Transitions in a time-dependent field: why shouldn't the field change rapidly?

$$\newcommand{\ddt}[1]{\frac{d #1}{dt}}$$ This is excerpted from Feynman's lectures' The Ammonia Maser:

[...] Inside the cavity, there will be a time-varying electric field, so the next problem we must discuss is the behavior of a molecule in an electric field that varies with time. We have a completely different kind of a problem—one with a time-varying Hamiltonian. Since $H_{ij}$ depends upon $E,$ the $H_{ij}$ vary with time, and we must determine the behavior of the system in this circumstance. To begin with, we write down the equations to be solved: $$iℏ\ddt{C_1}=(E_0+μE)C_1−AC_2,\\iℏ\ddt{C_2}=−AC_1+(E_0−μE)C_2.\tag{9.36}$$ To be definite, let’s suppose that the electric field varies sinusoidally; then we can write $$E=2E_0\cosωt=E_0(e^{iωt}+e^{−iωt}).\tag{9.37}$$ [...]The best way to solve our equations is to form linear combinations of $C_1$ and $C_2$ as we did before. So we add the two equations, divide by the square root of $2,$ and use the definitions of $C_I$ and $C_{II}$ that we had in Eq. $(9.13).$ We get $$iℏ\ddt{C_{II}}=(E_0−A)C_{II}+μEC_I.\tag{9.38}$$ You’ll note that this is the same as Eq. $(9.9)$ with an extra term due to the electric field. Similarly, if we subtract the two equations $(9.36),$ we get $$iℏ\ddt{C_I}=(E_0+A)C_I+μEC_{II}.\tag{9.39}$$ Now the question is, how to solve these equations? They are more difficult than our earlier set, because $E$ depends on $t;$ and, in fact, for a general $E(t)$ the solution is not expressible in elementary functions. However, we can get a good approximation so long as the electric field is small. First we will write $$C_I=γ_Ie^{−i(E_0+A)t/ℏ}=γ_Ie^{−i(E_I)t/ℏ},\\ C_{II}=γ_{II}e^{−i(E_0−A)t/ℏ}=γ_{II}e^{−i(E_II)t/ℏ}.\tag{9.40}$$ If there were no electric field, these solutions would be correct with $γ_I$ and $γ_{II}$ just chosen as two complex constants. In fact, since the probability of being in state $|I⟩$ is the absolute square of $C_I$ and the probability of being in state $|II⟩$ is the absolute square of $C_{II},$ the probability of being in state $|I⟩$ or in state $|II⟩$ is just $∣γ_I∣^2$ or $∣γ_{II}|^2.$ For instance, if the system were to start originally in state $|II⟩$ so that $γ_I$ was zero and $∣γ_{II}|^2$ was one, this condition would go on forever. There would be no chance, if the molecule were originally in state $|II⟩,$ ever to get into state $|I⟩.$ Now the idea of writing our equations in the form of Eq. $(9.40)$ is that if $μE$ is small in comparison with $A,$ the solutions can still be written in this way, but then $γ_I$ and $γ_{II}$ become slowly varying functions of time—where by “slowly varying” we mean slowly in comparison with the exponential functions. That is the trick. We use the fact that $γ_I$ and $γ_{II}$ vary slowly to get an approximate solution. We want now to substitute $C_I$ from Eq. $(9.40)$ in the differential equation $(9.39),$ but we must remember that $γ_I$ is also a function of $t.$ We have $$iℏ\ddt{C_I}=E_Iγ_Ie^{−iE_It/ℏ}+iℏ\ddt{γ_I}e^{−iE_It/ℏ}.$$ The differential equation becomes $$\left(E_Iγ_I+iℏ\ddt{γ_I}\right)e^{−(i/ℏ)E_I t}=E_Iγ_Ie^{−(i/ℏ)E_I t}+μEγ_{II}e^{−(i/ℏ)E_{II}t}.\tag{9.41}$$ Similarly, the equation in dCII/dt becomes $$\left(E_{II}γ_{II}+iℏ\ddt{γ_{II}}\right)e^{−(i/ℏ)E_{II}t}=E_{II}γ_{II}e^{−(i/ℏ)E_{II}t}+μEγ_Ie^{−(i/ℏ)E_It}.\tag{9.42}$$ Now you will notice that we have equal terms on both sides of each equation. We cancel these terms, and we also multiply the first equation by $e^{+iE_It/ℏ}$ and the second by $e^{+iE_{II}t/ℏ}.$ Remembering that $(E_I−E_{II})= 2A= ℏω_0,$ we have finally, $$iℏ\ddt{γ_I}=μE(t)e^{iω_0t}γ_{II},\\iℏ\ddt{γ_{II}}=μE(t)e^{−iω_0t}γ_I.\tag{9.43}$$ Now we have an apparently simple pair of equations—and they are still exact, of course. The derivative of one variable is a function of time $μE(t)e^{iω_0t},$ multiplied by the second variable; the derivative of the second is a similar time function, multiplied by the first. Although these simple equations cannot be solved in general, we will solve them for some special cases. We are, for the moment at least, interested only in the case of an oscillating electric field. Taking $E(t)$ as given in Eq. $(9.37),$ we find that the equations for $γ_I$ and $γ_{II}$ become $$iℏ\ddt{γ_I}=μE_0[e^{i(ω+ω_0)t}+e^{−i(ω−ω_0)t}]γ_{II},\\iℏ\ddt{γ_{II}}=μE_0[e^{i(ω−ω_0)t}+e^{−i(ω+ω_0)t}]γ_I.\tag{9.44}$$ Now if $E_0$ is sufficiently small, the rates of change of $γ_I$ and $γ_{II}$ are also small. The two $γ$’s will not vary much with $t,$ especially in comparison with the rapid variations due to the exponential terms. These exponential terms have real and imaginary parts that oscillate at the frequency ω+ω0 or ω−ω0. The terms with ω+ω0 oscillate very rapidly about an average value of zero and, therefore, do not contribute very much on the average to the rate of change of γ. So we can make a reasonably good approximation by replacing these terms by their average value, namely, zero. We will just leave them out, and take as our approximation: $$iℏ\ddt{γ_I}=μE_0e^{−i(ω−ω_0)t}γ_{II},\\iℏ\ddt{γ_{II}}=μE_0e^{i(ω−ω_0)t}γ_I.\tag{9.45}$$ Even the remaining terms, with exponents proportional to $(ω−ω_0),$ will also vary rapidly unless $ω$ is near $ω_0.$ Only then will the right-hand side vary slowly enough that any appreciable amount will accumulate when we integrate the equations with respect to t. In other words, with a weak electric field the only significant frequencies are those near $ω_0.$

