The Ladder-Operator Method for Angular Momentum

We used the letter L for orbital angular momentum. Here we will use the letter M to indicate that we are dealing with any kind of angular momentum. We have three linear 

We begin by evaluating the commutators of with its components, using Eqs. (5.107) and (5.108). The work is identical with that used to derive Eqs. (5.49) and (5.50), and we have 

Next we define two new operators, the raising operator M+ and the lowering operator M  

These are examples of ladder operators. The reason for the terminology will become clear shortly. We have 

If we operate on (5.117) with the lowering operator and apply (5.115), we find in the same manner 

Of course, there is nothing special about the z axis all directions of space are equivalent. If we had chosen to specify and (rather than L ), we would have gotten the same eigenvalues for as we found for L. However, it is easier to solve the L  

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Section 5.4 The Ladder-Operator Method for Angular Momentum 115... 

We now solve equation (6.24) by means of ladder operators, analogous to the method used in Chapter 4 for the harmonic oscillator and in Chapter 5 for the angular momentum. We define the operators Ax and Bx as... 

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