Derivation of 1st and 2nd Order Perturbation Equations

To keep track of powers of the perturbation in this derivation we will make the substitution $H_1\to\lambda H_1$ where $\lambda$ is assumed to be a small parameter in which we are making the series expansion of our energy eigenvalues and eigenstates. It is there to do the book-keeping correctly and can go away at the end of the derivations.

To solve the problem using a perturbation series, we will expand both our energy eigenvalues and eigenstates in powers of $\lambda$.

$$\begin{eqnarray*} E_n&=&E_n^{(0)}+\lambda E_n^{(1)} +\lambda^2 E_n^{(2)}+... ... ..._{nk}(\lambda)&=&\lambda c_{nk}^{(1)}+\lambda^2 c_{nk}^{(2)}+... \end{eqnarray*}$$

The full Schrödinger equation is

$$\bgroup\color{black}(H_0+\lambda H_1)\left(\phi_n+\sum\limits... ...\phi_n+\sum\limits_{k\neq n}c_{nk}(\lambda)\phi_k\right)\egroup$$

where the $N(\lambda)$ has been factored out on both sides. For this equation to hold as we vary $\lambda$, it must hold for each power of $\lambda$. We will investigate the first three terms.

$\lambda^0$	$H_0\phi_n=E_n^{(0)}\phi_n$
$\lambda^1$	$\lambda H_1\phi_n+H_0\lambda\sum\limits_{k\neq n}c_{nk}^{(1... ...E_n^{(1)}\phi_n+\lambda E_n^{(0)}\sum\limits_{k\neq n}c_{nk}^{(1)}\phi_k$
$\lambda^2$	$H_0\sum\limits_{k\neq n}\lambda^2c_{nk}^{(2)}\phi_k+ \lamb... ...m\limits_{k\neq n}\lambda c_{nk}^{(1)}\phi_k+ \lambda^2 E_n^{(2)}\phi_n$

The zero order term is just the solution to the unperturbed problem so there is no new information there. The other two terms contain linear combinations of the orthonormal functions $\phi_i$. This means we can dot the equations into each of the $\phi_i$ to get information, much like getting the components of a vector individually. Since $\phi_n$ is treated separately in this analysis, we will dot the equation into $\phi_n$ and separately into all the other functions $\phi_k$.

The first order equation dotted into $\phi_n$ yields

$$\bgroup\color{black}\langle\phi_n\vert\lambda H_1\vert\phi_n\rangle=\lambda E_n^{(1)}\egroup$$

and dotted into $\phi_k$ yields

$$\bgroup\color{black} \langle\phi_k\vert\lambda H_1\vert\phi_n... ...)}\lambda c_{nk}^{(1)}= E_n^{(0)}\lambda c_{nk}^{(1)}.\egroup$$

From these it is simple to derive the first order corrections

$$\begin{eqnarray*} \lambda E_n^{(1)}=\langle\phi_n\vert\lambda H_1\vert\phi_n\ran... ...\vert\lambda H_1\vert\phi_n\rangle\over E_n^{(0)}-E_k^{(0)}} \\ \end{eqnarray*}$$

The second order equation projected on $\phi_n$ yields

$$\bgroup\color{black}\sum\limits_{k\neq n}\lambda c_{nk}^{(1)}... ...\vert\lambda H_1\vert\phi_k\rangle= \lambda^2E_N^{(2)}.\egroup$$

We will not need the projection on $\phi_k$ but could proceed with it to get the second order correction to the wave function, if that were needed. Solving for the second order correction to the energy and substituting for $c_{nk}^{(1)}$, we have

$$\bgroup\color{black}\lambda^2 E_n^{(2)}=\sum\limits_{k\neq n}... ...H_1\vert\phi_n\rangle\vert^2 \over E_n^{(0)}-E_k^{(0)}}.\egroup$$

The normalization factor $N(\lambda)$ played no role in the solutions to the Schrödinger equation since that equation is independent of normalization. We do need to go back and check whether the first order corrected wavefunction needs normalization.

$$\begin{eqnarray*} & {1\over N(\lambda)^2}=\langle\phi_n+\sum\limits_{k\neq n}\la... ...\over 2}\sum\limits_{k\neq n}\lambda^2 \vert c_{nk}^{(1)}\vert^2 \end{eqnarray*}$$

The correction is of order $\lambda^2$ and can be neglected at this level of approximation.

These results are rewritten with all the $\lambda$ removed in section 22.1.