First-Order Perturbation Theory
Most potentials cannot be solved exactly, so a small extra term is treated as a correction to a solved problem. The first-order energy shift is the expectation value of that term in the unperturbed state.
What a learner can do afterwards
- Splits a Hamiltonian into a solved part and a small perturbation
- Computes a first-order energy shift as an expectation value
- States when the method is trustworthy and what makes a perturbation small
1 · Read
Split the full energy rule into a solved part and a small extra push. The solved part is a system you know exactly, like a square well or an oscillator. The extra part is the perturbation.
Add a gentle slope to a square well. The first order shift equals the average of that slope inside the known unperturbed state. You compute the average with the old wave, not a new one.
Trust the recipe only when the extra push is tiny beside the gaps between known levels. A push near a close pair of levels mixes them, and the simple average fails. Small means small against the level spacing.
Work in order: pick the solved system, write down the extra term, average it in the known state, and add the shift. Never solve the full problem from scratch.
Split off the small extra, average it in the known state, and check it is tiny beside the level gaps.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.