Solving General QM Problem: Finding Commuting Operators

[vincebs]

Hmm well this question isn't general except in the sense that I have no idea about how to solve it and hence have no idea what physics it involves!

Two particles

particle one obeys H_1 = (p_1)^2/(2m_1) + V1(r_1)
particle two obeys H_2 = (p_2)^2/(2m_2) + V2(r_2)

The system obeys H = H_1 + H_2 + V(|r1 - r2|)

(the latter is the relative distance between the two particles' positions)
r_1 and r_2 are scalars, r1 and r2 are the respective vectors

1.) Find an operator that commutes with H_1 and state its equation of motion
2.) Find an operator that commutes with H_2 and state its equation of motion
3.) Find an operator that commutes with H (and prove that it does)

How do I go about doing this? I can only think of trivial examples, like H_2 commutes with H_1 and vice-versa, and nothing for H.

There is no other information given, so I can't assume this is the helium atom or something.

 

[meopemuk]

Hmm well this question isn't general except in the sense that I have no idea about how to solve it and hence have no idea what physics it involves!

Two particles

particle one obeys H_1 = (p_1)^2/(2m_1) + V1(r_1)
particle two obeys H_2 = (p_2)^2/(2m_2) + V2(r_2)

The system obeys H = H_1 + H_2 + V(|r1 - r2|)

(the latter is the relative distance between the two particles' positions)
r_1 and r_2 are scalars, r1 and r2 are the respective vectors

1.) Find an operator that commutes with H_1 and state its equation of motion
2.) Find an operator that commutes with H_2 and state its equation of motion
3.) Find an operator that commutes with H (and prove that it does)

How do I go about doing this? I can only think of trivial examples, like H_2 commutes with H_1 and vice-versa, and nothing for H.

There is no other information given, so I can't assume this is the helium atom or something.

One non-trivial example of operator satisfying condition 1.) is angular momentum
$\mathbf{L}_1 = [\mathbf{r}_1 \times \mathbf{p}_1]$. This may give you a clue on how to approach questions 2.) and 3.).

Eugene.

 

[vincebs]

Thanks! Ugh! In fact I had considered angular momentum but I thought that its form was particular to the hydrogen atom. But I guess since there is no phi-dependence on the potential it should work fine. So L1 and L2 work for 1.) and 2.), but how about #3? Will the total angular momentum be conserved? If so, how is it defined? Is it just L = L1 + L2? I tried writing out the Hamiltonians in terms of reduced mass but that didn't seem to do anything.

Is there any classical mechanics reasoning I can use? e.g. with Hamilton's equations?

 

[meopemuk]

but how about #3? Will the total angular momentum be conserved? If so, how is it defined? Is it just L = L1 + L2?

Yes, I think L = L1+L2 should work. You can try to calculate [L,H]=0 explicitly. There is, however, a nicer approach. Try to prove the following equivalent condition

$$H = \exp(\frac{i}{\hbar} (\mathbf{L} \cdot \vec{\phi})) H \exp(-\frac{i}{\hbar} (\mathbf{L} \cdot \vec{\phi}))$$

by noticing that for any vector operator $\mathbf{a}$

$$\exp(\frac{i}{\hbar} (\mathbf{L} \cdot \vec{\phi})) \mathbf{a} \exp(-\frac{i}{\hbar} (\mathbf{L} \cdot \vec{\phi}))$$

is the result of rotation of this vector around axis $\vec{\phi}$.

Eugene.