Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: the perturbation expansion, first-order energies and states, second-order energies and level repulsion, degenerate perturbation theory, the exactly solvable two-level problem and the Stark effect. No worked example sits above the answers.

Foundations8 min readRevision 1

Read freely. Sign in when you want to save your place.

Sign in to save progress

H^= H^_{0} + λ V^

E_{n} = E_{n}^{(0)} + λ E_{n}^{(1)}

E_{n}^{(1)} = ⟨ n ∣ V^∣ n ⟩

∣ ψ ⟩ = ∣ ψ^{(0)} ⟩ + λ ∣ ψ^{(1)} ⟩

Chapter navigation

Intuition

Bring the chapter together: the perturbation expansion, first-order energies and states, second-order energies and level repulsion, degenerate perturbation theory, the exactly solvable two-level problem and the Stark effect. No worked example sits above the answers.

A surveyor measures a field with a rough map and small corrections, one layer at a time. Perturbation theory is surveying in quantum mechanics.

Before you practise

$E_{n}^{(1)}=V_{nn}$ ; $E_{n}^{(2)}=\sum_{m\neq n}|V_{mn}|^{2}/(E_{n}^{(0)}-E_{m}^{(0)})$ ; admixtures $V_{mn}/(E_{n}^{(0)}-E_{m}^{(0)})$ ; degenerate levels diagonalise $W$ within the level.

E_{\pm}=\bar E\pm\sqrt{\Delta^{2}+w^{2}}

Reminders

Second order always lowers the ground state; coupled levels repel.
Odd perturbations have no first-order effect on states of definite parity.
Hydrogen: quadratic Stark shift of 1s, $-\tfrac12\alpha F^{2}$ ; linear shifts $\pm3eFa_{0}$ in $n=2$ .
The series in $w$ converges for $|w|<\Delta$ , half the gap.

Applications

Practice

Try it

An oscillator with $\hbar\omega=1$ , $x_{0}=1$ gains $0.1\hat{x}^{2}$ . What is its ground-state shift to first order?