1. Not finding help here? Sign up for a free 30min 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!

Variation Method

  1. Aug 22, 2008 #1
    1. The problem statement, all variables and given/known data
    This is the problem 8.10 from Levine's Quantum Chemistry 5th edition:
    Prove that, for a system with nondegenerate ground state, [tex]\int \phi^{*} \hat{H} \phi d\tau>E_{1}[/tex], if [tex]\phi[/tex] is any normalized, well-behaved function that is not equal to the true ground-state wave function. Hint: Let [tex]b[/tex] be a positive constant such that [tex]E_{1}+b<E_{2}[/tex]. Turn (8.4) into an inequality by replacing all [tex]E_{k}[/tex]'s except [tex]E_{1}[/tex] with [tex]E_{1}+b[/tex].


    2. Relevant equations

    Equation (8.4):
    [tex]\int \phi^{*} \hat{H} \phi d\tau=\sum_{k}a^{*}_{k}a_{k}E_{k}=\sum_{k}|a_{k}|^{2}E_{k}[/tex]​

    Other relevant equations:

    [tex]\phi=\sum_{k}a_{k}\psi_{k}[/tex]​

    where

    [tex]\hat{H}\psi_{k}=E_{k}\psi_{k}[/tex]​


    [tex]1=\sum_{k}|a_{k}|^{2}[/tex]


    [tex]E_{1}<E_{2}<E_{3}...[/tex]​

    3. The attempt at a solution

    [tex]\int \phi^{*} \hat{H} \phi d\tau=|a_{1}|^{2}E_{1}+\sum^{\infty}_{k=2}|a_{k}|^{2}E_{k}>|a_{1}|^{2}E_{1}+\sum^{\infty}_{k=2}|a_{k}|^{2}\left(E_{1}+b\right)=|a_{1}|^{2}E_{1}+E_{1}\sum^{\infty}_{k=2}|a_{k}|^{2}+b\sum^{\infty}_{k=2}|a_{k}|^{2}=E_{1}\sum_{k}|a_{k}|^{2}+b\sum^{\infty}_{k=2}|a_{k}|^{2}[/tex]
    [tex]\int \phi^{*} \hat{H} \phi d\tau>E_{1}+b\sum^{\infty}_{k=2}|a_{k}|^{2}[/tex]

    I don't know how to apply the condition that [tex]\phi\neq \psi_{1}[/tex] to complete the proof, also I'm not sure if this is the right way to start but that's how I understand the hint given. If you need more information or something is not clear, please tell me so I can do the proper correction.
     
    Last edited: Aug 22, 2008
  2. jcsd
  3. Aug 22, 2008 #2

    Avodyne

    User Avatar
    Science Advisor

    You just need to show that your final sum is not zero. If it was zero, what would that tell us about each [itex]a_k[/itex], [itex]k\ge 2[/itex]? And what would that tell us about [itex]\phi[/itex]?

    Minor point: your [itex]>[/itex] sign should really be [itex]\ge[/itex] to account for this case.
     
  4. Aug 22, 2008 #3
    But if the last sum is not zero that mean that there is a mistake somewhere since the purpose is to obtain [tex]\int \phi^{*} \hat{H} \phi d\tau>E_{1}[/tex] right?
     
  5. Aug 22, 2008 #4
    I think that it is easy to show that the last sum is not zero because if it was zero that would mean that [tex]\phi = \psi_{1}[/tex] according to the equations
    [tex]1=\sum_{k}|a_{k}|^{2}[/tex]

    [tex]\phi = \sum_{k}a_{k}\psi_{k}[/tex]​

    But as the problem statement says, [tex]\phi\neq \psi_{1}[/tex], so the sum can't be zero. So far, I have not been able to find the mistake (since as I said before the second term must be eliminated to complete the proof). Any help would be appreciated.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Variation Method
  1. Method of variation (Replies: 1)

  2. Variational method. (Replies: 0)

Loading...