Perturbation Theory Basics Made Easy with Simple Examples

Perturbation theory is a practical method for solving quantum problems that are almost, but not quite, exactly solvable. We split the Hamiltonian into a known part we can solve perfectly and a small extra part, the perturbation, which slightly shifts the energies and states. Corrections are then added step by step, like adjusting a good estimate.

Very few real systems, perhaps only the box, oscillator and hydrogen atom, have exact solutions. Real atoms have interacting electrons, real molecules sit in electric fields, and real devices have imperfections. Perturbation theory is how physicists still get accurate answers for all of these, making it one of the most used tools in quantum mechanics.

Key Concepts

• Unperturbed system Ĥ₀: the exactly solvable part with known energies E_n⁰ and states ψ_n⁰.
• Perturbation Ĥ′: the small additional term, so the full Hamiltonian is Ĥ = Ĥ₀ + Ĥ′. Small means its effects are much weaker than the level spacings of Ĥ₀.
• First-order correction: the leading shift in energy, usually the dominant improvement over the unperturbed answer.
• Non-degenerate vs degenerate: the simple formulas assume each E_n⁰ belongs to a single state. Degenerate levels need a modified treatment.

Formulas

First-order energy: E_n ≈ E_n⁰ + ⟨ψ_n⁰| Ĥ′ |ψ_n⁰⟩. The correction is the expectation value of the perturbation in the unperturbed state, in joules.

First-order state: ψ_n ≈ ψ_n⁰ + Σ_{m≠n} [⟨ψ_m⁰| Ĥ′ |ψ_n⁰⟩ / (E_n⁰ − E_m⁰)] ψ_m⁰. The mixing is small when the numerator is small compared with the energy denominator. Both formulas fail if Ĥ′ is not small or if levels are degenerate.

Worked Example

An electron sits in the ground state of a 1 nm box with E₁⁰ = 0.38 eV. A weak uniform bump adds a constant perturbation Ĥ′ = 0.02 eV everywhere inside the box. Find the corrected energy.

1. The unperturbed ground state is normalised, so ⟨ψ₁⁰| Ĥ′ |ψ₁⁰⟩ = 0.02 eV × ∫|ψ₁⁰|² dx = 0.02 eV × 1 = 0.02 eV.
2. Corrected energy E₁ ≈ 0.38 + 0.02 = 0.40 eV.
3. A constant shift raises every level by the same 0.02 eV, so the wave functions stay unchanged here, while a position-dependent perturbation would also mix the states slightly.

Common Misconceptions

• Perturbation theory gives exact answers. Wrong. It gives an approximation that improves as more orders are added, and only works when the perturbation is genuinely small.
• It can fix any hard problem. Wrong. Strong perturbations, such as chemical bond formation from free atoms, need other methods because the series may not converge.
• The first-order state correction changes the energy most. Wrong. Energy corrections at first order need only the unperturbed state; state mixing is a smaller, higher refinement.

Real-World Applications

Perturbation theory predicts fine structure of atomic spectra, Stark and Zeeman shifts in electric and magnetic fields, and corrections to semiconductor energy bands. Chemists use it to estimate how molecules respond to fields, and engineers model small defects and impurities in quantum devices before building them.

To summarise, start from an exactly known system, add the small real-world effect as an expectation value, and obtain accurate energies without solving the full problem.