How to Derive Psi(x,t) from Psi(x,0) in Quantum Mechanics?

[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.

Thanks in advance, Rob.

 

[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.

 

[neo2478]

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

 

[nrqed]

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

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

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