Though I could easily conceive the equations, I couldn't understand why Feynman always emphasised on a weak electric field that varies slowly with time; $\gamma_I$ & $\gamma_{II}$ must vary slowly according to him, but why? What is the necessity for these conditions? Can anyone please explain why Feynman wanted a slowly-varying weak electric field?

• He does not seem to be saying that $E$ should be slowly varying. Rather, he seems to be saying that the size of $E$ should be small and that the frequency of $E$ should be close to $\omega_0$, which I guess is something like the resonance frequency of the system. Close to resonance, the auxiliary quantities $\gamma_I$ and $\gamma_{II}$ will be slowly varying. Oct 13, 2015 at 13:37
• @user2309840: This should be the answer to me.
– user36790
Oct 13, 2015 at 15:13

why Feynman wanted a slowly-varying weak electric field?

There are two separate relevant requirements pointed out in the transcribed text:

(1): the electric field having small magnitude, whereby the probabilities will vary only "slowly"; i.e. at rates which are

• accessible to practical/experimental study, and therefore relevant; and also

• allowing the approximations/simplifications of the formulas as shown; and separately:

(2): the right hand sides of the last equations, $(9.45)$, varying only "slowly".

I believe that you understood requirement (1) correctly; let's focus on requirement (2):

Clearly the right-hand sides of equations $(9.45)$ do involve the electric field. But comparing to eq. $(9.37)$ which presents the electric field itself you see that these right-hand sides of equations $(9.45)$ are not identical to the electric field. Instead these are functions which involve both $\omega$, the oscillating frequency of the electric field, as well as $\omega_0$, which characterizes the energy difference between the two states.

So by requirement (2) Feynman does not "want a slowly-varying electric field" at all (but instead precisely what's stated there).

What is the necessity for these conditions?

I can give some analogy to illustrate that requirement (2) is sensible on top of requirement (1):

Suppose that you're in the middle of a long flight of stairs. You're supposed to climb upwards or downwards very softly, leisurely, say taking only one stairstep per second. (This restriction on your pace is effectively corresponding to requirement (1) by itself.)

Now, always taking one stairstep per second, you may decide to

• either change the direction of your climbs quite rapidly, say every two seconds. In this case you're going up two steps and immediately down two steps again, and so on. Following this routine, you're not getting very far along the stairway, "effectively".

• or change the direction of your climbs rather less frequently, say only every thirty seconds. In this case you're actually traversing along a much larger stretch of the staircase, taking "large swings", as you alternatingly climb up and down again, at your leisurely pace.

Likewise, requirement (2) leads to large oscillations of the probabilities, which are all the better accessible experimentally. Mathematically, of course, requirement (2) means that $\omega$ should be close or equal to $\omega_0$; the electric field oscillating "in resonance" with the characteristic frequency of the two-state system.

And in this resonance condition "large swings" can be observed, i.e. large differences between the extreme values of the alternating probabilities of one state, or the other.