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

<br /> l=\sqrt{\frac{2m}{\hbar^2}(V_0-E)}<br />

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.