Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.4.2.
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We can now generalize our treatment of rotation, originally studied in two dimensions, to three dimensions. We’ll view a 3-d rotation as a combination of rotations about the , and axes, each of which can be represented by a matrix. These matrices are as follows:
We’re interested in infinitesimal rotations, for which we retain terms up to first order in the rotation angle , so that and . This gives the infinitesimal rotation matrices as
We now consider the series of rotations as follows: first, a rotation by , then by , then by and finally by . Because the various rotations don’t commute, we don’t end up back where we started. We can calculate the matrix products to find the final rotation.
To get the third line, we multiplied the first two matrices in the second line, and the last two matrices in the second line. In the final result, we can discard terms containing or to get
Thus the result of the four rotations about the and axes is a single rotation about the axis.
To convert this to quantum operators, we define the operator by comparison with the procedure we used for 2-d rotations. That is, the operator is given by the corresponding angular momentum operator , or as
We can get the commutation relation by matching coefficients of on each side of this equation. On the RHS, the coefficient is . On the LHS, we can pick out the terms involving to get
The first term on the LHS comes from the term in the first in 15 multiplied by the term in the second (with the term in the other two s); the second term on the LHS comes from the term in the first in 15 multiplied by the term in the fourth , and so on.
Matching the two sides, we get
By comparison with the classical definitions of the three components of , we can write the quantum operators in terms of position and momentum operators as
From the commutators of position and momentum and so on, we can verify 17 from these relations as well.
The third line follows because . The other two commutation relations follow by cyclic permutation of , and :