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Solving a Piecewise Schrodinger equation

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  1. Jun 7, 2017 #1
    1. The problem statement, all variables and given/known data
    I was trying to solve the time-independent Schrodinger's equation for this well: http://i.imgur.com/C9QrvkX.png
    First I tried to look at cases where the energy of a particle is ##E < V_1##.

    2. Relevant equations
    Schrodinger's equation:
    $$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x) - V(x)\psi(x) = E\psi(x)$$

    3. The attempt at a solution
    I thought the normal way to deal with these square barriers is to break the equation up piece-wise and then match the conditions at the boundary. But in region III, there's a problem. Schrodinger's equation requires an exponential solution because ##V_1>E##. The wave function is

    $$\psi_{III} (x) = Fe^{-\mu x} + Ge^{\mu x},$$

    where ##\mu = \frac{\sqrt{2m(V_1-E)}}{\hbar}##. But the boundary condition requires that at ##x = a_3##, ##\psi_{III} (a_3) = 0##. This isn't possible unless either F = G = 0, or the wave function is actually a sin/cos wave function here. What am I missing?
     
  2. jcsd
  3. Jun 7, 2017 #2

    DrDu

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    Science Advisor

    Are you sure? How about ##F=\exp \mu a_3## and ## G=-\exp -\mu a_3##?
     
  4. Jun 7, 2017 #3
    Doh, I didn't think hard enough. Thanks
     
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