Results of Stationary Perturbation Theory

Review: Results of Stationary Perturbation Theory

For reference, we recapitulate some elementary results from perturbation theory. Assume that the Hamiltonian of a system may be written as the sum of two parts

$H=H_{0}+H^{\prime }$

and that the eigenstates and eigenvalues of $H_{0}$ are known:

$H_{0}|n^{(0)}\rangle =E_{n}^{(0)}|n^{(0)}{\rangle }$

If it is not possible to find the eigenvalues of $H$ exactly, it is possible to write power series expressions for them that converge over some interval. If $H^{\prime }$ is time independent, the problem is stationary and the appropriate perturbation theory is Rayleigh- Schrodinger stationary state perturbation theory, described in most texts in quantum mechanics. We write

$E_{n}=E_{n}^{(0)}+E_{n}^{(1)}+E_{n}^{(2)}+\cdots$

$|n\rangle =|n^{(0)}\rangle +|n^{(1)}\rangle +\cdots$

and express the $(i+1)^{\rm {th}}$ order perturbation in terms of $E^{(i)}$ and $|n^{(i)}{\rangle }$. The energies are given by

$E_{n}^{(m)}=\langle n^{(0)}|H^{\prime }|n^{(m-1)}{\rangle }$

We shall only use the lowest two orders here. The first order results are

$E_{n}^{(1)}=\langle n^{(0)}|H^{\prime }|n^{(0)}{\rangle }$

$|n^{(1)}\rangle =\sum '_{m}{\frac {|m^{(0)}\rangle \langle m^{(0)}|H^{\prime }|n^{(0)}{\rangle }}{E_{n}^{(0)}-E_{m}^{(0)}}}$

The symbol $\sum ^{\prime }$ indicates that the term $m=n$ is excluded. It is understood that the sum extends over continuum states. Note that the state function is nor properly normalized, but that the error is quadratic in $H^{\prime }$.

The second order results are

${\begin{array}{rcl}E_{n}^{(2)}&=&\sum '_{m}{\frac {|\langle m^{(0)}|H^{\prime }|n^{(0)}\rangle |^{2}}{E_{n}^{(0)}-E_{m}^{(0)}}}\\|n^{(2)}{\rangle }&=&\sum '_{m}|m\rangle {\left[{\frac {\langle m|H^{\prime }|n{\rangle }}{E_{n}-E_{m}}}{\left[1-{\frac {\langle n|H^{\prime }|n{\rangle }}{E_{n}-E_{m}}}\right]}+\sum '_{p}{\frac {\langle m|H^{\prime }|p\rangle \langle p|H^{\prime }|n{\rangle }}{(E_{n}-E_{m})(E_{n}-E_{p})}}\right]}\end{array}}$

In second order perturbation theory the effect of a coupling of $n$ and $m$ by $H^{\prime }$ is to push the levels apart, independent of the value of $H_{nm}^{\prime }$. Consequently, states coupled by $H^{\prime }$ always repel each other.