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Perturbation theory
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Approximate solutions for small changes to solvable systems.
Approximate the energy and states of a system with small extra interaction: E ≈ E⁰ + ⟨ψ⁰|H'|ψ⁰⟩. It's the main tool for real atoms and molecules.
Helium atom treated as hydrogen-like plus electron repulsion.
Fails for strong perturbations.
In more detail
Perturbation theory approximates solutions to problems that are close to exactly solvable ones by adding small corrections. It is one of the most useful tools in quantum mechanics.
Real-world uses
- Corrections to hydrogen energy levels are calculated with perturbation theory.
- Quantum chemistry uses perturbation methods for molecular energies.
- Physicists calculate small corrections to atomic energy levels using perturbation theory, which handles weak extra effects.
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Frequently asked questions
When does it work?
When the perturbation is small compared with the unperturbed energy scale.
Does it always converge?
Series often converge for weak perturbations, but not always.
When does perturbation theory work?
When the extra term is small compared with the main part of the Hamiltonian, so the corrections converge quickly.
Related topics in Quantum Mechanics
Spin–orbit coupling · Zeeman effect · Stark effect · Variational principle · Dirac notation · Superposition