2.1: Small changes to the Hamiltonian
( \newcommand{\kernel}{\mathrm{null}\,}\)
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
ˆH=ˆH0+ˆV
where ˆH0 is the ‘main bit’ of the Hamiltonian of an exactly solvable system, for which we know the eigenvalues, En, and eigenfunctions, |n⟩, and ˆV is a small, time-independent perturbation. ˆH, ˆH0 and ˆV are Hermitean operators. Using perturbation theory, we can get approximate solutions for ˆH using as basis functions eigenstates of the similar, exactly solvable system ˆH0.
Assuming that ˆH and ˆH0 possess discrete, non-degenerate eigenvalues only, we write
ˆH0|ni⟩=Ei|ni⟩
in Dirac notation. The states |ni⟩ 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 |n0⟩ will become |ϕ0⟩ and E0 will shift to E0+ΔE0, where
ˆH|ϕ0⟩=E0+ΔE0|ϕ0⟩
WLOG, expanding |ϕ0⟩ in the basis set |ni⟩ with coefficients ci0 and premultiplying by ⟨n0|
⟨nk|(ˆH0+ˆV)∑i=0,∞ci0|ni⟩=(E0+ΔE0)⟨n0|∑i=0,∞ci0|ni⟩
Which after a little algebra and cancellation yields the exact result:
ΔE0=⟨n0|ˆV|n0⟩+∑i=1,∞(ci0/c00)⟨n0|ˆV|ni⟩
Similarly, expanding |ϕ0⟩ in the basis set |ni⟩ and premultiplying by another state ⟨nk|
⟨nk|(ˆH0+ˆV)∑i=0,∞ci0|ni⟩=(E0+ΔE0)⟨nk|∑i=0,∞ci0|ni⟩
leading to |ϕ0⟩ having a component of |nk⟩
ck0(E0+ΔE0−Ek)=∑i=0,∞ci0⟨nk|ˆV|ni⟩
Note that although we have denoted the unperturbed state as |n0⟩, it is not necessarily the ground state.