Determining eigenvalue problem

[thepopasmurf]

I'm trying to teach myself quantum mechanics using a book I got. I made an attempt at one of the questions but there are no solutions or worked examples so I'm wondering if I got it right.

Here it goes

Homework Statement

Suppose an observable quantity corresponds to the operator $$\hat{B}= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$$.

For a particular system, the eigenstates of this operator are
$$\Psi(x)=Asin\frac{n\pi x}{L}$$, where n = 1,2,3,...; A is the normalisation constant

Determine the eigenvalues of $$\hat{B}$$ for this case

Homework Equations

$$\hat{A}\psi_{j}=a_{j}\psi_{j}$$ I think

The Attempt at a Solution

I used the operator on $$\psi$$ and differenciated twice to get
$$\frac{\hbar^2 n^2 \pi^2}{2mL^2}ASin\frac{n\pi x}{L}$$
this corresponds to $$a_j\psi_j$$ so my answer for the eigenvalues is

$$\frac{\hbar^2 n^2 \pi^2}{2mL^2}$$

This is my first attempt at anything like this so any help is welcome

 

[kuruman]

You are 100% correct. Congratulations on your first successful attempt.