---
title: "First-Order Perturbation Theory"
description: "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 st"
canonical: https://lightmysky.com/learn/science/first-order-perturbation-theory-mt_UsbiiQFOAt
source: https://lightmysky.com/learn/science/first-order-perturbation-theory-mt_UsbiiQFOAt.md
retrieved: 2026-09-12
---

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# 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.

Subject: Science · Area: Quantum & Modern Physics · Ages 21 to 22
Page: https://lightmysky.com/learn/science/first-order-perturbation-theory-mt_UsbiiQFOAt

## Ready when they can

- 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

## Lesson: Small pushes, small shifts

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Split off the small extra, average it in the known state, and check it is tiny beside the level gaps.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Exclusion Principle and Multi-Electron Atoms](https://lightmysky.com/learn/science/the-exclusion-principle-and-multi-electron-atoms-mt_4tLa-3Av0A)
- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)
- [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO)

## Opens up

- [Degenerate Perturbation Theory and the Variational Method](https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy)