A particle with spin 1/2 in a potential well

kisdrA
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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)
 
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@kisdrA you will need to post any relevant equations and show your attempt at a solution.
 
1734856353093.png
 
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 Schrodinger 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?
 
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)##
 
kisdrA said:
@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.
 
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It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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