The idea of perturbation theory · Quantum Mechanics

Intuition

When a Hamiltonian is one we can solve plus a small extra piece, the answer is the solvable answer plus small corrections. Write the extra piece with a parameter $\lambda$ that switches it on, expand every energy and state in powers of $\lambda$ , and demand that the Schrödinger equation hold at each power separately. The first power gives the first-order corrections, the second power the second, and so on. The expansion works while the extra piece is small compared with the gaps between the unperturbed levels.

A planet’s orbit round the Sun is an ellipse; the pull of the other planets is small, so astronomers compute the ellipse first and then the small corrections to it, order by order. Quantum perturbation theory is the same method, and borrowed its name from astronomy.

The ground energy of an oscillator with an extra $\lambda m\omega^{2}x^{2}$ , in units of $\hbar\omega$ : exactly $\tfrac12\sqrt{1+2\lambda}$ . The first-order line and the second-order parabola follow it closely for small and drift away as grows.

The perturbation expansion

Split the Hamiltonian into a solved part and a small perturbation, and expand each eigenvalue and eigenstate in powers of the strength $\lambda$ of the perturbation.

\begin{aligned}\hat{H}&=\hat{H}_{0}+\lambda\hat{V},\qquad \hat{H}_{0}\ket{n^{(0)}}=E_{n}^{(0)}\ket{n^{(0)}}\\ E_{n}&=E_{n}^{(0)}+\lambda E_{n}^{(1)}+\lambda^{2}E_{n}^{(2)}+\dots\end{aligned}

Properties

The states expand the same way: $\ket{n}=\ket{n^{(0)}}+\lambda\ket{n^{(1)}}+\dots$ , usually with for .

The equations order by order

\begin{aligned}&(\hat{H}_{0}+\lambda\hat{V})\ket{n}=E_{n}\ket{n}\\ &\Longrightarrow\ (\hat{H}_{0}-E_{n}^{(0)})\ket{n^{(1)}}=(E_{n}^{(1)}-\hat{V})\ket{n^{(0)}}\end{aligned}

Put the two expansions into the eigenvalue equation and multiply out. The equation must hold for every small $\lambda$ , so the coefficient of each power of $\lambda$ must balance on its own. The zeroth power is the unperturbed problem; the first power is an equation for the first corrections.

Proof steps

\begin{aligned}&(\hat{H}_{0}+\lambda\hat{V})\bigl(\ket{n^{(0)}}+\lambda\ket{n^{(1)}}+\dots\bigr)\\ &\quad=\bigl(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\dots\bigr)\bigl(\ket{n^{(0)}}+\lambda\ket{n^{(1)}}+\dots\bigr)\end{aligned}

Put the expansions in.

\lambda^{0}:\quad \hat{H}_{0}\ket{n^{(0)}}=E_{n}^{(0)}\ket{n^{(0)}}

The unperturbed problem, already solved.

\begin{aligned}&\lambda^{1}:\\ &\hat{H}_{0}\ket{n^{(1)}}+\hat{V}\ket{n^{(0)}}=E_{n}^{(0)}\ket{n^{(1)}}+E_{n}^{(1)}\ket{n^{(0)}}\end{aligned}

Collect the terms with one power of $\lambda$ .

(\hat{H}_{0}-E_{n}^{(0)})\ket{n^{(1)}}=(E_{n}^{(1)}-\hat{V})\ket{n^{(0)}}

The first-order equation, which the next two lessons solve.

Applications

Practice

Small Compared With What

Perturbation theory needs the extra piece to be small compared with the gaps between the unperturbed levels it connects, not small in some absolute sense.

Try it

When is perturbation theory reliable?