Probability distribution in a quantum well

[moonkey]

Homework Statement

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

Homework Equations

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?

 

[mark726]

You are on the right track! Your calculations are correct so far. The next step would be to simplify the terms involving the exponential functions. Remember that e^(ix) = cos(x) + i*sin(x). This will help you simplify the terms in your expression for |ψ(x,t)|^2. Once you have simplified it, you can substitute in the values for ψ1 and ψ2 and plot the graph for different times within the given range.