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Homework Help: Complex wave function

  1. Feb 12, 2009 #1
    Show that for ψ1 , ψ2 ∈ C,
    |ψ1 + ψ2 |^2
    can take any value in [0, 4α], where
    α = |ψ1 |^2 = |ψ2 |^ 2

    I think the solution has something to do with triangular identies but I am not sure how to start this problem at all.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 13, 2009 #2

    Tom Mattson

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    Staff Emeritus
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    Gold Member

    I think you can get away without using the triangle inequality. If [itex]\psi_1[/itex] and [itex]\psi_2[/itex] both have constant modulus [itex]\sqrt{a}[/itex] then the functions must be of the following form (assuming 1 spatial dimension and time independence).

    [tex]\psi_1(x)=\sqrt{a}\exp\left(ik_1x+i\phi_1\right)[/tex]

    [tex]\psi_1(x)=\sqrt{a}\exp\left(ik_2x+i\phi_2\right)[/tex]

    Here [itex]k_1,k_2,\phi_1,\phi_2\in\mathbb{R}[/itex].

    Can you do something with that?
     
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