QM: time evolution in an infinite well

[liorda]

Homework Statement

http://img379.imageshack.us/img379/1864/screenshothw4pdfapplicamd7.png

Homework Equations

$$H|\psi > = E_n |\psi >$$

The Attempt at a Solution

About part 1 of the question: I can find the eigenfunctions of psi_1 by comparing coefficients with the well known eigenfunctions of the Hamiltonian in an infinite well, using trigonometric identities, but is there any simple way to find the eigenfunctions of psi_2, without actually solving the Schrödinger equation?

Thanks :)

 

[Mindscrape]

How, exactly, did you solve for psi_1? Did you normalize?

I assume that since you used it, you're familiar with bra-ket notation, so just do $\langle \psi_1 | \psi_1 \rangle=1$ and similar for the second wavefunction.

I'm a little confused by the "Hamiltonian eigenfunction" saying, but I think all it means is that the observable postulate holds true and no kind of perturbation is necessary.

 

[liorda]

Hi Mindscrape, thanks for answering.

I didn't normalize psi_1, because the expression is a superposition of two eigenstates of a particle in a infinite box (it's just a sum of two sines). I compared the coefficients of $$sin(a) cos(b) = \frac{1}{2} \left[ sin(a+b)+sin(a-b)\right]$$ to the well known coefficient sqrt(2/L) for particle in a box.

as for psi_2, the main problem is to find out what are the eigenfunctions. I didn't find any way other than using Fourier series for the polynomial expression and getting a sum of sines...