Quantum Mechanics · Lesson 10

Chapter practice

Bring the chapter together: occupation numbers and Fock space, creation and annihilation operators for bosons and for fermions, one-body operators and bands, quantised modes, the scalar and the spinor field, and particles as the quanta of fields. No worked example sits above the answers.

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F = n = 0 ⨁ \infty H^{(n)}

a^^{†} ∣ n ⟩ = n + 1 ∣ n + 1 ⟩

N^= i \sum a^_{i}^{†} a^_{i}

ψ^(x) = i \sum a^_{i} ϕ_{i} (x)

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Note

Prerequisites

Complete Particles as field quanta first.

Intuition

Bring the chapter together: occupation numbers and Fock space, creation and annihilation operators for bosons and for fermions, one-body operators and bands, quantised modes, the scalar and the spinor field, and particles as the quanta of fields. No worked example sits above the answers.

A census counts how many people live in each house without naming them. Second quantisation is a census of particles, taken again whenever one is born or leaves.

Before you practise

$\hat{a}^{\dagger}\ket{n}=\sqrt{n+1}\ket{n+1}$ with $[\hat{a},\hat{a}^{\dagger}]=1$ for bosons; $\{\hat{c},\hat{c}^{\dagger}\}=1$ and $(\hat{c}^{\dagger})^{2}=0$ for fermions; $\hat{F}=\sum\bra{i}f\ket{j}\hat{a}^{\dagger}_{i}\hat{a}_{j}$ .

\hat{H}=\sum_{\mathbf{k}}\hbar\omega_{k}\,\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}},\qquad \hbar\omega_{k}=\sqrt{\hbar^{2}k^{2}c^{2}+m^{2}c^{4}}

Reminders

$N$ bosons in $d$ states: $\binom{N+d-1}{N}$ states; $N$ fermions: $\binom{d}{N}$ ; the fermion Fock space has $2^{d}$ .
The Dirac field needs anticommutators; its vacuum has no electrons and no positrons.
Emission into a mode holding $n$ photons goes as $n+1$ .

Applications