- #1
jbergman
- 339
- 141
I am trying to do some self-study and plow through Ballentine's book on Quantum Mechanics. I thought I was following the majority of it until I got to Sec. 3.3, in particular the derivation of the multiples of identity for the commutators of these generators.
For example, the commutator of two rotations is given as
[tex][J_\alpha,J_\beta]=i\epsilon_{\alpha\beta\gamma}J_\gamma + i\epsilon_{\alpha\beta\gamma}b_\gamma I[/tex]
He then states that one can redefine the phase of certain vectors to transform the phase term to 0 and does so with the substitutions
[tex]J_\alpha + b_\alpha I \rightarrow J_\alpha[/tex]
leading to
[tex][J_\alpha,J_\beta]=i\epsilon_{\alpha\beta\gamma}J_\gamma[/tex]
I feel I am missing something key in what is going on here because I am not understanding why one can't just arbitrarily say the extra phase piece in the first equation is just 0 by setting [tex]b_\gamma = 0[/tex]. Where are these phase differences coming from that one needs to change your generator as Ballentine does?
For example, the commutator of two rotations is given as
[tex][J_\alpha,J_\beta]=i\epsilon_{\alpha\beta\gamma}J_\gamma + i\epsilon_{\alpha\beta\gamma}b_\gamma I[/tex]
He then states that one can redefine the phase of certain vectors to transform the phase term to 0 and does so with the substitutions
[tex]J_\alpha + b_\alpha I \rightarrow J_\alpha[/tex]
leading to
[tex][J_\alpha,J_\beta]=i\epsilon_{\alpha\beta\gamma}J_\gamma[/tex]
I feel I am missing something key in what is going on here because I am not understanding why one can't just arbitrarily say the extra phase piece in the first equation is just 0 by setting [tex]b_\gamma = 0[/tex]. Where are these phase differences coming from that one needs to change your generator as Ballentine does?