Klein-Gordon Finite Well

[Pengwuino]

Homework Statement

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

Homework Equations

$$V = 0 -a<x<a$$
$$V = V_o$$ elsewhere

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

The Attempt at a Solution

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

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

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:

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

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

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?

 

[mark726]

I would like to offer some feedback on your solution attempt. First, it is important to clearly state the problem and the given conditions. In this case, the problem is to solve the Klein-Gordon equation for a finite well with a constant potential, under the condition that V > E + mc^2.

Next, you have correctly identified the solutions to the Klein-Gordon equation in the different regions of the well, denoted as 1, 2, and 3. However, you have not explained how you arrived at these solutions or how they are related to the given conditions. It would be helpful to provide some explanation or derivation for these solutions.

Furthermore, you mention the concept of bound energy states, but do not explain what these are or how they relate to the problem. It would be important to clarify this for a better understanding of the solution. Additionally, you ask if it is possible to solve for the coefficients in the wave function outside the well. This is a valid question, but again, it would be helpful to provide some explanation or reasoning behind it.

In terms of the scattering problem, it is possible to define transmission and reflection coefficients for particles with high potential barriers, as long as the particles have enough energy to overcome the barrier. However, it is not clear how this relates to the given problem or the solutions you have provided.

Overall, your solution attempt shows some understanding of the problem, but it would benefit from further explanation and clarification. it is important to clearly state the problem, provide a thorough explanation of the solution, and clearly connect the solution to the given conditions. I hope this feedback is helpful in your understanding of the problem and in improving your solution attempt.