chapter together: relativistic energies, Lorentz symmetry, the Klein–Gordon and Dirac equations, gamma matrices, spinors, antiparticles, the non-relativistic limit and hydrogen’s fine structure. No worked example sits above the answers.

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E^{2} = p^{2} c^{2} + m^{2} c^{4}

(□ + m^{2}) ψ = 0

(i ℏ γ^{μ} \partial_{μ} - m c) ψ = 0

ψ ˉ = ψ^{†} γ^{0}

Chapter navigation

Complete Fine structure first.

Intuition

Bring the chapter together: relativistic energies, Lorentz symmetry, the Klein–Gordon and Dirac equations, gamma matrices, spinors, antiparticles, the non-relativistic limit and hydrogen’s fine structure. No worked example sits above the answers.

A good map has insets: the whole country small, and the cities large. Relativistic quantum mechanics is the full map; the Schrödinger equation is one of its insets, accurate where speeds are small.

Before you practise

$E^{2}=p^{2}c^{2}+m^{2}c^{4}$ ; $(i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi=0$ with $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$ ; $\chi\approx\frac{\vec{\sigma}\cdot\vec{\pi}}{2mc}\varphi$ .

\Delta E_{nj}=-\frac{\alpha^{4}mc^{2}}{2n^{4}}\Bigl(\frac{n}{j+\frac12}-\frac34\Bigr)

Reminders

Klein–Gordon: spin zero, a density that can be negative.
Dirac: four components, spin $\tfrac12$ built in, $g=2$ , and a positive density.
Pair creation needs $2mc^{2}$ ; annihilation at rest gives two photons of $mc^{2}$ .

Applications