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Both Ballentine in "Quantum Mechanics - a modern development" pages 160-162 and Sakurai in "Modern Quantum Mechanics" pages 193-196 use essentialy the same argument to show the existence of a set of eigenvectors of J² and J_z with integer spaced values of the J_z eigenvalues for fixed J² eigenvalue.

I get everything up to the result -b_max <= a <= b_max (following Sakurai's notation). But then (in Ballentine) it is claimed that the difference between b_max and -b_max must be an integer. In Sakurai the equivalent (actually stronger) claim is "Clearly, we must be able to reach |a, b_max> by applying J_+ succesively to |a, b_min>".

It is not obvious to me that this should be so. Can anyone help me see this?

Thanks,

A_B