1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding Eigenvalue

  1. Oct 18, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose:

    Ĥ = - (ħ2/(2m))(delta)2 - A/r
    where r = (x2+y2+z2)
    (delta)2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
    A = a constant

    Then, show that a function of the form,

    f(r) = Ce-r/a

    with a, C as constants, is an EIGENFUNCTION of Ĥ provided that the constant a is chosen correctly. Find the correct a and give the eigenvalue.

    2. Relevant equations
    Given above

    3. The attempt at a solution
    Because Ĥ is an energy (Hamiltonian) operator, I put E as an eigenvalue in the following equation: Ĥf(r) = Ef(r)
    So....
    - (ħ2/(2m))(delta)2[Ce-r/a] - A/r = ECe-r/a
    - (ħ2/(2m)) [Ce-r/a/a2 - 2Ce-r/a/(ra)] - A/r = E*Ce-r/a
    ..was what I've been doing...so:

    E = - {(ħ2/(2m)) [Ce-r/a/a2 - 2Ce-r/a/(ra)] - A/r}/(Ce-r/a)

    ..but I am at loss as to if it is the right E eigenvalue and as how to get the "a" value? Also, a question--is {-A/r} in the Hamiltonian operator a potential energy part?
     
  2. jcsd
  3. Oct 18, 2014 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Hint: The energy eigenvalue must be a constant independent of the coordinate r.
     
  4. Oct 18, 2014 #3
    Independent of r? So, does it mean E should be some number based on a, C, and A?

    I have been trying to eliminate "r", but I keep failing:

    - (ħ2/(2m))(delta)2[Ce-r/a] - A/r = ECe-r/a
    - (ħ2/(2m)) (C/a) [1/a - 2/r] - A/(re-r/a) = E*C
    - Cħ2/(2ma) (1/a - 2/r) - A/(re-r/a) = EC

    ...I still haven't managed to eliminate r.......
     
    Last edited: Oct 18, 2014
  5. Oct 18, 2014 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Well, C is a normalization constant and will not matter. You must determine a such that E is independent of r. Being a constant independent of the coordinates is the entire point of E being an eigenvalue to H.
     
  6. Oct 18, 2014 #5
    Wait, I am getting slightly confuse--I thought a was supposed to be some value, without "r" in it?
     
  7. Oct 18, 2014 #6

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Yes, a is also a constant independent of r.

    Note that you have also forgotten the f(r) in the potential energy term in the original post ...
     
  8. Oct 18, 2014 #7
    Could you please clarify by what you mean by:

    I thought -A/r was potential energy term?
     
  9. Oct 18, 2014 #8

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    In this expression, you have included Ce-r/a in all terms except the -A/r term. This term is also a part of the Hamiltonian and must also act on the wave function. In the end, you should be able to divide out this term.
     
  10. Oct 18, 2014 #9
    Ohhhhhhhh, so Hf(x) = - (ħ2/(2m))(delta)2[Ce-r/a] - A[Ce-r/a]/r ? Plus, -Af(x)/r is the potential term, then?

    Thank you for your patience with me!
     
  11. Oct 18, 2014 #10

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Yes. After performing the derivatives you should now be able to fix a such that E is a constant independent of r.
     
  12. Oct 18, 2014 #11
    So, a = Am/ħ2 and E = -ħ2/(2ma2) = -ħ6/(2m3A2)?
     
  13. Oct 18, 2014 #12

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I believe you have made an arithmetic error. You should obtain the inverse of your expression for a. Otherwise I think you now have the correct idea.
     
  14. Oct 18, 2014 #13
    fixed my mistake and got a = ħ2/(Am), and E = -A2m/(2ħ2)

    By the way, just on my own (aside from hw-related question previously), I sketched the function f(r) = Ce-r(Am/ħ2) as a function of r and I am curious if the graph does correspond to the ground state?
     
    Last edited: Oct 18, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted