The fact that the angular momentum ladders of the previous section must have a top and a bottom rung restricts the possible values that angular momentum can take. This section will show that the azimuthal quantum number can either be a nonnegative whole number or half of one, but nothing else. And it will show that the magnetic quantum number must range from to in unit increments. In other words, the bosonic and fermionic example ladders in figures 12.1 and 12.2 are representative of all that is possible.
To start, in order for a ladder to end at a top rung
, has to be zero for
. More specifically, its magnitude
must be zero. The square magnitude is
given by the inner product with itself:
The lowest rung on the ladder goes the same way; you get
And in fact, it is more than that; it also limits what the quantum
numbers and can be. For, since each step on a ladder
increases the magnetic quantum number by one unit, you have
for the total number of steps up from bottom to top:
Integer values occur, for example, for the spherical harmonics of orbital angular momentum and for the spin of bosons like photons. Half-integer values occur, for example, for the spin of fermions such as electrons, protons, neutrons, and particles.
Note that if is a half-integer, then so are the corresponding values of , since starts from and increases in unit steps. See again figures 12.1 and 12.2 for some examples. Also note that ladders terminate just before -momentum would exceed total momentum.
It may also be noted that ladders are distinct. It is not possible to go up one ladder, like the first one in figure 12.1 with and then come down the second one using . The reason is that the states are eigenstates of the operators , (12.5), and , (12.7), so going up with and then down again with , or vice-versa, returns to the same state. For similar reasons, if the tops of two ladders are orthonormal, then so is the rest of their rungs.