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5.4: Rotation Operators in Spin Space

  • Page ID
    1207
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    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 $ \frac{\hbar}{2} \,\exp(\,{\rm i}\,S_z\, {\mit\Delta}\varphi/\hbar...
...rt-\rangle \langle +\vert\,)\, \exp(-{\rm i}\,S_z \,{\mit\Delta}\varphi/\hbar),$ \ref{444}

    or

    $ \frac{\hbar}{2} \left( {\rm e}^{\,{\rm i}\,{\mit\Delta}\varphi/2}...
...ert-\rangle \langle +\vert\,{\rm e}^{\,-{\rm i}\,{\mit\Delta}\varphi/2}\right),$ \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

    $ 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,$ \ref{448}

    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],$ \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

    $ T_z({\mit\Delta}\varphi)\,\vert A\rangle = {\rm e}^{-{\rm i}\,{\m...
... e}^{\,{\rm i}\,{\mit\Delta}\varphi/2}\, \langle -\vert A\rangle \vert-\rangle.$ \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)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 5.4: Rotation Operators in Spin Space is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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