QM - Shankar 12.6.1

1. Mar 10, 2008

jsc314159

1. The problem statement, all variables and given/known data

Particle described by wavefunction psi(r,theta,phi) = A*Exp[-r/a0] (a0 = constant)

(1) What is the angular momentum content of the state
(2) Assuming psi is an eigenstate in a potential that vanishes as r -> infinity, find E (match leading terms in Schrodinger's equation)
(3) Having found E, consider finite r and find V(r)

2. Relevant equations
Schrodinger's equation in spherical coordinates.

3. The attempt at a solution

(1) The term A, in the wavefunction, is not given to be a function of theta or phi. I am thinking it is a normalization constant. Therefore, apparently there is no theta or phi dependence, l = 0. Does this seem reasonable.

(2) If l = 0, then Schrodinger's equation becomes:

(-h_bar^2/(2*mu) *(1/r^2 * partial/partial_r * r^2 partial/partial_r) + V(r))psi = E*psi

Let V(r) = 0 and plug in the given psi.

The answer if get is E = -h_bar^2/(2*mu*ao^2*r^2) * (r^2 - 2*r*a0).
The book's answer is E = -h_bar^2/(2*mu*ao^2)

The additional terms in my solution come from carring out the differentiation operations on psi. The (1/r^2 * partial/partial_r * r^2 partial/partial_r) on psi gets me 1/r^2)* (r^2 - 2*r*a0). Somehow the book solution eliminates the 2*r*ao term but I do not see how.

If I can figure out where I am going off track on part 2, I think I can manage part 3. Can you help?

jsc

2. Mar 10, 2008

kdv

The potential is NOT zero. they just say that it goes to zero as r goes to infinity. So your solution is only valid when r goes to infinity. Take this limit in your answer and you will agree with the book.

3. Mar 10, 2008

jsc314159

kdv,

Thanks, that is the solution.

How can I learn to see these types of things more effectively?

jsc

4. Mar 10, 2008

kdv

In this case, you simply had to read very carefully the question. That was the key: that they say thatV goes to zero at infinity.

5. Mar 10, 2008

jsc314159

I will keep that in mind.

Thanks again.