---
title: "Degenerate Perturbation Theory and the Variational Method"
description: "When unperturbed states share an energy, the correction has to be found by diagonalising the perturbation inside that subspace. The variational method gives a second handle: any trial state puts an up"
canonical: https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy
source: https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy.md
retrieved: 2026-09-12
---

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# Degenerate Perturbation Theory and the Variational Method

When unperturbed states share an energy, the correction has to be found by diagonalising the perturbation inside that subspace. The variational method gives a second handle: any trial state puts an upper bound on the ground state energy.

Subject: Science · Area: Quantum & Modern Physics · Ages 23 to 24
Page: https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy

## Ready when they can

- Explains why the first-order formula breaks down for degenerate levels and what replaces it
- Diagonalises a small perturbation matrix to get the split levels
- Uses a trial function to bound a ground state energy and judges how tight the bound is

## Lesson: When levels split and how to bound them

The usual first-order formula assumes the unperturbed levels stand apart. When states share one energy, that formula breaks down, because it was built on gaps that are no longer there. Inside that shared-energy subspace, you must diagonalise the perturbation to find the corrections.

Diagonalising a small perturbation matrix gives the split levels directly. Each result is one level the degenerate group breaks into once the perturbation is switched on.

**Example.** Two states share an energy and a small perturbation links them. Write its two-by-two matrix inside that pair, diagonalise it, and read off the two split levels: one pushed up, one pushed down.

**Tip.** The variational method gives a second handle on the ground state: any trial state you write down puts an upper bound on its energy. A better trial state tightens the bound, so judge each bound by how tight it is.

**Recap.** Diagonalise inside the shared subspace to split degenerate levels, and bound the ground state from above with trial states.

## Practice

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

## Needs first

- [Identical Particles and the Symmetry of a Many-Body State](https://lightmysky.com/learn/science/identical-particles-and-the-symmetry-of-a-many-body-state-mt_hns-eEGeHU)
- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)
- [First-Order Perturbation Theory](https://lightmysky.com/learn/science/first-order-perturbation-theory-mt_UsbiiQFOAt)

## Opens up

- [Time-Dependent Perturbation Theory and Fermi's Golden Rule](https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC)