D.39 Time-dependent perturbation theory

The equations to be solved are

$${\rm i}\hbar \dot c_1 = \langle{E_1}\rangle c_1 + H_{12} c... ... {\rm i}\hbar \dot c_2 = H_{21} c_1 + \langle{E}_2\rangle c_2$$

To simplify the use of perturbation theory, it is convenient to use a trick that gets rid of half the terms in these equations. The trick is to define new coefficients $\bar{c}_1$ and $\bar{c}_2$ by

$$\bar c_1 = c_1 e^{{\rm i}\int \langle{E}_1\rangle {\,\rm d... ...= c_2 e^{{\rm i}\int \langle{E}_2\rangle {\,\rm d}t/\hbar} %$$ (D.20)

The new coefficients $\bar{c}_1$ and $\bar{c}_2$ are physically just as good as $c_1$ and $c_2$. For one, the probabilities are given by the square magnitudes of the coefficients, and the square magnitudes of $\bar{c}_1$ and $\bar{c}_2$ are exactly the same as those of $c_1$ and $c_2$. That is because the exponentials have magnitude one. Also, the initial conditions are unchanged, assuming that you choose the integration constants so that the integrals are initially zero.

The evolution equations for $\bar{c}_1$ and $\bar{c}_2$ are

$$\fbox{$\displaystyle {\rm i}\hbar \dot {\bar c}_1 = H_{1... ...{21} e^{ {\rm i}\int E_{21}{\,\rm d}t/\hbar} \bar c_1 $} %$$ (D.21)

with $E_{21}=\langle{E}_2\rangle-\langle{E}_1\rangle$. Effectively, the two energy expectation values have been turned into zero. However, the matrix element is now time-dependent, if it was not already. To check the above evolution equations, just plug in the definition of the coefficients.

It will from now on be assumed that the original Hamiltonian coefficients are independent of time. That makes the difference in expectation energies $E_{21}$ constant too.

Now the formal way to perform time-dependent perturbation theory is to assume that the matrix element $H_{21}$ is small. Write $H_{21}$ as ${\varepsilon}H_{21}^0$ where $\varepsilon$ is a scale factor. Then you can find the behavior of the solution in the limiting process $\varepsilon\to0$ by expanding the solution in powers of $\varepsilon$. The definition of the scale factor $\varepsilon$ is not important. You might identify it with a small physical parameter in the matrix element. But in fact you can take $H_{21}^0$ the same as $H_{21}$ and $\varepsilon$ as an additional mathematical parameter with no meaning for the physical problem. In that approach, $\varepsilon$ disappears when you take it to be 1 in the final answer.

But because the problem here is so trivial, there is really no need for a formal time-dependent perturbation expansion. In particular, by assumption the system stays close to state $\psi_1$, so the coefficient $\bar{c}_2$ must remain small. Then the evolution equations above show that $\bar{c}_1$ will hardly change. That allows it to be treated as a constant in the evolution equation for $\bar{c}_2$. That then allows $\bar{c}_2$ to be found by simple integration. The integration constant follows from the condition that $c_2$ is zero at the initial time. That then gives the result cited in the text.

It may be noted that for the analysis to be valid, $H_{21}t/\hbar$ must be small. That ensures that $\bar{c}_2$ is correspondingly small according to its evolution equation. And then the change in $\bar{c}_1$ from its original value is small of order $(H_{21}t/\hbar)^2$ according to its evolution equation. So the assumption that it is about constant in the equation for $\bar{c}_2$ is verified. The error will be of order $(H_{21}t/\hbar)^3$.

To be sure, this does not verify that this error in $\bar{c}_2$ decays to zero when $E_{21}t/2\hbar$ tends to infinity. But it does, as can be seen from the exact solution,

$$\vert c_2\vert^2 = \left(\frac{\vert H_{21}\vert t}{\hbar}... ... \tilde E_{21} \equiv \sqrt{E_{21}^2 + \vert H_{21}\vert^2}$$

By splitting it up into ranges $\vert E_{21}\vert t/\hbar$ no larger than $\vert H_{21}\vert t/\hbar$ and $\vert E_{21}\vert t/\hbar$ no larger than 1, you can see that the error is never larger than order $(H_{21}t/\hbar)^2$ for $\vert E_{21}\vert t/\hbar$ no larger than 1. And it is of order $(H_{21}t/\hbar)^2/(\vert E_{21}\vert t/\hbar)^2$ outside that range.

Finally, consider the case that the state cannot just transition to one state $\psi_2$ but to a large number $N$ of them, each with its own coefficient $\bar{c}_2$. In that case, the individual contributions of all these states add up to change $\bar{c}_1$. And $\bar{c}_1$ must definitely stay approximately constant for the above analysis to be valid. Fortunately, if you plug the approximate expressions for the $\bar{c}_2$ into the evolution equation for $\bar{c}_1$, you can see that $\bar{c}_1$ stays approximately constant as long as the sum of all the transition probabilities does. So as long as there is little probability of any transition at time $t$, time-dependent perturbation theory should be OK.