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Schrodinger equation and free particles

  1. Mar 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Show whether the functions
    psi_I = A cos(kx - wt)
    psi_II = A sin(kx - wt)
    are solutions of Schrodinger equation for a free particle


    2. Relevant equations
    Schrodinger equation

    3. The attempt at a solution
    For psi_I = A cos(kx - wt),
    d2psi_I/dx2 = -Ak2psi[/SUB]I[/SUB]
    dpsi_I/dt = Awsin(kx - wt)
    Substituting into S.E,
    ih_bar Awsin(kx - wt) = ((h_bar)2/2m) Ak2psi_I + 0 psi (free particle so V=O)
    So it doesn't satisfy the S.E.

    For psi_II = A sin(kx - wt),
    d2psi_II/dx2 = -Ak2psi[/SUB]II[/SUB]
    dpsi_II/dt = -Awcos(kx - wt)
    Substituting into S.E,
    ih_bar -Awcos(kx - wt) = ((h_bar)2/2m) Ak2psi_II + 0 psi
    So it doesn't satisfy the S.E either.

    Is this correct? Or is there some way that I'm supposed to manipulate both sides to equal each other? :S
     
  2. jcsd
  3. Mar 19, 2009 #2

    lanedance

    User Avatar
    Homework Helper

    hi kehler

    as you've shown, they will not be a solution of the Time Independent SE for a free particle by themselves...

    How about a linear combination of the 2 states? ie
    [tex] \psi = C\psi_1 + D\psi_2 [/tex]

    what do C & D have to staisfy to be a solution...?
     
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