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Probability distribution in a quantum well

  1. Nov 21, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a mixed state comprising equal components of the first two energy levels in an infinite QW of width L. These have (normalised) wavefunctions ψ1 and ψ2. The wavefunction for the mixed state will be

    ψ(x,t)=(1/√2)ψ1e^(iw1t)+(1/√2)ψ2e^(iω2t)

    a) Calculate the probability distribution |ψ|2 at t=0 and a full cycle later, at
    t = h/[6(E2−E1)].

    b) Now do the same for a range of different times within this range, and see how
    |ψ (x)|2 changes with time. Plot a graph showing |ψ (L/2)|2




    2. Relevant equations



    3. The attempt at a solution

    |ψ(x,t)|2=ψ(x,t)ψ*(x,t)

    =((1/√2)ψ1e^(iw1t)+(1/√2)ψ2e^(iω2t))((1/√2)ψ1e^(-iw1t)+(1/√2)ψ2e^(-iω2t))

    =(1/2)ψ12+(1/2)ψ22+(1/2)ψ1ψ2e^(iw1t)e^(-iω2t)+(1/2)ψ1ψ2e^(-iw1t)e^(iω2t)

    where ψ1=√(2/L)sin(πx/L)
    ψ2=√(2/L)sin(2πx/L)

    I'm just wondering if I'm heading in the right direction or did I make a mistake somewhere along the line?
     
  2. jcsd
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