Degeneracy of hydrogen energy levels

[Silicon-Based]

Homework Statement

Find the energy degeneracy of a hydrogen atom with infinite potential barrier

Relevant Equations

$|m| \leq l < n$
$P = (-1)^l$

I'm considering a hydrogen atom placed in an infinite potential on one side of the nucleus, i.e. $V(x) = +\infty$ for $x < 0$. I require the wavefunctions to be odd in order to satisfy the boundary condition at $x=0$. By parity of the spherical harmonics only states with $l$ odd are allowed, so the ground state and first excited state should respectively have $n=2$ and $n=4$. Since even $l$ are excluded, the number of possible values of $l$ is $\lfloor{n/2}\rfloor$, so the degeneracy is:

$$
g_n = \sum_{l=1,\, l \,\text{odd}}^{n-1 \, (n\, \text{even}),\,n-2 \, (n \,\text{odd})} (2l+1) = \lfloor{n/2}\rfloor(\lfloor{n/2}\rfloor+1)
$$

Is my reasoning above correct? Is it reasonable to assume that the energy eigenvalues have the same form as for the usual hydrogen atom?

 

[Abhishek11235]

Yes. The differential equation is same as original Hydrogen atom problem with the restriction. As you have correctly determined only odd values are the solutions so as to satisfy boundary conditions. The degeneracy is now over those restricted $l$