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Particle in a potential well of harmonic oscillator

  1. May 19, 2013 #1
    1. The problem statement, all variables and given/known data
    I have a similar problem to this one on Physicsforum from a few years ago.


    2. Relevant equations
    Cleggy has finished part a) saying he gets the answer as
    Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iwt/2+ iψ3(x)exp(-7iwt/2)

    OK
    classical angular frequency ω0 = √C/m for period of oscillation T = 2∏ / ω0
    I note that E0 = ½ ħ ω0




    3. The attempt at a solution
    I have, for a wave packet with equal coefficients for 1st & 2nd stationary state wave function;
    ψA(x,t) = 1/√2 (ψ0(x) e-iwt/2 + ψ1(x) e-3iwt/2)

    This question asks for vaidity for t>0

    I don't get how he got there.
    Could someone expand this please
     
    Last edited: May 19, 2013
  2. jcsd
  3. May 19, 2013 #2
    Suppose [tex]\psi_n(x,t)[/tex] are the eigenstates with energy [tex]E_n[/tex].

    Then a generic state (general solution of the Schrodinger equation) can be written as the superposition of these states:

    [tex]\psi(x,t)=\sum_{n}a_n\psi_n(x,t)=\sum_{n}a_n\psi_n(x)e^{-iE_n t/\hbar}[/tex]

    Now match this with the initial condition to find the coefficients a_n and use the energy states for the harmonic oscillator:

    [tex]E_n=\hbar\omega\left(n+\frac12\right)[/tex]
     
  4. May 19, 2013 #3
    Ah! Yes, I hadn't done reading ahead enough, so hadn't got to that bit.
    Thanks for the pointer, though.
     
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