Potential in center of mass for Hydrogen atom

In summary, to write the potential in terms of the center of mass and reduced mass, you need to substitute the definitions into the given potential, resulting in a potential that is only dependent on ##r## and ##R_c##.
  • #1
Yoni V
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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
 
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  • #2
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.
 

FAQ: Potential in center of mass for Hydrogen atom

1. What is the definition of center of mass for a Hydrogen atom?

The center of mass for a Hydrogen atom is the point in space where the average position of the atom's mass is located. It is the point around which the atom's mass is evenly distributed.

2. How is the center of mass calculated for a Hydrogen atom?

The center of mass for a Hydrogen atom can be calculated by taking the average of the positions of the proton and the electron in the atom. Since the proton is much more massive than the electron, the center of mass will be closer to the proton's position.

3. Why is the center of mass important for a Hydrogen atom?

The center of mass is important for a Hydrogen atom because it helps us understand the atom's behavior and interactions with other particles. It is also a key concept in quantum mechanics and studying the energy levels of the atom.

4. How does the center of mass affect the potential energy of a Hydrogen atom?

The center of mass plays a role in determining the potential energy of a Hydrogen atom. The closer the electron is to the center of mass, the lower the potential energy will be. This is because the electron and proton are closer together, resulting in a stronger attractive force between them.

5. Can the center of mass for a Hydrogen atom be changed?

No, the center of mass for a Hydrogen atom cannot be changed. It is a fixed point in space that is determined by the positions of the proton and electron, and these positions cannot be altered without changing the atom's fundamental properties.

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