Getting Psi(x,t) from Psi(x,0)

1. Mar 1, 2007

neo2478

Hey guys, first time poster.

I am doing some quantum physics homework, and I came across the following problem:

A particle in an infinite square well has the initial wave function
Psi(x,0) = Ax 0?x?a/2
A(a-x) a/2 ?x?a

Find Psi (x,t)

Now after normalizing it, I tried plugging it in Schrödinger's equation, however I'm still having problems.

2. Mar 1, 2007

Meir Achuz

Psi(x,t)=Sum exp{-iE_n hbar t}phi_n(x)<phi_n|Psi(x,0)>,
where phi_n are the estates and E_n the eignevalues of the square well.

3. Mar 1, 2007

neo2478

Why did you do the summation of the function instead of integrating it??

4. Mar 1, 2007

nrqed

The energies are discrete, hence the summation.

You must write
$$\Psi(x,t=0) = \sum_n c_n \psi_n(x)$$
where the $\psi_n(x)$ are the energy eigenstates, $\sqrt{2/a} \, sin(n \pi x/a)$ (for a well located between x=0 and x=a). What you have to do is to find the coefficients c_n using the orthonormality of the sine wavefunctions. Once you have that, the wavefunction at any time is

$$\Psi(x,t) = \sum_n c_n e^{-i E_n t / \hbar} \psi_n(x)$$