5.4: Rotation Operators in Spin Space
( \newcommand{\kernel}{\mathrm{null}\,}\)
Let us, for the moment, forget about the spatial position of the particle, and concentrate on its spin state. A general spin state A is represented by the ket
Rz(Δφ) that rotates the system through an angle z -axis in position space. Can we also construct an operator Δφ about the Tz(Δφ)=exp(−iSzΔφ/ℏ). ???Thus, after rotation, the ket |AR⟩=Tz(Δφ)|A⟩.
???To demonstrate that the operator ??? really does rotate the spin of the system, let us consider its effect on ⟨Sx⟩→⟨AR|Sx|AR⟩=⟨A|T\dagzSxTz|A⟩.
???Thus, we need to compute
Sx given in Equation ???. We find that Equation ??? becomes
or

which reduces to
exp(iGλ)Aexp(−iGλ) +(i3λ33!)[G,[G,[G,A]]]+⋯, ???where λ a real parameter. The proof of this lemma is left as an exercise. Applying the Baker-Hausdorff lemma to Equation ???, we obtain
![$ S_x + \left(\frac{{\rm i}\,{\mit\Delta}\varphi}{\hbar}\right) [S_...
...\frac{{\rm i}\,{\mit\Delta}\varphi}{\hbar}\right)^2 [S_z, [S_z, S_x]] + \cdots,$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1090.png)
which reduces to
![$ S_x\left[ 1- \frac{({\mit\Delta}\varphi)^2}{2!} + \frac{({\mit\De...
...{\mit\Delta}\varphi)^3}{3!}+ \frac{({\mit\Delta}\varphi)^5}{5!} +\cdots\right],$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1091.png)
or
⟨Sx⟩→⟨Sx⟩cosΔφ−⟨Sy⟩sinΔφ ???under the action of the rotation operator ???. It is straightforward to show that
⟨Sz⟩→⟨Sz⟩, ???because S by an angle z -axis. In fact, the expectation value of the spin operator behaves like a classical vector under rotation:
Rkl are the elements of the conventional rotation matrix for the rotation in question. It is clear, from our second derivation of the result ???, that this property is not restricted to the spin operators of a spin one-half system. In fact, we have effectively demonstrated that Jk are the generators of rotation, satisfying the fundamental commutation relation k th axis is written Rk(Δφ)=exp(−iJkΔφ/ℏ) .Consider the effect of the rotation operator ??? on the state ket ???. It is easily seen that

Consider a rotation by |A⟩→Tz(2π)|A⟩=−|A⟩.
???Note that a ket rotated by 4π radians is needed to transform a ket into itself. The minus sign does not affect the expectation value of S is sandwiched between |A⟩, both of which change sign. Nevertheless, the minus sign does give rise to observable consequences, as we shall see presently.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)