A Weyl Fermion in an infinite well

[Paul159]

TL;DR

I try to solve the Weyl equation in a infinite well with infinite mass condition.

Hello everyone,

I have a problem with bounds states of the 1D Weyl equation. I want to solve the Dirac equation

$−i\hbar \partial _x\Psi+m(x)\sigma _z \Psi=E\Psi$ with the mass $m(x)=0,0<x<a$, $m(x)=\infty,x<0,x>a$. $\Psi=(\Psi_1,\Psi_2)^T$ is a two component spinor. Outside the well, $\Psi=0$. Inside the well we have the plane wave equation
$$\Psi(x)=A e^{ikx} \begin{pmatrix} 1\\1 \end{pmatrix}+Be^{−ikx} \begin{pmatrix} -i\\i \end{pmatrix}$$. Of course the "wave function" is discontinuous at $x=0,x=a$. I found this article where they talk about this problem. The condition they choose is that the Noether current is 0 at the well boundary. It is quite simple to find that we get from that $|A|=|B|$. So we can write$B=Ae^{-i\phi}$ where $\phi$ is real. After that I used eq. 33 of the article : at each boundary we must have $\Psi_2/\Psi_1=ie^{i\alpha}$ where $\alpha=\pi$ at $x=0$ and $\alpha=0$ at $x=a$.

The condition at $x=0$ gives me that $\cos \phi = \sin \phi -1$, so $\phi = \pi/2$. The second condition gives me that $e^{2ika} = -i$, so $k_n = (n + 3/4) \pi /a$. The problem is that in the article they found $k_n = (n + 1/2) \pi /a$.

If someone have already done this exercise, can you help me ?


Thanks !

 

[Paul159]

Ok I get it. You have to take two different spinors for $x = 0$ and $x = a$. The first condition at $x = 0$ will give you the trivial property $-i = -i$. The condition at $x = a$ will give you $e^{2ika} = -1$, such that $k_n = (n + 1/2) \pi /a$.
You can lock this topic thanks.