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Potential barrier. Schroedinger equation.

  1. Mar 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Schroedinger equation for potential barrier.
    What if ##V_0=E##
    First region. Particles are free.
    ##\psi_1(x)=Ae^{ikx}+Be^{-ikx}##
    In third region
    ##\psi_3(x)=Ce^{ikx}##


    2. Relevant equations
    ##\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(V_0-E)\psi=0##
    where ##V_0## is height of barrier.
    For region II

    3. The attempt at a solution
    In second region
    ##\frac{d^2 \psi}{dx^2}=0##
    from that
    ##\frac{d\psi}{dx}=C_1##
    ##\psi(x)=C_1x+C_2##
    Boundary condition
    ##A+B=C_2##
    ##C_1a+C_2=Ce^{ika}##
    ##ikA-ikB=C_1##
    ##C_1=ikCe^{ika}##
    System 4x4
    Is this correct?
    Could you tell me in this case do I have bond state?
     
    Last edited: Mar 29, 2014
  2. jcsd
  3. Mar 29, 2014 #2
    Yes, it seems correct. Bond states have negative energy.
     
  4. Mar 30, 2014 #3
    Bound states have negative energy? Can you explain me this. In case of this problem.
     
  5. Mar 31, 2014 #4
    The oscillatory solutions of regions 1 and 3 happen because energy is positive and the wave is free to propagate to infinity. (unbounded particle). If the energy is negative you get a exponentially decaying wave function and the wave does not propagate to infinity (bounded particle).
     
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