# 2.1: Small changes to the Hamiltonian

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There are very few problems in quantum mechanics which can be solved exactly. However, we are often interested in the effect of a small change to a system, and in such cases we can proceed by assuming that this causes only a small change in the eigenstates. Perturbation theory provides a method for finding approximate energy eigenvalues and eigenfunctions for a system whose Hamiltonian is of the form

$\hat{H} = \hat{H}_0 + \hat{V} \nonumber$

where $$\hat{H}_0$$ is the ‘main bit’ of the Hamiltonian of an exactly solvable system, for which we know the eigenvalues, $$E_n$$, and eigenfunctions, $$|n\rangle$$, and $$\hat{V}$$ is a small, time-independent perturbation. $$\hat{H}$$, $$\hat{H}_0$$ and $$\hat{V}$$ are Hermitean operators. Using perturbation theory, we can get approximate solutions for $$\hat{H}$$ using as basis functions eigenstates of the similar, exactly solvable system $$\hat{H}_0$$.

Assuming that $$\hat{H}$$ and $$\hat{H}_0$$ possess discrete, non-degenerate eigenvalues only, we write

$\hat{H}_0 |n_i \rangle = E_i |n_i \rangle \nonumber$

in Dirac notation. The states $$|n_i\rangle$$ are orthonormal. WLOG, consider a state $$i = 0$$: the effect of the perturbation will be to modify the state and its corresponding energy slightly; The eigenstate $$|n_0 \rangle$$ will become $$|\phi_0 \rangle$$ and $$E_0$$ will shift to $$E_0 + \Delta E_0$$, where

$\hat{H} |\phi_0 \rangle = E_0 + \Delta E_0 | \phi_0 \rangle \nonumber$

WLOG, expanding $$|\phi_0 \rangle$$ in the basis set $$|n_i\rangle$$ with coefficients $$c_{i0}$$ and premultiplying by $$\langle n_0|$$

$\langle n_k|(\hat{H}_0 + \hat{V} ) \sum_{i=0,\infty} c_{i0}|n_i \rangle = (E_0 + \Delta E_0) \langle n_0| \sum_{i=0, \infty} c_{i0} |n_i \rangle \nonumber$

Which after a little algebra and cancellation yields the exact result:

$\Delta E_0 = \langle n_0| \hat{V} |n_0 \rangle + \sum_{i=1,\infty} (c_{i0}/c_{00}) \langle n_0|\hat{V}|n_i \rangle$

Similarly, expanding $$|\phi_0 \rangle$$ in the basis set $$|n_i \rangle$$ and premultiplying by another state $$\langle n_k|$$

$\langle n_k|(\hat{H}_0 + \hat{V} ) \sum_{i=0,\infty} c_{i0}|n_i \rangle = (E_0 + \Delta E_0) \langle n_k| \sum_{i=0, \infty} c_{i0} |n_i \rangle \nonumber$

leading to $$|\phi_0 \rangle$$ having a component of $$|n_k \rangle$$

$c_{k0} (E_0 + \Delta E_0 − E_k) = \sum_{i=0, \infty} c_{i0} \langle n_k | \hat{V} |n_i \rangle$

Note that although we have denoted the unperturbed state as $$|n_0 \rangle$$, it is not necessarily the ground state.

This page titled 2.1: Small changes to the Hamiltonian is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.