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.