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One-dimensional linear harmonic oscillator perturbation

  1. Jan 29, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state

    2. Relevant equations

    ψn(x) = [itex]\frac{α}{√π*2n*n!}[/itex]1/2 * e2x2[itex]\frac{1}{2}[/itex]

    E1n = <ψ0n|H'|ψ0n>

    3. The attempt at a solution


    E10 = <ψ00|H'|ψ00> =

    ∫[itex]\frac{α}{√π*2n*n!}[/itex]1/2 * e2x2[itex]\frac{1}{2}[/itex]*[itex]\frac{α}{√π*2n*n!}[/itex]1/2 * e2x2[itex]\frac{1}{2}[/itex]*H'* dx

    Is this right ? What is α in this case ?
     
  2. jcsd
  3. Jan 29, 2014 #2

    vela

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    Your expression for the n-th wavefunction is missing the Hermite polynomial Hn(x). It should be
    $$\psi_n(x) = \left(\frac{a}{\sqrt{\pi}2^n n!}\right)^{1/2} e^{-a^2x^2/2} H_n(ax).$$ Your expression for the first-order energy correction for the ground state turns out to be fine because H0(x)=1. The quantity ##a## should be defined in your notes or textbook. It's the characteristic length scale for the harmonic oscillator.
     
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