How Does a Delta Function Potential Affect a Quantum Particle's Radial Equation?

[maria clara]

Homework Statement

write the radial equation for a particle with mass m and angular momentum l=0 which is under the influence of the following potential:
V(r)=-a*delta(r-R)
a,R>0
write all the conditions for the solution of the problem.

Homework Equations

Schroedinger's equation:
Hu=Eu
Hamiltonian: H=p/2m +V = pr/2m+L^2/2mr^2+V(r)

The Attempt at a Solution

since the angular momentum is zero, the radial equation appears as:
(-hbar/2m)(d^2u/dr^2)-a*delta(r-R)u=Eu
the conditions I can think of are:
1) continuity of u
2) u(infinity)= 0 (for u to be square integrable)
3) from integration of Schroedinger's equation on the interval [R-epsilon, R+epsilon] the jump in the first derivative of u at r=R should be -2mau(R)/hbar^2

but there is another condition according to the answers, that is, u(0)=0.
where does this condition come from?

 

[maria clara]

anyone?

 

[malawi_glenn]

you mean why u(0) = 0 ?

This is due to that $$\Psi (r) = \frac{u(r)}{r}$$ so $$u(r)$$ must go to zero faster than r, in order to have a bounded wave function $$\Psi (r)$$.