General Time Dependent Perturbations

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

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

We will expand $\psi$ in terms of the eigenfunctions: $\psi(t)=\sum\limits_k c_k(t)\phi_ke^{-iE_kt/\hbar}$ with $c_k(t)e^{-iE_kt/\hbar}=\langle\phi_k\vert\psi(t)\rangle$. 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 $\langle\phi_n\vert$ 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 $t=0$, we are in an initial state $\psi(t=0)=\phi_i$ and hence all the other $c_k$ are equal to zero: $c_k=\delta_{ki}$.

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

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

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

$$c_n^{(1)}(t)={1\over i\hbar}\int\limits_0^te^{i\omega_{ni}t'}{\cal V}_{ni}(t')dt'$$

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 $t$ is small enough that $c_i$ 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 $\omega_{ni}$ 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 $\phi_k$, then a transition to $\phi_n$. We just put the first order $c_k^{(1)}(t)$ into the sum.

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

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

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