1. The problem statement, all variables and given/known data
Hi! My issue here is that I need to find the bound states (if any) of the potential:
[tex] U(r)=-C\frac{s_1\cdot \hat{r}\, s_2\cdot \hat{r}-s_1\cdot s_2}{r}.[/tex]
Here [itex]s_1[/itex] and [itex]s_2[/itex] are the spins of the two spin-one particles involved in this interaction. The two particles have the same mass.
3. The attempt at a solution
My initial reaction here is to try to diagonalize the numerator, since once this is done we essentially just have the hydrogen atom potential, so the bound states would be trivial to find given the solution to the hydrogen atom.
The second term is easy to do, we can define [itex]J=(s_1+s_2)[/itex] to get [itex]s_1\cdot s_2=\frac{J^2-s_1^2-s_2^2}{2}[/itex]. The other term has been giving me more trouble though. It's possible my approach here is completely wrong.
Any suggestions? I'd prefer an analytic solution, but if it has to be done numerically that's okay too.
Hi! My issue here is that I need to find the bound states (if any) of the potential:
[tex] U(r)=-C\frac{s_1\cdot \hat{r}\, s_2\cdot \hat{r}-s_1\cdot s_2}{r}.[/tex]
Here [itex]s_1[/itex] and [itex]s_2[/itex] are the spins of the two spin-one particles involved in this interaction. The two particles have the same mass.
3. The attempt at a solution
My initial reaction here is to try to diagonalize the numerator, since once this is done we essentially just have the hydrogen atom potential, so the bound states would be trivial to find given the solution to the hydrogen atom.
The second term is easy to do, we can define [itex]J=(s_1+s_2)[/itex] to get [itex]s_1\cdot s_2=\frac{J^2-s_1^2-s_2^2}{2}[/itex]. The other term has been giving me more trouble though. It's possible my approach here is completely wrong.
Any suggestions? I'd prefer an analytic solution, but if it has to be done numerically that's okay too.
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