- #1
paco_uk
- 22
- 0
Homework Statement
I am trying to understand the allowed eigenvalues for the angular momentum operators J and L. In particular why, mj can take integer and half-integer values whereas ml can take only integer values.
Homework Equations
I have learned about angular momentum operators as generators of rotations.
So for complete rotation of the system:
[tex]
U_{rotation}( \bold{\alpha}) = \exp ( - i \bold{\alpha} \cdot \bold{J} )
[/tex]
where U is a rotation operator and J, the total angular momentum, is the generator of this rotation.
And for circular translation:
[tex]
U_{circular translation}( \bold{\alpha}) = \exp ( - i \bold{\alpha} \cdot \bold{L} )
[/tex]
where U generates circular translations and L, the orbital angular momentum, is the generator of this motion.
I also have:
[tex]
\bold{L} = \bold{r} \times \bold{p}
[/tex]
Which can be shown to be consistent with the previous equation. ( I am mostly using Binney: http://www-thphys.physics.ox.ac.uk/people/JamesBinney/QBhome.htm)
The Attempt at a Solution
I am happy with the proof that the eigenvalues of Lz and Jz must take half integer or integer values. I am confused about the argument that ml, the eigenvalues of Lz must take only integer values.
The argument is that moving the system all the way round in a circle ( so alpha is 2 pi radians) must leave the state unchanged. Plugging this into the equation involving L we find that ml must take only integer values unlike mj which can take half-integer values.
I don't understand why we can't apply the same argument to rotations through 2 pi and then argue that mj must take only integer values for the same reasons.