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Quantum Mechanics Variational Method

  1. Oct 29, 2011 #1

    cep

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    1. The problem statement, all variables and given/known data
    Consider a particle in a box in the interval [-a,a]. Use the trial wavefunction

    ψT = x(a-x2)

    to obtain an approximate energy for the first excited state of the box as a function of a.

    2. Relevant equations

    Schrodinger equation, Hamiltonian for atomic units is 1/2(d2/dx2)

    Normalized energy E = ∫-aaψ*Hψdx/∫-aaψ*ψdx

    3. The attempt at a solution

    Right, so I just plugged in ψT to the energy equation, and after evaluating the integrals, got E = 1-(3/5)a2 / [a(a2/7-1/15)]. The problem is, this function is discontinuous. It is discontinuous at 0, which makes sense to me, and at a = √(7/15), which I don't understand. I'm pretty sure my integrations were done properly... can anyone either explain this discontinuity to me, or tell me why my set-up was flawed?

    Thanks!
     
  2. jcsd
  3. Oct 29, 2011 #2

    cep

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    Okay, I found one error in my integration. The revised function is E = 1-(3/5)a / [a(a^2/7-1/15)]. Still has the same discontinuity problems, though.
     
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