A particle with spin 1/2 in a potential well

[kisdrA]

Hello everyone. Help me solve the problem. I don't understand how to handle this type of task.

Find the energy levels of a spin 1/2 particle in a potential well: V(r)+W(r)*(l,s), where V(r<a)=-U, V(r>a)=0, W(r) = q*δ(r-a)

 

[PeterDonis]

Moderator's note: Thread moved to advanced physics homework help.

@kisdrA you will need to post any relevant equations and show your attempt at a solution.

 

[kisdrA]

 

[kisdrA]

It is clear that the problem in the potential $V(r)+W(r)*(l,s)$, but without the (l,s) term: $V(r)+W(r)$, simply reduces to solving the radial Schrödinger equation. And I understand how to find energy levels in such a task. But what to do when ls interaction also appears in the delta layer?

 

[kisdrA]

A vector product can be written like this.
$(\overrightarrow{l}, \overrightarrow{s}) = \frac{1}{2} (\overrightarrow{j}^2 - \overrightarrow{l}^2 - \overrightarrow{s}^2)$
Then maybe just transform the stitching condition?
$\Psi^{'}_{II}(a+0) -\Psi^{'}_{I}(a-0) = \frac{2m}{\hbar^{2}}*q*\frac{1}{2}(j(j+1)-l(l+1)-s(s+1)) \Psi_{I,II}(a\pm 0)$

 

[PeterDonis]

View attachment 354756

@kisdrA posting text and equations in images is not allowed here. Please post text and equations directly; use the PF LaTeX feature for equations. There is a LaTeX Guide link at the bottom left of each post window.