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Extended 1D well to 3D

  1. Mar 22, 2009 #1
    1. Quantum well structure can be realized by sandwiching layers of semiconductor and two insulators.
    This sandwiching problem was often treated with 1 D infinite well. Suppose now the problem is 3 dimensional well with length L at z direction from 0 to L, at which

    V(x,y,z)= 0 when 0<z<L
    infinity otherwise
    where V(x) and V(y) is 0. We assume x and y be infinitely large.

    Wht is the total energy and the wave function of the electron in such well?



    2. I have done the separation of varibales in the Schrodinger equation and obtained the three independent wavefunction.
    [tex]-\frac{\hbar^{2}}{2m}\psi_{x_{i}}=E \psi_{x_{i}} [/tex]
    But i dont know what is the boundary condition of the x and y.
    I only got psi(z) is the psi of the one d wavefunction.
    [tex]\psi_{z}=\sqrt{\frac{2}{L}}sin(\frac{n \pi z}{L})[/tex]
    I just cannot get the constant for the wavefunctions for x and y.
    I know that
    [tex]\psi(x)=\psi(x+2\pi)[/tex]
    but I still cannot get the value of the constant and the energy.
     
    Last edited: Mar 22, 2009
  2. jcsd
  3. Mar 22, 2009 #2
    Why do you assume periodic boundary conditions?

    It is an infinite well. Remember, that means the wavefunction is zero at all boundaries of the well.
     
  4. Mar 22, 2009 #3
    But now V(x,y,z)=V(z), where V is not a function of x and y anymore.
    With this condition, in x and y, the wavefunction is certainly not zero, it is a free particle.
    [tex]\psi_{x}=e^{i(kx-wt)}[/tex]
     
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