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Physics LibreTexts

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

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+ΔE0Ek)=i=0,ci0nk|ˆV|ni

Note that although we have denoted the unperturbed state as |n0, 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.

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