Fermi's golden rule (transition rate) for two widely separated states

Question

My problem has to do with quantum (e.g. electronic) transitions of a single particle between two orthogonal states. I know, for example, that light can couple two orthogonal states in a Hydrogen atom and cause transitions from one state to the other. The electron transition rate going from the 1s state to the 2p in the presence of monochromatic radiation (frequency $\omega $) can be computed from Fermi's golden rule, and is proportional to the factor

$ < 1s|\sin \omega t|2p > $.

The term between the bra and the ket is interaction Hamiltonian ${H_I}\left( {r,t} \right) = {H_I}\left( t \right) \propto \sin \omega t.$ In this specific example, ${H_I}$ is independent of the position variable $r$, though in general ${H_I}$ might depend on r.

In my problem, I wish to consider transitions between two states that are widely separated in space. Not only are these states orthogonal, but they exist in physically distinct regions of space. IOW, if ${\psi _1}\left( r,t \right)$ and ${\psi _2}\left( r,t \right)$ are the spatial representations of the two states, then at $ t = 0 $

${\psi _1}\left( r,0 \right)\;\psi _2^*\left( r,0 \right) = 0$

for all values of $r$, by construction.

In the presence of an interaction Hamiltonian $ {H_I}\left( {r,t} \right) $, Fermi’s golden rule says the transition rate is proportional to

$ \[ < {\psi _1}|{H_I}|{\psi _2} > \quad = \quad \int\limits_{{\rm{all \ space}}} {{\psi _1}\left( r,t \right)\;H\left( {r,t} \right)} \;{\psi _2}\left( r,t \right)\;dr\] $.

My question has two parts:

1. In the special case ${H_I}\left( {r,t} \right) = {H_I}\left( t \right)$, the transition probability is zero when $ t=0 $. I believe that the transition probability remains zero for all subsequent times (but I can't prove it). Is this correct?

2. In the case where ${H_I}$ does depend on $r$, I think the transition probability can be greater than zero. Is this correct? Or is the transition probability always zero even in this case? I am having trouble deciding between these two options.

If the answer in part 2) is that transitions are allowed, then could someone provide an example of an interaction Hamiltonian that makes the transition rate nonzero? Thank you.

Answer 1

Looking back now, it seems obvious to me that if the Hamiltonian is a function of position HI(r,t) and has nonzero overlap with the otherwise widely separated & orthogonal states, then of course transitions between such states is possible. The position variable is just like any other variable, such as angular momentum, and Fermi's golden rule doesn't treat r any differently than another variable. Note, also true for momentum states, though that one is easier to see.

Suppose you have a Hamiltonian that is constant over all space and a function of time: HI=HI(t) . As noted by #Ján Lalinský, if there is no spatial dependence then the Hamiltonian cannot couple two widely separated states.