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Paul159

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- TL;DR Summary
- 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 !

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 !

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