Quantum Mechanics · Chapter 14

Time-independent perturbation theory

When a problem is a solvable one plus something small, the answer is the solvable answer plus corrections, order by order. The first-order energy shift and state correction, the second-order shift that always pushes the ground state down, what to do when levels are degenerate, the two-level problem that shows exactly where the expansion breaks, and the Stark effect in hydrogen.

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)} ⟩

Lessons

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05Degenerate perturbation theoryWhen several states share an energy, the ordinary formulas divide by zero. The way out is to choose the unperturbed states wisely. Within the degenerate level any combination is an eigenstate of $\hat{H}_{0}$, and the perturbation picks out particular combinations — those that diagonalise it within the level. On them the first-order shifts are the eigenvalues of the small matrix of the perturbation in the level, and the level splits.Read lesson →

06Two levels, exactly and approximatelyTwo coupled levels can be solved exactly, which makes them the best place to see what perturbation theory gets right and where it breaks. When the coupling is small against the gap, the exact energies agree with the second-order ones; when the gap is zero the exact answer is the degenerate one, split by twice the coupling. In between, the exact formula interpolates smoothly, and the perturbation series — whose radius of convergence is set by the ratio of coupling to half the gap — gives out.Read lesson →

07The Stark effectPut a hydrogen atom in a uniform electric field. The ground state has definite parity, so the first-order shift vanishes, and the energy drops only as the square of the field: the quadratic Stark effect, measuring the atom’s polarisability. The $n=2$ level is different. 2s and 2p have the same energy and opposite parity, the field connects them, and degenerate perturbation theory splits the level linearly in the field — a signature of hydrogen’s accidental degeneracy.Read lesson →

08Chapter practiceBring 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.Read lesson →

Mapping the lesson.

Time-independent perturbation theory

Build the foundation