(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am trying to understand the allowed eigenvalues for the angular momentum operatorsJandL. In particular why, m_{j}can take integer and half-integer values whereas m_{l}can take only integer values.

2. Relevant 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)

3. The attempt at a solution

I am happy with the proof that the eigenvalues of L_{z}and J_{z}must take half integer or integer values. I am confused about the argument that ml, the eigenvalues of L_{z}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 m_{l}must take only integer values unlike m_{j}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 m_{j}must take only integer values for the same reasons.

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# Spectrum of angular momentum operators

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