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QM - Shankar 12.6.1

  1. Mar 10, 2008 #1
    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?

  2. jcsd
  3. Mar 10, 2008 #2


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    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.
  4. Mar 10, 2008 #3

    Thanks, that is the solution.

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

  5. Mar 10, 2008 #4


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    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.
  6. Mar 10, 2008 #5
    I will keep that in mind.

    Thanks again.
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