00:01
All right, so this question gives us a wave function at time zero and asks us to time evolve it if we assume that these are two stationary state solutions.
00:12
So in general, when we have stationary state solutions, so psi of one or two of x and now in time is just going to equal the stationary part with an added time dependent.
00:36
Exponential imaginary factor and this is just from the separation of the schrodinger equation so therefore if we want to write time evolution of our total psi it's going to equal c1, psi 1 and i'm going to suppress the x e to the negative i e1 t over h bar plus c2, psi 2, e to the negative i e2 t over h bar so it's dependent on the energy and then plank's constant so that is our time evolved wave function so now for probability density so our probability density is just given by psi of x and t complex squared and so that is just going to be si x and t complex conjugated times si x and t and we're told that's c1 c2 psi i i two are all real so that means that this complex conjugate isn't going to do anything to those pieces of our equation.
02:22
So writing this out then, we're going to have si star, which is going to be c1 star, psi 1 star.
02:38
But those are just the same.
02:40
And then e.
02:41
But this time, right, our negative sign is going to change under the complex conjugation.
02:50
So instead of negative, we're going to have positive.
02:54
The exponential is positive.
02:57
And then similarly here, these stars, i'm just writing for completeness, but they aren't actually going to change anything.
03:07
The only thing that's going to change is the sign of our exponential.
03:12
And then we're going to multiply this, right? we have to multiply this by the regular wave function to the negative ie1t over h bar plus c2, side 2, d to the negative i b2t over h bar and so because these are all real we know that c1 2 star equals c1 or 2 and also that psi 1 or 2 star just equal si i 1 2 squared and then e to the negative i e1t over each bar cancels the positive so that's going to be our total term in our next term so if we take this and this we're going to get plus c1 because c1 star is just equal to c1.
04:44
So i1 again same thing side 2 and now we're going to get an e to the i e1 minus e2t over h bar.
05:05
And then for our third term, we need to take these two and multiply them together...