- #1
da_willem
- 599
- 1
How can I find the momentum density in the dirac field? Can someone show me, tell me how or give me a reference?
Preferably not in relativistically covariant notation; I found this expression for the momentum density G, and want to know where it comes from:
[tex]\mathbf{G}=\frac{\hbar}{4i}[\psi^\dagger \nabla \psi + \psi^\dagger \mathbf{\alpha} (\mathbf{\alpha} \cdot \nabla)\psi]+hc [/tex]
with hc the hermitian conjugate of teh expression. This can be written using the commutation relations for the matrices [itex]\alpha_k[/itex]:
[tex]\mathbf{G}=\frac{\hbar}{2i}[\psi^\dagger \nabla \psi -(\nabla \psi ^\dagger)\psi]+\frac{\hbar}{4}\nabla \times (\psi^\dagger \mathbf{\sigma} \psi)[/tex]
Preferably not in relativistically covariant notation; I found this expression for the momentum density G, and want to know where it comes from:
[tex]\mathbf{G}=\frac{\hbar}{4i}[\psi^\dagger \nabla \psi + \psi^\dagger \mathbf{\alpha} (\mathbf{\alpha} \cdot \nabla)\psi]+hc [/tex]
with hc the hermitian conjugate of teh expression. This can be written using the commutation relations for the matrices [itex]\alpha_k[/itex]:
[tex]\mathbf{G}=\frac{\hbar}{2i}[\psi^\dagger \nabla \psi -(\nabla \psi ^\dagger)\psi]+\frac{\hbar}{4}\nabla \times (\psi^\dagger \mathbf{\sigma} \psi)[/tex]
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