(adsbygoogle = window.adsbygoogle || []).push({}); 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 ψ_{1}and ψ_{2}. The wavefunction for the mixed state will be

ψ(x,t)=(1/√2)ψ_{1}e^(iw_{1}t)+(1/√2)ψ_{2}e^(iω_{2}t)

a) Calculate the probability distribution |ψ|^{2}at t=0 and a full cycle later, at

t = h/[6(E_{2}−E_{1})].

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)ψ_{1}e^(iw_{1}t)+(1/√2)ψ_{2}e^(iω_{2}t))((1/√2)ψ_{1}e^(-iw_{1}t)+(1/√2)ψ_{2}e^(-iω_{2}t))

=(1/2)ψ_{1}^{2}+(1/2)ψ_{2}^{2}+(1/2)ψ_{1}ψ_{2}e^(iw_{1}t)e^(-iω_{2}t)+(1/2)ψ_{1}ψ_{2}e^(-iw_{1}t)e^(iω_{2}t)

where ψ_{1}=√(2/L)sin(πx/L)

ψ_{2}=√(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?

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# Homework Help: Probability distribution in a quantum well

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