Orbital angular momentum problem

[lhcQFT]

Hi, I have a question related to the orbital angular momentum.

In the referring to Arfken & Weber Mathematical Methods for physicists-6th edition page 267,

"In the relativistic Dirac equation, orbital angular momentum is no longer conserved, but J=L+S is conserved,"

Here, I want to know why orbital angular momentum isn't conserved in relativistic Dirac equation.

 

[Simon Bridge]

There is no "why" - it just isn't.

 

[DEvens]

Total angular momentum is conserved. But one component is not. That means that you can have orbital angular momentum exchanged for spin. For example, a spin "up" particle could convert to a spin "down" particle. That would change the total orbital angular momentum by one. L changed, meaning it was not conserved. But J=L+S did not change, meaning it was conserved.

In other words, S is angular momentum as well as L.

 

[dextercioby]

Orbital angular momentum is not the whole angular momentum of the particle, in case of a Dirac wavefunction. Spin angular momentum is on equal footing with the orbital one and one can only prove that only Lx1+1xS is conserved (don't forget they act in different vector spaces).