quantum mechanics - How does Kohn's theorem demonstrate that a rotating microwave field can only connect the ground state with the cyclotron mode?

This is a follow-up question to Proof of Kohn's theorem.

I am confused about a point in the answer given by @NowIGetToLearnWhatAHeadIs. It is noted that the perturbing Hamiltonian in Equation 12 of Kohn's paper is not Hermitian. We see in the answer that the rotating microwave field Hamiltonian is actually the Hermitian part of $H'$. We then conclude that since $H'$ is proportional to $P_+$, a rotating microwave field can only excite the cyclotron mode as demonstrated by Kohn.

I am missing a step in this logic. In particular, since the rotating microwave field Hamiltonian is not actually $H'$ but only the Hermitian part, it seems that the phrase in Kohn "We see that the perturbation connects the state $\Psi_0$ with and only with, the state $\Psi_1$" is false. How can the rotating microwave field connect these two states when it is only a part of $H'$ and presumably not proportional to $P_+$?

Answer

To understand what is going on, you need to understand something called unitarity. Unitarity basically just says that anything that can happen in forwards in time can also happen backwards in time. So in this case, unitarity means that if the particle can go from $\Psi_0$ to $\Psi_1$, then it can also go from $\Psi_1$ to $\Psi_0$. Now what does that have to do with your question?.

Well when you add the hamiltonian $H'$, you are only adding the piece proportional to $P_+$. Therefore you are only adding the potential for the system to go from $\Psi_0$ to $\Psi_1$, but not the other way around. This a problem because now our system can go only one way, so it is not unitary. We will see that by taking the hermitian peice, we will automatically add in the reverse process and so the theory will become unitary.

So let's take the Hermitian part of $H'$. The important part of $H'$ to focus on is $P_+$. Now remember taking the Hermitian part is equivalent to adding the hermitian conjugate. In this case, the hermitian conjugate of $P_+$ is $P_-$. $P_-$ gives you the transition from $\Psi_1$ to $\Psi_0$. Therefore we can directly see that adding the hermitian conjugate gives the reverse transition that the original $H'$ had. Thus it restored unitarity. This is consistent with the fact that we expected it to restore unitarity by making the hamiltonian hermitian.

Now many authors, including this one, take unitarity for granted. So when they want to add a new transition, they will only explicity mention one direction (e.g., this author only gave $H'$ and not its hermitian conjugate). However, they implicity mean to add the reverse transition as well (i.e., the hermitian conjugate of $H'$), but they do not explicity say it because the reverse transition is implied by unitarity.

But notice that the presence of this reverse transition does not affect his argument at the end. All he needs to say is that $H'$ connects $\Psi_0$ and $\Psi_1$. He doesn't specifically say that $H'$ gives a transition from $\Psi_0$ to $\Psi_1$. This is of course because he knows that the theory needs to be unitary so you can't have such a directional transition. The author knows $H'$ gives a transition from $\Psi_1$ to $\Psi_0$ as well as a transition from $\Psi_0$ to $\Psi_1$. But this is all he needs to say.