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Applying particle in a box boundaries to an eigenfunction

  1. Jan 27, 2017 #1
    1. The problem statement, all variables and given/known data
    Use the eigenfunction Ψ(x) =A'eikx + B'e-ikx rather than Ψ(x)=Asinkx + Bcoskx to apply the boundary conditions for the particle in a box. A. How do the boundary conditions restrict the acceptable choices for A’ and B’ and for k? B. Do these two functions give different probability density if each is normalized?

    2. Relevant equations
    ?

    3. The attempt at a solution
    I know that the limits are Ψ(0)=Ψ(a)=0, Ψ(x)=0

    When I apply that to the equation
    Ψ(0) =A'eik0 + B'e-ik0 = A'+B'= 0
    and
    Ψ(a) =A'eika + B'e-ika

    And I don't know what to do from here. Any help would be appreciated
     
  2. jcsd
  3. Jan 27, 2017 #2

    TSny

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    Gold Member

    Ψ(0)=Ψ(a)=0 is good. But why Ψ(x)=0?

    Use what you found from Ψ(0) = 0 to simplify Ψ(a)=0.
     
  4. Jan 27, 2017 #3
    I've got further since I posted this.
    From A'+B'=0
    B'=-A'
    Ψ(a)=A'(eika-e-ika
    eika=e-ika

    But I'm stuck again.
     
  5. Jan 27, 2017 #4

    TSny

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    Gold Member

    If you know how to plot the complex numbers eika and e-ika in the complex plane, then you should be able to see the condition for them to be equal.

    However, a better way is to recall how to write eika and e-ika in terms of the trig functions sin(ka) and cos(ka), or vice versa.
     
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