Klein-Gordon Finite Well

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Pengwuino
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Homework Statement



The problem is basically solving the Klein-Gordon equation for a finite well for a constant potential under the condition [tex]V > E + mc^2[/tex]

Homework Equations



[tex] V = 0 -a<x<a[/tex]
[tex] V = V_o [/tex] elsewhere

KG Equation: [tex][\nabla^2 + \left({{V-E} \over {\hbar c}}\right)^2 - k_c^2]\phi(x) = 0[/tex]

The Attempt at a Solution



So we're looking at the situation where the potential is high enough so that [tex]V > KE + 2mc^2[/tex]

We have solutions of the form [tex]\psi(x) = Ae^{ikx}+Be^{-ikx}[/tex] where
[tex]k = \sqrt{\left({{V-E}\over{\hbar c}}\right)^2 - k_c^2}[/tex]
[tex]k_c = {{m} \over {\hbar c}}[/tex]

For this case, the group velocities are such that the solutions in the left, center, and right regions, which i'll denote as 1, 2, and 3, are as follows:

[tex]\phi_1(x) = Ae^{ik_1x}[/tex]
[tex]\phi_{2}(x) = Ccos(k_{2}x) + Dsin(k_{2}x)[/tex]
[tex]\phi_{3}(x) = Ge^{-ik_{3}x}[/tex]

where
[tex]k_{1/3} = \sqrt{\left({{V-E}\over{\hbar c}}\right)^2 - k_c^2}[/tex]
[tex]k_{2} = \sqrt{\left({{E}\over{\hbar c}}\right)^2 - k_c^2}[/tex]

The problem is I have no idea how far this problem can be taken. I assume you can figure out the bound energy states for the particle. However, considering the wave function outside the well are sinusoidal, are the coefficients solvable? In scattering problems you can define a transmission and reflection coefficient but does it make sense to talk about "incoming" particles if the potential is so high?
 

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