- #1
mathman44
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Dirac "bubble potential"
Consider a radially symmetric delta potential V(r) = −Vo * δ(r − a) with l=0. How many bound states does this system admit?
With l=0, the radial equation reduces to the one dimensional TISE. So, solving the 1D TISE with a delta potential V(r) = −Vo * δ(r − a):
I have [tex]R_{in} = A\exp{kr}[/tex] for r < a
[tex]R_{out} = A\exp{k(2a-r)}[/tex] for r > a
which I obtained my matching the condition R_in = R_out at r=a. Also, the "discontinuity equation" gives me that
[tex]k = \frac{mV_o}{\hbar^2}[/tex]
meaning that there is only one energy and only one bound state. I don't believe this to be correct... especially since the question hints that the number of bound states should depend on "a".
Any help please?
Homework Statement
Consider a radially symmetric delta potential V(r) = −Vo * δ(r − a) with l=0. How many bound states does this system admit?
The Attempt at a Solution
With l=0, the radial equation reduces to the one dimensional TISE. So, solving the 1D TISE with a delta potential V(r) = −Vo * δ(r − a):
I have [tex]R_{in} = A\exp{kr}[/tex] for r < a
[tex]R_{out} = A\exp{k(2a-r)}[/tex] for r > a
which I obtained my matching the condition R_in = R_out at r=a. Also, the "discontinuity equation" gives me that
[tex]k = \frac{mV_o}{\hbar^2}[/tex]
meaning that there is only one energy and only one bound state. I don't believe this to be correct... especially since the question hints that the number of bound states should depend on "a".
Any help please?