Silicon-Based
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- 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?
$$
g_n = \sum_{l=1,\, l \,\text{odd}}^{n-1 \, (n\, \text{even}),\,n-2 \, (n \,\text{odd})} (2l+1) = \lfloor{n/2}\rfloor(\lfloor{n/2}\rfloor+1)
$$
Is my reasoning above correct? Is it reasonable to assume that the energy eigenvalues have the same form as for the usual hydrogen atom?