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Schrodinger Equation for a central 2D potential

  1. Mar 24, 2009 #1
    Hello,
    I would like to ask something about central potentials. When I am working in 3D, I haven´t got any problem solving the schrodinger equation since I use the following change of variables:
    [tex]-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V(r)\Psi=E\Psi[/tex]
    [tex]\Psi=\frac{\chi}{r}[/tex]
    With this change of variables, I can solve and understand the equation, but when I am working in 2D( usig r and [tex]\varphi[/tex]) I can't solve the equation because the laplacian is different. The concrete potential distribution I want to study is the following:
    [tex]V(\vec{r})=0 \stackrel{if}{\rightarrow} r<a[/tex]
    [tex]V(\vec{r})=\infty \stackrel{if}{\rightarrow} r>a[/tex]
    And I would like to solve the equation for the first and the second energy levels of the system. I would appreciate some tips, thank you.
     
  2. jcsd
  3. Mar 24, 2009 #2

    clem

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    Science Advisor

    It's not as easy as 3D. The bound states are Bessel functions:
    [tex]\psi=J_m(kr)\cos(m\theta)[/tex], with the energy determined by
    [tex]J_m(ka)=0[/tex].
     
  4. Mar 25, 2009 #3
    Thank you very much clem, that will be really useful for me.
     
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