Suppose that the simultaneous eigenkets of and are completely specified by two quantum numbers, and . These kets are denoted . The quantum number is defined by
Thus, is the eigenvalue of divided by . It is possible to write such an equation because has the dimensions of angular momentum. Note that is a real number, because is an Hermitian operator.
We can write
without loss of generality, where is some real dimensionless function of and . Later on, we will show that . Now,
assuming that the have unit norms. However,
It is easily demonstrated that
where is a general ket, and is an Hermitian operator. The proof follows from the observation that
where , plus the fact that for a general ket [see Equation (21)]. It follows from Equations (316)-(318) that
Consider the effect of the shift operator on the eigenket . It is easily demonstrated that
where use has been made of Equation (315), plus the fact that and commute. It follows that the ket has the same eigenvalue of as the ket . Thus, the shift operator does not affect the magnitude of the angular momentum of any eigenket it acts upon. However,
where use has been made of Equation (311). The above equation implies that is proportional to . We can write
where is a number. It is clear that if the operator acts on a simultaneous eigenstate of and then the eigenvalue of remains unchanged, but the eigenvalue of is increased by . For this reason, is called a raising operator.
Using similar arguments to those given above, it is possible to demonstrate that
Hence, is called a lowering operator.
The shift operators, and , respectively step the value of up and down by unity each time they operate on one of the simultaneous eigenkets of and . It would appear, at first sight, that any value of can be obtained by applying these operators a sufficient number of times. However, according to Equation (320), there is a definite upper bound to the values that can take. This bound is determined by the eigenvalue of [see Equation (315)]. It follows that there is a maximum and a minimum possible value which can take. Suppose that we attempt to raise the value of above its maximum value . Since there is no state with , we must have
This implies that
so Equation (326) yields
The above equation can be rearranged to give
Comparison of this equation with Equation (315) yields the result
But, when operates on it generates , , etc. Since the lowering operator does not change the eigenvalue of , all of these states must correspond to the same value of , namely . Thus,
At this stage, we can give the unknown quantum number the value , without loss of generality. We can also write the above equation in the form
It is easily seen that
However, we also know that
where use has been made of Equations (323) and (324). It follows that
Consider the following:
where use has been made of the fact that and are Hermitian. The above equation reduces to
with the aid of Equations (323) and (324).
Equations (336) and (338) can be combined to give
The solution of the above equation is
Note that is undetermined to an arbitrary phase-factor [i.e., we can replace , given above, by , where is real, and we still satisfy Equation (339)]. We have made the arbitrary, but convenient, choice that is real and positive. This is equivalent to choosing the relative phases of the eigenkets . According to Equation (338),
We have already seen that the inequality (320) implies that there is a maximum and a minimum possible value of . The maximum value of is denoted . What is the minimum value? Suppose that we try to lower the value of below its minimum value . Because there is no state with , we must have
According to Equation (324), this implies that
It can be seen from Equation (341) that . We conclude that can take a ``ladder'' of discrete values, each rung differing from its immediate neighbors by unity. The top rung is , and the bottom rung is . There are only two possible choices for . Either it is an integer (e.g., , which allows to take the values ), or it is a half-integer (e.g., , which allows to take the values ). We shall prove in the next section that an orbital angular momentum can only take integer values of .
In summary, using just the fundamental commutation relations (297)-(299), plus the fact that , , and are Hermitian operators, we have shown that the eigenvalues of can be written , where is an integer, or a half-integer. We have also demonstrated that the eigenvalues of can only take the values , where lies in the range . Let denote a properly normalized simultaneous eigenket of and , belonging to the eigenvalues and , respectively. We have shown that
where are the so-called shift operators.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)