User:Espen180/Quantum Mechanics/Time-Independent Perturbation Theory

In all but a small number of cases, a given quantum mechanical system cannot be solved analytically. However, if the system is "close" to some analytically solvable system, we can approximate its solution. In this chapter and the following we will study the methods to accomplish this, called perturbation theory.

Let our system be $H|\psi \rangle =E|\psi \rangle$, and let $H=H_{0}+H_{1}$, where $H_{0}|n\rangle =E_{n}^{0}|n\rangle$ is analytically solvable. We view $H_{1}=H-H_{0}$ as a perturbation of the unperturbed Hamiltonian $H_{0}$. Assume that we know the energies $E_{n}^{0}$ and the eigenstates $|n\rangle$ of the unperturbed system.

We seek the energies and eigenstates of the perturbed Hamiltonian

$H=H_{0}+\lambda H_{1}$

where we have introduced the parameter $\lambda$ to make the bookkeeping simpler.

Perturbation Theory for Non-Degenerate States

Assume first that the states $|n\rangle$ are non-degenerate. We have $H|\psi _{n}\rangle =E_{n}|\psi _{n}\rangle$, giving

$(H_{0}+\lambda H_{1}-E_{n})|\psi _{n}\rangle =0$.

We expand $E_{n}$ and $|\psi _{n}\rangle$ in powers of $\lambda$:

$E_{n}=E_{n}^{0}+\lambda E_{n}^{(1)}+\lambda ^{2}E_{n}^{(2)}+...$,

$|\psi _{n}\rangle =|n\rangle +\lambda |n^{(1)}\rangle +\lambda ^{2}|n^{(2)}\rangle +...$.

Inserting this, we obtain

$(H_{0}+\lambda H_{1}-E_{n}^{0}-\lambda E_{n}^{(1)}-\lambda ^{2}E_{n}^{(2)}-...)(|n\rangle +\lambda |n^{(1)}\rangle +\lambda ^{2}|n^{(2)}\rangle +...)=0$.

By performing the multiplication we obtain a power series, and we require that each of the coefficients are zero. For $\lambda ^{0}$, this gives us

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

which we already knew. The $\lambda ^{1}$ term becomes

$(H_{0}-E_{n}^{0})|n^{(1)}\rangle +(H_{1}-E_{n}^{(1)})|n\rangle =0$.

Multiplying by $\langle n|$ and using that the set of all $|n\rangle$'s form an orthonormal basis set, we get

$\langle n|H_{0}-E_{n}^{0}|n^{(1)}\rangle +\langle n|H_{1}|n\rangle =E_{n}^{(1)}$.

Finally, since $\langle n|H_{0}-E_{n}^{0}|n^{(1)}\rangle =\langle n^{(1)}|H_{0}-E_{n}^{0}|n\rangle ^{*}=\langle n^{(1)}|E_{n}^{0}-E_{n}^{0}|n\rangle ^{*}=0$, we obtain

${\underline {E_{n}^{(1)}=\langle n|H_{1}|n\rangle }}$.

Thus, the first order correction of the $n$-th energy level is simply the expectation value of $H_{1}$ in the $n$-th unperturbed state. We will now find $|n^{(1)}\rangle =\langle m|n^{(1)}\rangle |m\rangle$. To do this, we multiply the $\lambda ^{1}$ term by $\langle m|$, where $m\neq n$. Doing this, we get

$\langle m|H_{0}-E_{n}^{0}|n^{(1)}\rangle +\langle m|H_{1}-E_{n}^{(1)}|n\rangle =0$,

which becomes

$\langle m|n^{(1)}\rangle ={\frac {\langle m|H_{1}|n\rangle }{E_{m}^{0}-E_{n}^{0}}}$.

In addition, we require $\langle n|n^{(1)}\rangle =0$.

Thus, we obtain

${\underline {|n^{(1)}\rangle =\sum _{m\neq n}{\frac {\langle m|H_{1}|n\rangle }{E_{m}^{0}-E_{n}^{0}}}|m\rangle }}$.

Thus we have found the first order approximations $E_{n}\approx E_{n}^{0}+E_{n}^{(1)}$ and $|\psi _{n}\rangle \approx |n\rangle +|n^{(1)}\rangle$.

Moving on to the $\lambda ^{2}$ term, we get

$(H_{0}-E_{n}^{0})|n^{(2)}\rangle +(H_{1}-E_{n}^{(1)})|n^{(1)}\rangle -E_{n}^{(2)}|n\rangle =0$.

Again multiplying by $\langle n|$, we see that the first term vanishes, and we are left with

$E_{n}^{(2)}=\langle n|H_{1}-E_{n}^{(1)})|n^{(1)}\rangle =\langle n|H_{1}|n^{(1)}\rangle$.

If we now insert the expression for $|n^{(1)}\rangle$, we get

${\underline {E_{n}^{(2)}=\sum _{m\neq n}{\frac {\langle n|H_{1}|m\rangle \langle m|H_{1}|n\rangle }{E_{m}^{0}-E_{n}^{0}}}=\sum _{m\neq n}{\frac {(\langle m|H_{1}|n\rangle )^{2}}{E_{m}^{0}-E_{n}^{0}}}}}$

where we have used the fact that $\langle n|H_{1}|m\rangle$ is real.

It is obvious that this appreach does not work for degenerate states because of the energy differences in the denominators. We also get a rough estimate for when our approximation is valid. Namely when

$\lambda |\langle m|H_{1}|n\rangle |<<|E_{m}^{0}-E_{n}^{0}|$.

Perturbation Theory for Degenerate States

There is a simple way to get the above scheme to work for degenerated energy values. Given a $g$ times degenerate energy value with orthonormal states $|\psi _{g}\rangle$, we can choose a new orthonormal set $|\phi _{n}\rangle =\sum _{k=1}^{g}a_{kn}|k\rangle$ such that $\langle \phi _{n}|H_{1}|\phi _{m}\rangle =0$ when $n\neq m$. This removes the terms with zero in the denominator, such that the perturbation is properly defined.