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Schroedinger's equation with positive energy

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider a particle (which mass is [tex]m[/tex]) and the following unidimensional potential:

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

    Let [tex]E[/tex] be positive. Find the spatial autofunction.
    2. Relevant equations
    I'm convinced that I have to use Schroedinger's equation.


    3. The attempt at a solution
    I found:
    [tex]\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}[/tex]
    where [tex]k=\frac{\sqrt(2mE)}{\bar{h}}[/tex] and [tex]l=\frac{\sqrt{2m(E+V_0)}}{\bar{h}}[/tex].
    Is this correct?
     
  2. jcsd
  3. Nov 23, 2009 #2
    you may want to check your signs on [itex]l[/itex].
     
  4. Nov 23, 2009 #3
    also, to get [itex]\hbar[/itex] in latex, you have to use \hbar inside the tex environment. Using \bar{h} gives you a small line over h: [tex]\bar{h}[/tex]
     
  5. Nov 25, 2009 #4
    What do you mean? I don't understand where is the problem: I know that [tex]E>-V_0[/tex], so [tex]l\in\mathbb{R}[/tex].
     
  6. Nov 25, 2009 #5
    For some reason, I missed the [itex]-V_0[/itex] and thought that you should have

    [tex]
    l=\sqrt{\frac{2m}{\hbar^2}(V_0-E)}
    [/tex]

    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 [itex]x=a[/itex] and [itex]x=0[/itex]) to find your constants [itex]A,\,B,\,C[/itex] and [itex]D[/itex].
     
  7. Nov 25, 2009 #6
    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.
     
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