cc94
- 19
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Homework Statement
Suppose we have the standard rectangular potential barrier in 1D, with
$$
V =
\left\{
\!
\begin{aligned}
0 & \,\text{ if } x<0, x>d\\
V_0 & \,\text{ if } x>0,x<d\\
\end{aligned}
\right.
$$
The standard approach to solve for tunneling through the barrier is to match the wavefunctions at the boundaries of the barrier, and solve for the ratio of amplitudes of the outgoing and incoming waves. My issue is that this problem can't be solved numerically, since it is open on the ends, so the spectrum is continuous. Or so I thought, until I found a paper that says that one can turn this problem into a boundary value problem. If you pick some arbitrary $$x_a, x_b$$ to the left and right of the barrier, respectively, then one can use the following equations as boundary conditions to numerically solve the ODE. My first question is, how were these equations derived? And then if it's possible to answer, does this method work in 2D? From simulating some 2D potentials with these boundary conditions, it looks like it works, but I need to know more specifics.
Homework Equations
$$
\left\{
\!
\begin{aligned}
\frac{d\psi}{dx}\Big|_{x=x_a} &+ ik\psi(x_a) = 2ikA\text{exp}(ikx_a),\\
\frac{d\psi}{dx}\Big|_{x=x_b} &- ik\psi(x_b) = 0
\end{aligned}
\right.
$$
where A is amplitude of inc. wave.
The Attempt at a Solution
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To derive those equations, I'm assuming you follow the same piecewise procedure to match wavefunctions, with the potential being V=0 on both sides. But that doesn't give an answer. Also I don't get why, for example, the first condition only depends on the amplitude of the incoming wave and not the amplitude of the reflected wave, R.