Perturbations in Quantum Mechanics

This is the quantum companion to Perturbations in Classical Mechanics.

Setting up perturbation theory

Suppose a Hamiltonian \(\hat H\) cannot be diagonalised exactly, but is close to one that can, \(\hat H_0\). Then we can approximate its spectrum systematically. Write

\[\hat{H}=\hat{H}_0+\lambda\hat{H}',\]

where \(\hat H'\) is the perturbation and \(\lambda\) is a bookkeeping parameter that we set to \(1\) at the end.

For simplicity, assume the spectrum of \(\hat H_0\) is non-degenerate: each eigenvalue \(E_n^{(0)}\) has a one-dimensional eigenspace, spanned by \(\lvert n^{(0)}\rangle\).

We expand the exact eigenvalues and eigenstates as power series in \(\lambda\):

\[E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2 E_n^{(2)}+\cdots\]

and

\[\lvert n\rangle=\left|n^{(0)}\right\rangle+\lambda\left|n^{(1)}\right\rangle+\lambda^2\left|n^{(2)}\right\rangle+\cdots,\]

where the superscript \((j)\) labels the \(j\)-th order correction. (The states \(\lvert n\rangle\) written this way are not normalised in general.)

Substituting into \(\hat H\lvert n\rangle = E_n\lvert n\rangle\):

\[\begin{aligned} & \left(\hat{H}_0+\lambda \hat{H}^{\prime}\right)\left(\left|n^{(0)}\right\rangle+\lambda\left|n^{(1)}\right\rangle+\lambda^2\left|n^{(2)}\right\rangle+\cdots\right) \\ & \quad=\left(E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2 E_n^{(2)}+\cdots\right)\left(\left|n^{(0)}\right\rangle+\lambda\left|n^{(1)}\right\rangle+\lambda^2\left|n^{(2)}\right\rangle+\cdots\right). \end{aligned}\]

First-order energy shift

Comparing the terms of order \(\lambda^1\) gives

\[\left(\hat{H}_0-E_n^{(0)}\right)\left|n^{(1)}\right\rangle =\left(E_n^{(1)}-\hat{H}^{\prime}\right)\left|n^{(0)}\right\rangle .\]

Take the inner product with \(\langle k^{(0)}\rvert\). Since \(\hat H_0\) is Hermitian, \(\langle k^{(0)} \rvert \hat{H}_0 \lvert n^{(1)} \rangle = E_k^{(0)} \langle k^{(0)} \mid n^{(1)} \rangle\), so

\[\left(E_k^{(0)}-E_n^{(0)}\right)\left\langle k^{(0)} \mid n^{(1)}\right\rangle=E_n^{(1)} \delta_{k n}-\left\langle k^{(0)}\right|\hat{H}^{\prime}\left| n^{(0)}\right\rangle .\]

Choosing \(k=n\) makes the left-hand side vanish, giving

\[E_n^{(1)}=\left\langle n^{(0)}\right|\hat{H}^{\prime}\left| n^{(0)}\right\rangle .\]

This is an expectation value: to first order, the shift in energy is the expectation value of the perturbation in the unperturbed state. (Choosing \(k \neq n\) instead gives the first-order correction to the state.)

A classic example

Consider a particle in an infinite square well on \(-a \le x \le a\), perturbed by

\[V(x)=V_0 \cos\left(\frac{\pi x}{2a}\right).\]

What is the first-order shift in the ground-state energy?

The unperturbed ground state is

\[\psi_1(x)=\frac{1}{\sqrt{a}} \cos\left(\frac{\pi x}{2a}\right),\]

so

\[\begin{aligned} \Delta E_1 &= \langle \psi_1 \rvert V \lvert \psi_1 \rangle = \int_{-a}^{a} \psi_1^*(x)\, V(x)\, \psi_1(x)\, dx \\ &=\frac{V_0}{a} \int_{-a}^{a} \cos^3 \left(\frac{\pi x}{2 a}\right) dx \\ &=\frac{2 V_0}{\pi} \int_{-\pi / 2}^{\pi / 2} \cos^3 \theta \, d\theta\\ &=\frac{8 V_0}{3 \pi}, \end{aligned}\]

using the substitution \(\theta = \pi x/2a\). Filling in the integral is left as an exercise.

Credits

These notes are heavily inspired by Prof. Arjun Berera’s courses Introductory Dynamics (PHYS08052) and Principles of Quantum Mechanics (PHYS10094) at the University of Edinburgh.

Changes on 25 October 2026

What was wrong. The expansion parameter \(\lambda\) appeared in the series without being introduced, and the worked example labelled the ground-state shift inconsistently (\(\Delta E_n\), \(\psi^{(n)}\) and \(\psi^1\) for the same thing).

What changed. \(\lambda\) is defined at the start, the example uses \(\psi_1\) and \(\Delta E_1\) throughout, and the bra–ket notation is cleaned up. Previously titled “Perturbations - Quantum Mechanics”.