12.4 Possible values of angular momentum

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:

Now because of the complex conjugate that is used in the left hand side of an inner product, (see chapter 2.3), goes to the other side of the product as , and you must have

That operator product can be multiplied out:

but is the square angular momentum except for , and the term within the parentheses is the commutator which is according to the fundamental commutation relations equal to , so

The effect of each of the operators in the left hand side on a state is known and the inner product can be figured out:

The question where angular momentum ladders end can now be answered:

There are two possible solutions to this quadratic equation for , to wit or . The second solution is impossible since it already would have the square angular momentum exceed the total square angular momentum. So unavoidably,

That is one of the things this section was supposed to show.

The lowest rung on the ladder goes the same way; you get

and the only acceptable solution for the lowest rung on the ladders is

It is nice and symmetric; ladders run from up to , as the examples in figures 12.1 and 12.2 already showed.

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:

But the number of steps is a whole number, and so the azimuthal quantum must either be a nonnegative integer, such as 0, 1, 2, ..., or half of one, such as , , ...

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.