7.3: Non-Degenerate Perturbation Theory
- Page ID
- 1218
Let us now generalize our perturbation analysis to deal with systems possessing more than two energy eigenstates. The energy eigenstates of the unperturbed Hamiltonian, \( H_0\) , are denoted
where \( n\) runs from 1 to \( N\) . The eigenkets \( \vert n\rangle\) are orthogonal, form a complete set, and have their lengths normalized to unity. Let us now try to solve the energy eigenvalue problem for the perturbed Hamiltonian:
We can express \( \vert E\rangle\) as a linear superposition of the unperturbed energy eigenkets,
where the summation is from \( k=1\) to \( N\) . Substituting the above equation into Equation \ref{602}, and right-multiplying by \( \langle m\vert\) , we obtain
where
Let us now develop our perturbation expansion. We assume that
for all \( m\neq k\) , where \( \epsilon\ll 1\) is our expansion parameter. We also assume that
for all \( m\) . Let us search for a modified version of the \( n\) th unperturbed energy eigenstate, for which
and
for \( m\neq n\) . Suppose that we write out Equation \ref{604} for \( m\neq n\) , neglecting terms that are \( O(\epsilon^2)\) according to our expansion scheme. We find that
giving
Substituting the above expression into Equation \ref{604}, evaluated for \( m=n\) , and neglecting \( O(\epsilon^3)\) terms, we obtain
Thus, the modified \( n\) th energy eigenstate possesses an eigenvalue
and a eigenket
Note that
Thus, the modified eigenkets remain orthogonal and properly normalized to \( O(\epsilon^2)\) .
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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