Finding V(x) of a given wave function?

[Rumor]

Homework Statement

"A wave function is given by Aexp[(-x²)/(2L²)] with an energy of E = h-bar²/2mL². Assuming this is a solution to the time-independent Schroedinger equation,
a) What is V(x)? Make an accurate sketch of V vs. x with labeled axes
b) What sort of classical potential has this form?

Homework Equations

The Schroedinger time-independent equation: -(h-bar²/2m) * d²Psi/dx² + V * Psi = E * Psi

The Attempt at a Solution

I know that to solve this problem, I have to integrate the original Psi function twice in order to plug it into the Schroedinger equation. Or normalize it, in order to plug it into the equation. E also has to be determined, but I'm not sure how to go about that or what value of n to use. Basically, my biggest problem is my lack of ability to successfully integrate the psi equation and knowing how to go about figuring out E. Could someone help me, please?

 

[fzero]

The Schroedinger equation is a differential equation, you will need to differentiate \Psi twice, not integrate it. It is not necessary to normalize the wavefunction to solve this problem. Also, the energy of the state described by \Psi has been given to you in the problem.

 

[Rumor]

Ah, right! Wow, I've been working on physics problem long enough that I'm starting to mix them. But anyway, after differentiating the psi function twice and plugging it back into the Schroedinger equation, am I correct in thinking that that's all to be done? Aside from simplifying and everything.

 

[fzero]

Yes, it's mostly an algebra problem at its heart.

 

[Rumor]

Awesome. Thanks for settin' me straight.