--- title: "First-Order Perturbation Theory" description: "Split a Hamiltonian into a solved reference part and an added term. For an isolated nondegenerate level, the first-order energy correction is the added term's expectation in the reference state. Coupl" 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # First-Order Perturbation Theory Split a Hamiltonian into a solved reference part and an added term. For an isolated nondegenerate level, the first-order energy correction is the added term's expectation in the reference state. Coupling matrix elements compared with energy gaps provide a first check of weak mixing. 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 - Checks coupling-to-gap ratios, explains their limits, and states how degenerate reference levels require a separate treatment ## Lesson: Small pushes, small shifts Shell models describe levels; here we use state vectors. Write H=H0+V. H0 is solved: H0|n>=E_n^0|n>. V is the added self-adjoint term. Choose orthonormal reference eigenstates |n>, so =1. A matrix element is the overlap of |m> with V|n>; is V's expectation in |n>. A nondegenerate level differs in energy from every other reference level. First order keeps the energy correction linear in V. For an isolated nondegenerate level, ΔE_n= and E_n≈E_n^0+ΔE_n. For a position-dependent potential, ΔE_n=∫|psi_n(x)|² V(x) dx over the allowed region. The reference density supplies the weights. Thus V=F x gives ΔE_n=F. For V=C I, with I the identity, normalization gives ΔE_n=C. This constant offset shifts every energy by C exactly and leaves the eigenstates unchanged. **Example.** An infinite well runs from x=0 to x=L. Each reference energy state's density is symmetric about L/2, so =L/2. Adding V(x)=F x gives ΔE_n=F L/2. For L=12 nm and F=0.02 eV/nm, the shift is 0.12 eV. If the reference energy is 1.8 eV, the corrected energy is 1.92 eV. In a well centered at zero, the specified potential F x instead has zero first-order expectation. *(drawing: Average the added potential in the reference state, then add that shift to the reference energy.)* For another level m, form r_m=||/|E_n^0-E_m^0|. Ratios much less than one are a first check of weak mixing; a ratio near one warns against that assumption. Zero diagonal expectation does not imply zero off-diagonal coupling. A constant offset has no such coupling, even if it exceeds a gap. With many states, combined mixing also matters. These ratios alone do not bound every error or guarantee that higher orders stay small. Reference states with equal energy are degenerate. Form V's matrix within their shared-energy subspace and diagonalize it. This finds combinations that this matrix maps to multiples of themselves. Its eigenvalues give first-order shifts; its eigenvectors select starting combinations. Check coupling to outside levels separately. Nearly equal levels need a joint treatment when their coupling is not small compared with their gap. A narrow gap alone does not force mixing. For a nondegenerate level, the first-order state correction adds reference states |m> with coefficients /(E_n^0-E_m^0), for m different from n. Their magnitudes are the mixing ratios. This state correction is separate from the first-order energy calculation. That energy shift uses in the known reference state. An average in a guessed corrected state does not implement this formula. **Recap.** For an isolated nondegenerate level, use ΔE_n=. Check mixing and higher-order effects. For degenerate levels, diagonalize V within their subspace. *Lesson, practice questions, worked solutions and flashcards written for this dataset.* ## 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) - [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO) ## Optional connections - [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk) ## Opens up - [Degenerate Perturbation Theory and the Variational Method](https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy)