2.1: Small changes to the Hamiltonian

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.