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# 11: Time-Independent Perturbation Theory

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Consider the following very commonly occurring problem. The Hamiltonian of a quantum mechanical system is written $H = H_0 + H_1.$ Here, $$H_0$$ is a simple Hamiltonian whose eigenvalues and eigenstates are known exactly. $$H_1$$ introduces some interesting additional physics into the problem, but is sufficiently complicated that when we add it to $$H_0$$ we can no longer find the exact energy eigenvalues and eigenstates. However, $$H_1$$ can, in some sense (which we shall specify more precisely later on), be regarded as small compared to $$H_0$$. Can we find approximate eigenvalues and eigenstates of the modified Hamiltonian, $$H_0+H_1$$, by performing some sort of perturbation expansion about the eigenvalues and eigenstates of the original Hamiltonian, $$H_0$$? Let us investigate.

Incidentally, in this chapter, we shall only discuss so-called time-independent perturbation theory, in which the modification to the Hamiltonian, $$H_1$$, has no explicit dependence on time. It is also assumed that the unperturbed Hamiltonian, $$H_0$$, is time independent.

11: Time-Independent Perturbation Theory is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.