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.