Stationary Schrodinger equation for a particle moving in a potential well has two solutions
psi1(x)=e^-ax^2 with energy E1 and
psi2(x)xe^-ax^2 with energy E2
At t=o, the particle is in the state psi(x)=psi1(x)+psi2(x)
Calculate the probability distribution as a function of time.
psi(x,t)=e^iEt/h-bar * psi(x) and
probability distribution function=[mod(psi(x,t)]^2
The Attempt at a Solution
Well I tried subbing in psi(x,t)=e^iEt/h-bar * psi(x) for psi1(x) and psi2(x) to get psi(x) as a function of t.
But then when I try to do the modulus squared to get the probability distribution, the t disappears since I get (x+1)^2 * (e^-2ax^2) * (cos^2(Et/h-bar) + sin^2(Et/h-bar).
So I'm just left with (x+1)^2 * (e^-2ax^2).
I'm obviously making a stupid mistake here, unless I've got the theory part wrong? I definitely need a t term in the answer because the next question asks to figure out the time at which the probability distribution returns to the initial.