Potential in center of mass for Hydrogen atom

[Yoni V]

Homework Statement

A Hydrogen atom is interacting with an EM plane wave with vector potential
$$\bar A(r,t)=A_0\hat e e^{i(\bar k \cdot \bar r -\omega t)} + c.c.$$
The perurbation to the Hamiltonian can be written considering the proton and electron separately as
$$V(t)=-\sum_{i=1,2}\frac{q_i}{2m}\bar P_i\cdot \bar A(R_i ,t)$$
Write the potential in terms of the center of mass and reduced mass $r,R_c$.

Homework Equations

The Attempt at a Solution

This is just part of the exercise, but the one I'm stuck at. I initially thought I only need to substitute
$$m_1\rightarrow \mu ,\; m_2 \rightarrow M,\;P_1\rightarrow p ,\; P_2 \rightarrow P_C$$ etc. but it doesn't seem to fit the results that follow in the rest of the exercise. Also, I don't know what to make of the charges - how are they supposed to be transformed? Thanks

 

[mark726]

for any help!

you should be familiar with the concept of reduced mass and center of mass. The reduced mass is a concept used in classical mechanics, while the center of mass is used in both classical and quantum mechanics. In this case, the reduced mass is defined as $$\mu=\frac{m_1m_2}{m_1+m_2}$$ and the center of mass is defined as $$R_c=\frac{m_1r_1+m_2r_2}{m_1+m_2}$$ where $m_1$ and $m_2$ are the masses of the proton and electron, respectively, and $r_1$ and $r_2$ are their positions.

To write the potential in terms of the center of mass and reduced mass, you need to substitute the above definitions into the given potential. This will result in a potential that is only dependent on $r$ and $R_c$. The charges do not need to be transformed, as they are already included in the definition of the reduced mass. In this case, the reduced mass takes into account the fact that the proton and electron have different masses and charges.