Set of Commuting Observables for pures states 2p-1,2px and 2s

[DielsAlder]

Hi,

Here I have a question, apparently easy, but that I think it is a bit tricky.

Homework Statement

Indicate how can a hydrogen atom be prepared in the pure
states corresponding to the state vectors ψ2p-1 and ψ2px and
ψ2s. It is assumed that spin-related observables are not
relevant.

Hint: The state ψ2px becomes the state ψ2pz = ψ2p0 by a rotation
of 90º of the cartesian axis system

Homework Equations

Just remind the Hamiltonian H, L² and L_z operators and their commutation relation.

The Attempt at a Solution

For the ψ2p-1 pure state, it is clear that H, L² and L_z form a CSCO.

For the ψ2px pure state, H and L² are commuting observables, but not L_z because px=1/2^1/2(p+1-p-1) and it is not an eigenfunction of Lz. Could yoy give me any additional hint in order to solve it?

For the ψ2s, L² is equal to zero, so no angular momentum can be measured. Which observable can I use apart form hamiltonian?

Thanks in advance

 

[mark726]

Hi there,

To prepare a hydrogen atom in the pure states ψ2p-1 and ψ2px, you can use the technique of optical pumping. This involves shining a circularly polarized light on the atom, which will cause the atom to absorb photons with a specific spin and angular momentum. By carefully choosing the frequency and polarization of the light, you can prepare the atom in the desired state.

For the ψ2s state, since L2 is equal to zero, you can use the Hamiltonian to prepare the atom in this state. You can apply a potential that corresponds to the ψ2s state, which will cause the atom to be in this pure state. Alternatively, you can also use a laser with a specific frequency and polarization to excite the atom to the ψ2s state.

I hope this helps! Let me know if you have any further questions.