Stuck on 6.16: Solving for Wavefunctions in Different Regimes

[ehrenfest]

Homework Statement

I am working on 6.16 at the following site:

http://mikef.org/files/phys_4241_hw14.pdf

I think that the solution given is given is wrong. I can get part a), however, I am just getting stuck on part b). So, the wavefunction in r < r_0 is

R(r) = A/r sin(k_1*r)

and the solution in r > r_0 is given by

R(r) = B/r exp(-k_2*r)

I have no idea how t do what they are asking in part b) since we have three unknowns, A, B and V_0 and only two equations: namely continuity at r_0 of R and R'. Is there something that I am missing?

Homework Equations

The Attempt at a Solution

 

[dwintz02]

What I'm not understanding is the fact that they tell you to assume you are in a bound state, but then they find oscillatory solutions even as r goes to infinity. Did they just say the nuclear force is binding for all r?? Since they are modeling the nuclear force, R(r) should decrease exponentially outside of r0. If you're dying to come up with a solution, use
m(D) = 2.014102 u
m(p) = 1.00727647 u
m(n) = 1.00866501 u

Where 931.502 MeV = 1 u
That should give you the actual binding energy of a deuterium nucleus, although I know this doesn't help you solve it the way they want you to.

 

[ehrenfest]

As I said, I think that solution is totally wrong as you can see from the statement

"Then we must have E + V_0 > 0 or E > V_0."

But I think there is still a way to do the problem...

Here is more of my work:

For r < r_0 the only solution is R(r) = A/r sin (k_1*r), where k_1 =
sqrt((E+V_0)2m)/h-bar.

For r > r_0, I get
R(r) = B/r sin(k_2 r) + C/r cos (k_2 r) if E> 0
and
R(r) = D/r exp(-k_2 r) if E< 0
, where k_2 = sqrt(E2m)/h-bar.

So, for part b, since E is less than 0, I can use R(r) = D/r exp(-k_2 r)
for r > r_0. But, then there are 3 unknowns, A, D and V_0, and I do not
understand how I can solve for all any of them using only continuity.

 

[dwintz02]

Yes, I think it is wrong. It says assume a bound solution but at the beginning of the last paragraph they assume that E > 0 which should not be true for a bound solution in a potential of -V.

But, have you tried using normalization, continuity, and smoothness on Psi to give you 3 equations for your 4 unknowns A, B, C, and V?

 

[ehrenfest]

Yes, I think it is wrong. It says assume a bound solution but at the beginning of the last paragraph they assume that E > 0 which should not be true for a bound solution in a potential of -V.

But, have you tried using normalization, continuity, and smoothness on Psi to give you 3 equations for your 4 unknowns A, B, C, and V?

Part a says that I should not normalize the solutions. Anyway, do you think what I wrote in my last post is correct, reducing it to 3 unknowns?

 

[ehrenfest]

anyone see what is going on here?

 

[ehrenfest]

someone, please!