# Inifinite potential barrier problem

(modified this to reflect a better understanding)

Stationary solutions to the wave equation mean that if I calculate <x> I will get a value independent of time. For a potential defined as V(x) =0 from x = 0 to a and infinite outside that range of x, the lowest energy solution is Psi(x,0) = A sin (pi x / a). I can do the
math and show that <x> = a/2 for this solution.

Now classically we know that a particle of energy greater than 0 will oscillate between the two boundaries. So any physical solution to PSI(x,t) must give such a solution. I am trying to show this for the simple case of PSI(x,t) = A sin(pi x/a) exp(-iwt) + Asin(2pi x/a)exp(-i2wt). I have 2 questions.

1. I am correct in expecting that |PSI(x,t)|^2 is still a stationary value? My book suggests it is time varying but I don't believe it.

2. When I form <x> for this wave function, I multiply by its complex conjugate and do the integral. The 'crossterms' have the time dependence and I get <x> = a * (0.5 - 0.18 * cos (3wt) ). This was
not exactly what I expected but when I graphed |PSI(x,0)|^2 it seems
reasonable. <x> = 0.32 a at t =0, and peak of |PSI(x,0)|^2 is 0.30.

3. When I calc. <H> = 5/2 * pi * pi * hbar *hbar / (2ma2

When I add up E1 + E2 for the eigenstates i get twice the value I calculated above. That seems strange to me but I can't find any faults in the math. Does anyone know if <H> is *not* supposed to equal E1 + E2?

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Hi mmwave!

1. The new state is not stationary. The book is right. Actually, any time-dependent state can be written as a sum of stationary states. 