5.4: Rotation Operators in Spin Space

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

$R_z({\mit\Delta}\varphi)$ that rotates the system through an angle $z$ -axis in position space. Can we also construct an operator ${\mit\Delta}\varphi$ about the $T_z({\mit\Delta}\varphi) = \exp(-{\rm i} \,S_z\, {\mit\Delta}\varphi/\hbar).$ $\ref{440}$

Thus, after rotation, the ket $\vert A_R\rangle = T_z({\mit\Delta}\varphi)\, \vert A\rangle.$

$\ref{441}$

To demonstrate that the operator $\ref{440}$ really does rotate the spin of the system, let us consider its effect on $\langle S_x\rangle \rightarrow \langle A_R\vert \,S_x\, \vert A_R \rangle = \langle A\vert \,T_z^{\dag }\, S_x \,T_z \,\vert A\rangle.$

$\ref{442}$

Thus, we need to compute

$S_x$ given in Equation $\ref{427}$. We find that Equation $\ref{443}$ becomes $\ref{444}$

or

$\ref{445}$

which reduces to

$\exp(\,{\rm i}\, G\,\lambda)\,A\, \exp(-{\rm i} \,G \,\lambda)$ $+ \left(\frac{{\rm i}^3\lambda^3}{3!}\right)[G, [G, [G,A]]]+\cdots,$ $\ref{447}$

where $\lambda$ a real parameter. The proof of this lemma is left as an exercise. Applying the Baker-Hausdorff lemma to Equation $\ref{443}$, we obtain

$\ref{448}$

which reduces to

$\ref{449}$

or

$\langle S_x \rangle \rightarrow \langle S_x\rangle \,\cos{\mit\Delta}\varphi - \langle S_y\rangle\,\sin{\mit\Delta}\varphi$ $\ref{451}$

under the action of the rotation operator $\ref{440}$. It is straightforward to show that

$\langle S_z \rangle \rightarrow\langle S_z\rangle,$ $\ref{453}$

because ${\bf S}$ by an angle $z$ -axis. In fact, the expectation value of the spin operator behaves like a classical vector under rotation:

$R_{k\,l}$ are the elements of the conventional rotation matrix for the rotation in question. It is clear, from our second derivation of the result $\ref{451}$, that this property is not restricted to the spin operators of a spin one-half system. In fact, we have effectively demonstrated that $J_k$ are the generators of rotation, satisfying the fundamental commutation relation $k$ th axis is written $R_k ({\mit\Delta}\varphi) = \exp(-{\rm i}\,J_k\, {\mit\Delta}\varphi/\hbar)$ .

Consider the effect of the rotation operator $\ref{440}$ on the state ket $\ref{439}$. It is easily seen that

$\ref{456}$

Consider a rotation by $\vert A\rangle \rightarrow T_z(2\pi)\,\vert A\rangle = -\vert A\rangle.$

$\ref{457}$

Note that a ket rotated by $4\pi$ radians is needed to transform a ket into itself. The minus sign does not affect the expectation value of ${\bf S}$ is sandwiched between $\vert A\rangle$, 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)

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