Rotation Operators in Spin Space

in spin space. In Section 4.3, we were able to construct an operator that rotates the system through an angle about the -axis in position space. Can we also construct an operator that rotates the system through an angle about the -axis in spin space? By analogy with Equation (4.62), we would expect such an operator to take the form

Thus, after rotation, the ket becomes

(5.25) |

To demonstrate that the operator (5.24) really does rotate the spin of the system, let us consider its effect on . Under rotation, this expectation value changes as follows:

(5.26) |

Thus, we need to compute

This goal can be achieved in two different ways.

First, we can use the explicit formula for given in Equation (5.11). We find that expression (5.27) becomes

(5.28) |

or

(5.29) |

which yields

where use has been made of Equations (5.11)-(5.13).

A second approach is to use the so-called *Baker-Campbell-Hausdorff lemma* [5,20,62]. This
takes the form

where and are operators, and a real parameter. The proof of this lemma is left as an exercise. (See Exercise 2.) Applying the Baker-Campbell-Hausdorff lemma to expression (5.27), we obtain

(5.32) |

which reduces to

(5.33) |

where use has been made of Equation (5.1). Thus,

(5.34) |

The second proof is more general than the first, because it only makes use of the fundamental commutation relation (5.1), and is, therefore, valid for systems with spin angular momentum greater than one-half.

For a spin one-half system, both methods imply that

under the action of the rotation operator (5.24). It is straightforward to show that

(5.36) |

Furthermore,

because commutes with the rotation operator. Equations (5.35)-(5.37) demonstrate that the operator (5.24) rotates the expectation value of through an angle about the -axis. In fact, the expectation value of the spin operator behaves like a classical vector under rotation. In other words,

(5.38) |

where the are the elements of the conventional rotation matrix for the rotation in question [92]. (Here, and in the following, , , et cetera, are indices that run from 1 to 3, with 1 corresponding to the -axis, 2 to the -axis, and so on.) It is clear, from our second derivation of the result (5.35), that this property is not restricted to the spin operators of a spin one-half system. In fact, we have effectively demonstrated that

(5.39) |

where the are the generators of rotation, satisfying the fundamental commutation relation , and the rotation operator about the th axis is written .

Consider the effect of the rotation operator (5.24) on the state ket (5.23). It is easily seen that

Consider a rotation by radians. We find that

Note that a ket rotated by radians differs from the original ket by a minus sign. In fact, a rotation by radians is needed to transform a ket into itself. The minus sign does not affect the expectation value of , because is sandwiched between and , both of which change sign. Nevertheless, the minus sign does give rise to observable consequences, as we shall see presently.