Normally, eigenstates are indeterminate by a complex number of magnitude one. If you so desire, you can multiply any normalized eigenstate by a number of unit magnitude of your own choosing, and it is still a normalized eigenstate. It is important to remember that in analytical expressions involving angular momentum, you are not allowed to do this.
As an example, consider a pair of spin 1/2 particles, call them
and , in the singlet state
, in which their
spins cancel and there is no net angular momentum. It was noted in
chapter 5.5.6 that this state takes the form
It all has to do with the ladder operators and .
They are very convenient for analysis, but to make that easiest, you
would like to know exactly what they do to the angular momentum
states . What you have seen so far is that
produces a state with the same square angular momentum,
and with angular momentum in the -direction equal to
. In other words, is some multiple of
a suitably normalized eigenstate ;
To resolve this conundrum, restrictions are put on the normalization factors of the angular momentum states in ladders. It is required that the normalization factors are chosen such that the ladder operator constants are positive real numbers. That really leaves only one normalization factor in an entire ladder freely selectable, say the one of the top rung.
Most of the time, this is not a big deal. Only when you start trying to get too clever with angular momentum normalization factors, then you want to remember that you cannot really choose them to your own liking.
The good news is that in this convention, you know precisely
what the ladder operators do {D.65}: