General Time Dependent Perturbations

Assume that we solve the unperturbed energy eigenvalue problem exactly: \bgroup\color{black}$H_0\phi_n=E_n\phi_n$\egroup. Now we add a perturbation that depends on time, \bgroup\color{black}${\cal V}(t)$\egroup. Our problem is now inherently time dependent so we go back to the time dependent Schrödinger equation.

\begin{displaymath}\bgroup\color{black} \left(H_0+{\cal V}(t)\right)\psi(t)=i\hbar{\partial\psi(t)\over\partial t} \egroup\end{displaymath}

We will expand \bgroup\color{black}$\psi$\egroup in terms of the eigenfunctions: \bgroup\color{black}$\psi(t)=\sum\limits_k c_k(t)\phi_ke^{-iE_kt/\hbar}$\egroup with \bgroup\color{black}$c_k(t)e^{-iE_kt/\hbar}=\langle\phi_k\vert\psi(t)\rangle$\egroup. The time dependent Schrödinger equations is

\begin{eqnarray*}
\sum\limits_k\left(H_0+{\cal V}(t)\right)c_k(t)e^{-iE_kt/\hbar...
...mits_k{\partial c_k(t)\over\partial t}e^{-iE_kt/\hbar}\phi_k \\
\end{eqnarray*}


Now dot \bgroup\color{black}$\langle\phi_n\vert$\egroup into this equation to get the time dependence of one coefficient.

\begin{eqnarray*}
\sum\limits_k{\cal V}_{nk}(t)c_k(t)e^{-iE_kt/\hbar}&=&i\hbar{\...
...bar}\sum\limits_k{\cal V}_{nk}(t)c_k(t)e^{i(E_n-E_k)t/\hbar} \\
\end{eqnarray*}


Assume that at \bgroup\color{black}$t=0$\egroup, we are in an initial state \bgroup\color{black}$\psi(t=0)=\phi_i$\egroup and hence all the other \bgroup\color{black}$c_k$\egroup are equal to zero: \bgroup\color{black}$c_k=\delta_{ki}$\egroup.

\begin{eqnarray*}
{\partial c_n(t)\over\partial t}
&=&{1\over i\hbar}\left({\cal...
...mits_{k\neq i}{\cal V}_{nk}(t)c_k(t)e^{i\omega_{nk}t}\right) \\
\end{eqnarray*}


Now we want to calculate transition rates. To first order, all the \bgroup\color{black}$c_k(t)$\egroup are small compared to \bgroup\color{black}$c_i(t)\approx=1$\egroup, so the sum can be neglected.

\begin{eqnarray*}
{\partial c_n^{(1)}(t)\over\partial t}
&=&{1\over i\hbar}{\cal V}_{ni}(t)e^{i\omega_{ni}t} \\
\end{eqnarray*}


\bgroup\color{black}$\displaystyle c_n^{(1)}(t)={1\over i\hbar}\int\limits_0^te^{i\omega_{ni}t'}{\cal V}_{ni}(t')dt'$\egroup
This is the equation to use to compute transition probabilities for a general time dependent perturbation. We will also use it as a basis to compute transition rates for the specific problem of harmonic potentials. Again we are assuming \bgroup\color{black}$t$\egroup is small enough that \bgroup\color{black}$c_i$\egroup has not changed much. This is not a limitation. We can deal with the decrease of the population of the initial state later.

Note that, if there is a large energy difference between the initial and final states, a slowly varying perturbation can average to zero. We will find that the perturbation will need frequency components compatible with \bgroup\color{black}$\omega_{ni}$\egroup to cause transitions.

If the first order term is zero or higher accuracy is required, the second order term can be computed. In second order, a transition can be made to an intermediate state \bgroup\color{black}$\phi_k$\egroup, then a transition to \bgroup\color{black}$\phi_n$\egroup. We just put the first order \bgroup\color{black}$c_k^{(1)}(t)$\egroup into the sum.

\begin{eqnarray*}
{\partial c_n(t)\over\partial t}
&=&{1\over i\hbar}\left({\cal...
...'}
\int\limits_0^{t''}dt'e^{i\omega_{ki}t'}{\cal V}_{ki}(t') \\
\end{eqnarray*}


\bgroup\color{black}$\displaystyle c_n^{(2)}(t)={-1\over \hbar^2}\sum\limits_{k\...
...omega_{nk}t''}
\int\limits_0^{t''}dt'e^{i\omega_{ki}t'}{\cal V}_{ki}(t')$\egroup

* Example: Transitions of a 1D harmonic oscillator in a transient E field.*

Jim Branson 2013-04-22