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Variational Principle Problem

  1. Nov 11, 2013 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m is in a potential of V(x) = Kx4 and the wave function is given as ψ(x)= e^-(ax2) use the variational principle to estimate the ground state energy.

    Part B:
    The true ground state energy wave function for this potential is a symmetric function of x i.e. ψ0(x)=ψ0(-x). Use the result that <ψ0lψ(β)> = 0 along with an approximately chosen wave function, to estimate the energy of the first excited state.



    2. Relevant equations



    3. The attempt at a solution

    Ok so I know how to compute variational method approximations and I have proven the identity <ψ0lψ(β)> = 0 earlier on my assignment and understand the identity as well. What I don't understand is the part that says "Use the result that <ψ0lψ(β)> = 0 along with an approximately chosen wave function, to estimate the energy of the first excited state."

    Again I know that when <ψ0lψ(β)> = 0 the variational principle becomes E1≤ <ψlHlψ>/<ψlψ> but does the problem want me to chose a different wavefunction? And if so how to I go about choosing this new wave-function?
     
  2. jcsd
  3. Nov 11, 2013 #2
    The point is to pick a wavefunction which is orthogonal to your first
    and use that as a trial to read the first excited energy [again variationally].
    The clue is telling you to look at the parity, an even wf. is orthogonal to an odd wf,
    so your trial function should be chosen odd.
     
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