Time-Independent Perturbation Theory

Time-independent perturbation theory finds how the energy levels and stationary states of a solvable quantum system shift when a small, constant perturbation is added, by expanding the corrections as a power series in the strength of the perturbation.

Definition

Time-independent perturbation theory is the method of expanding the energy eigenvalues and eigenstates of a Hamiltonian as a power series in a small static perturbation added to an exactly solvable Hamiltonian.

Scope

The topic covers the Rayleigh-Schrodinger expansion of energies and states in powers of the perturbation, the first-order energy shift as the expectation of the perturbation, the second-order shift involving sums over intermediate states, the breakdown for degenerate levels and its resolution by diagonalizing the perturbation within the degenerate subspace, and applications such as the Stark and Zeeman effects.

Core questions

• How is the energy shift computed to first and second order in the perturbation?
• How do the stationary states themselves change under the perturbation?
• Why does the standard expansion fail when levels are degenerate?
• How is degeneracy handled by diagonalizing the perturbation in the degenerate subspace?

Key concepts

• perturbation expansion
• first-order energy shift
• second-order energy shift
• energy denominators
• degenerate perturbation theory
• level splitting

Key theories

Rayleigh-Schrodinger expansion

The first-order energy correction is the expectation value of the perturbation in the unperturbed state, while the second-order correction sums contributions from all other states weighted by inverse energy gaps, capturing how the perturbation mixes states.

Degenerate perturbation theory

When several states share an energy the naive series diverges, so one first diagonalizes the perturbation within the degenerate subspace to find the correct zeroth-order states and the splitting of the level, the mechanism behind effects like the linear Stark effect in hydrogen.

Clinical relevance

Time-independent perturbation theory quantifies how external fields and small interactions shift atomic and molecular levels: it predicts the Stark splitting in electric fields, the Zeeman splitting in magnetic fields, and fine-structure corrections, all observable in precision spectroscopy and used to calibrate atomic standards.

History

Schrodinger adapted Rayleigh's classical perturbation methods to wave mechanics in 1926 and immediately applied them to the Stark effect; the framework was soon extended to degenerate cases and became the standard tool for computing spectral shifts.

Key figures

• Lord Rayleigh
• Erwin Schrodinger
• Johannes Stark
• Pieter Zeeman

Seminal works

• sakurai2017
• cohentannoudji2019

Frequently asked questions

What does the first-order energy correction represent?

It is simply the average value of the perturbing interaction in the unperturbed state, the leading estimate of how much the energy level moves, valid when the perturbation is weak compared with the spacing between levels.

Why does degeneracy require special treatment?

With degenerate levels the standard formulas contain vanishing energy denominators and become infinite; one must instead choose the correct linear combinations within the degenerate subspace by diagonalizing the perturbation there, which also reveals how the level splits.