Spinor Rotation Matrices

(5.95) |

by analogy with Equation (5.24), where is a unit vector pointing along the axis of rotation, and is the angle of rotation. Here, can be regarded as a trivial position operator. The rotation operator is represented

(5.96) |

in the Pauli scheme. The term on the right-hand side of the previous expression is the exponential of a matrix. This can easily be evaluated using the Taylor series for an exponential, plus the rules

(5.97) | ||||

(5.98) |

These rules follow trivially from the identity (5.93). Thus, we can write

(5.99) |

The explicit form of this matrix is

Rotation matrices act on spinors in much the same manner as the corresponding rotation operators act on state kets. Thus,

(5.101) |

where denotes the spinor obtained after rotating the spinor an angle about the axis . The Pauli matrices remain unchanged under rotations. However, the quantity is proportional to the expectation value of [see Equation (5.75)], so we would expect it to transform like a vector under rotation. (See Section 5.4.) In fact, we require

(5.102) |

where the are the elements of a conventional rotation matrix [92]. This is easily demonstrated, because

plus all cyclic permutations. The previous expression is the matrix analog of

(5.104) |

[See Equation (5.30).] Equation (5.103) follows from the Baker-Campbell-Hausdorff lemma, (5.31), which holds for matrices, in addition to operators.