Let us, for the moment, forget about the spatial position of the particle, and concentrate on its spin state. A general spin state is represented by the ket
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 (351), we would expect such an operator to take the form
Thus, after rotation, the ket becomes
To demonstrate that the operator (440) really does rotate the spin of the system, let us consider its effect on . Under rotation, this expectation value changes as follows:
Thus, we need to compute
This can be achieved in two different ways.
First, we can use the explicit formula for given in Equation (427). We find that Equation (443) becomes
which reduces to
where use has been made of Equations (427)-(429).
A second approach is to use the so called Baker-Hausdorff lemma. This takes the form
where is an Hermitian operator, and a real parameter. The proof of this lemma is left as an exercise. Applying the Baker-Hausdorff lemma to Equation (443), we obtain
which reduces to
where use has been made of Equation (417). The second proof is more general than the first, because it only uses the fundamental commutation relation (417), and is, therefore, valid for systems with spin angular momentum higher than one-half.
For a spin one-half system, both methods imply that
under the action of the rotation operator (440). It is straightforward to show that
because commutes with the rotation operator. Equations (451)-(453) demonstrate that the operator (440) rotates the expectation value of by an angle about the -axis. In fact, the expectation value of the spin operator behaves like a classical vector under rotation:
where the are the elements of the conventional rotation matrix for the rotation in question. It is clear, from our second derivation of the result (451), that this property is not restricted to the spin operators of a spin one-half system. In fact, we have effectively demonstrated that
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 (440) on the state ket (439). 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 , since is sandwiched between and , both of which change sign. Nevertheless, the minus sign does give rise to observable consequences, as we shall see presently.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)