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Solution to Schrodinger Equation

  1. Jun 5, 2014 #1
    1. The problem statement, all variables and given/known data

    I need Part B of this question

    http://physics.wustl.edu/classes/FL2013/217/homework/ps03.pdf [Broken]

    Recall that the free particle Schr¨odinger equation,
    i~

    ∂tψ(x, t) = −
    ~
    2
    2m

    2
    ∂x2
    ψ(x, t) (1)
    has solutions of the “plane wave” form
    ψk(x, t) = exp[ikx − iω(k)t] , (2)
    where ω(k) = ~k
    2/2m.
    (a) (10 points)
    Consider φ1(x, t) = cos(k0x − ω(k0)t). Show that φ1(x, t) is not a solution of the
    Schr¨odinger equation, i.e. when plugged into both sides of the equation, identity
    does not hold for all x and t as long as k0 6= 0.
    (b) (10 points)
    Find a solution φ2(x, t) to the Schr¨odingier equation that also satisfies φ1(x, 0) =
    φ2(x, 0). (Hint: Write cos(k0x) as the sum of a right moving and a left moving
    plane wave the way we did in class, and use the superposition principle. That
    is, if you know the correct time dependence for each term in the sum, see class
    notes, then you also know the correct time dependence of their sum.)

    2. Relevant equations



    3. The attempt at a solution

    I thought the solution may be φ(x, t)=Ae(-k1x-ω1t)+Ae(-k2x-ω2t)
    but I dont think it works out

    Anyone have any idea how to solve Part B?

    Thanks!
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Jun 5, 2014 #2

    Simon Bridge

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    Welcome to PF;
    How did you go for part (a)?
    You are provided with a hint for part (b), how did you apply it?
    Why did you think your proposed solution may be correct, and what leads you to think that it doesn't work out after all?

    How to type math equations on PF:
    https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

    Schrödinger - you can write the umlaut in with the compose key.
     
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