Schroedinger's equation with positive energy

[Matthew888]

Homework Statement

Consider a particle (which mass is $$m$$) and the following unidimensional potential:

$$V(x)=\begin{cases}+\infty & x<0\\ -V_0 & 0<x<a\\0 & x>a \end{cases}$$

Let $$E$$ be positive. Find the spatial autofunction.

Homework Equations

I'm convinced that I have to use Schroedinger's equation.

The Attempt at a Solution

I found:
$$\psi(x)=\begin{cases} 0 & x<0 \\ A\sin{lx}+B\cos{lx} & 0<x<a\\ C\exp{ikx}+D\exp{-ikx} & x>a\end{cases}$$
where $$k=\frac{\sqrt(2mE)}{\bar{h}}$$ and $$l=\frac{\sqrt{2m(E+V_0)}}{\bar{h}}$$.
Is this correct?

 

[jdwood983]

you may want to check your signs on $l$.

 

[jdwood983]

also, to get $\hbar$ in latex, you have to use \hbar inside the tex environment. Using \bar{h} gives you a small line over h: $$\bar{h}$$

 

[Matthew888]

you may want to check your signs on $l$.

What do you mean? I don't understand where is the problem: I know that $$E>-V_0$$, so $$l\in\mathbb{R}$$.

 

[jdwood983]

What do you mean? I don't understand where is the problem: I know that $$E>-V_0$$, so $$l\in\mathbb{R}$$.

For some reason, I missed the $-V_0$ and thought that you should have

$$l=\sqrt{\frac{2m}{\hbar^2}(V_0-E)}$$

You are correct though.

Are you translating this from another language because you should be looking for an eigenfunction, and from what I understand autofunction is a common translation error. But to finish off the problem, you will need to apply boundary conditions (at $x=a$ and $x=0$) to find your constants $A,\,B,\,C$ and $D$.

 

[Matthew888]

I am translating the problem from Italian :shy:
I found A,B,C,D using the boundary conditions (I don't have doubts about the result because I used a mathematical software). Thank you for helping me.