- #1
Gene Naden
- 321
- 64
I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen; the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 11, equation 1.3.20. The authors have defined an operator ##L_{\mu\nu} = i( x_\mu \partial \nu - x_\nu \partial \mu)##. They state that ##L_{23}## is the x component of angular momentum. Reviewing the Schrodinger equation, I see that the momentum operator ##\vec{p} = -i \nabla##, from which I get
##\partial_1=i p_{1}## and ##\partial_2 = i p_{2}##.
Substituting these in the definition for L, I get ##L_1 = L_{23} = i(x_2 i p_3 - x_3 i p_2) = - (\vec{r} \times \vec{p})_1##
So I get ##L_{1}## equal to minus the x component of the angular momentum. I am wondering where I went wrong. I have worked through about 20 equations in this reference and have never found an error!
##\partial_1=i p_{1}## and ##\partial_2 = i p_{2}##.
Substituting these in the definition for L, I get ##L_1 = L_{23} = i(x_2 i p_3 - x_3 i p_2) = - (\vec{r} \times \vec{p})_1##
So I get ##L_{1}## equal to minus the x component of the angular momentum. I am wondering where I went wrong. I have worked through about 20 equations in this reference and have never found an error!