Dirac Equation for a moving square potential well

[haaj86]

Hi, I have learned the Dirac equation recently and I managed to solve it for a free particle (following Greiner book “relativistic quantum mechanics” and Paul Strange book “Relativistic Quantum Mechanics”). I was asked to solve the Dirac equation in the stationary frame for a free particle (no potential and zero momentum) and transform the solution to that of a free particle with momentum. I found the solution for this on page 157 of Greiner’s book.

Now I have to do the same thing but with a square potential well, starting by a stationary potential well (with the solutions given in Strange’s Chap 9 page 263-267, Greiner Chap 9 page 197-199) which I understand, and then solve the Dirac equation for the same square potential moving at a constant velocity let’s say in the x-direction and find the transformation for the two solutions. I really have no clue on how to solve the Dirac equation for a moving potential well. I think that I have to set the boundary conditions moving at a constant velocity but I am not sure what I should do next.

I know that the calculation is nasty for this problem, but all I am asking for is if anybody know the strategy to use in order to solve the Dirac equation for a moving potential well, give me as much references as you can, papers, books.

 

[per.sundqvist]

Hi, I have learned the Dirac equation recently and I managed to solve it for a free particle (following Greiner book “relativistic quantum mechanics” and Paul Strange book “Relativistic Quantum Mechanics”). I was asked to solve the Dirac equation in the stationary frame for a free particle (no potential and zero momentum) and transform the solution to that of a free particle with momentum. I found the solution for this on page 157 of Greiner’s book.

Now I have to do the same thing but with a square potential well, starting by a stationary potential well (with the solutions given in Strange’s Chap 9 page 263-267, Greiner Chap 9 page 197-199) which I understand, and then solve the Dirac equation for the same square potential moving at a constant velocity let’s say in the x-direction and find the transformation for the two solutions. I really have no clue on how to solve the Dirac equation for a moving potential well. I think that I have to set the boundary conditions moving at a constant velocity but I am not sure what I should do next.

I know that the calculation is nasty for this problem, but all I am asking for is if anybody know the strategy to use in order to solve the Dirac equation for a moving potential well, give me as much references as you can, papers, books.

What happens to the Dirac equation it self when the coordinates in it (also in derivatives) are transformed according to the Lorentz transform? I was told, but never checked it, that Diracs equation is invariant under such transform, but it semes you think not. My tip: Do the transform in 1D and see what you get, and then see if that change the box-potential in some way

 

[haaj86]

Actually I know that the Dirac equation is invariant under the Lorentz transformation, and I went through the proof. But, the four components of the wave function do not form a 4-vector and so the solutions are not invariant under the Lorentz transformation. However, I know the transformation to use in order to transform the solution from one frame to another and I am going to try and do that for the square potential well, but my question is how to solve the Dirac equation starting with the moving potential which should at the end give the same answer to the transformed solution.