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5.3: Wavefunction of Spin One-Half Particle

  • Page ID
    1206
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    The state of a spin one-half particle is represented as a vector in ket space. Let us suppose that this space is spanned by the basis kets \( \vert x', y', z', \pm\rangle\) denotes a simultaneous eigenstate of the position operators \( y\), \( S_z\), corresponding to the eigenvalues \( y'\), \( \pm \hbar/2\), respectively. The basis kets are assumed to satisfy the completeness relation

    \( \vert x', y', z', +\rangle\) as the product of two kets--a position space ket \( \vert+\rangle\). We assume that such a product obeys the commutative and distributive axioms of multiplication: \( = \vert+\rangle \vert x', y', z'\rangle,\) \ref{432} \( = c'\, \vert x', y', z'\rangle \vert+\rangle+ c'' \,\vert x'', y'', z''\rangle \vert+\rangle,\) \ref{433} \( = c_+ \, \vert x', y', z'\rangle\vert+\rangle+ c_-\,\vert x', y', z'\rangle\vert-\rangle,\) \ref{434}

    where the \( L_z\) ) acting on the product $ \vert x', y', z'\rangle
\vert+\rangle$ by assuming that it operates only on the \( \vert+\rangle\) factor. Similarly, we can give a meaning to any spin operator (such as \( S_z\) ) acting on $ \vert x', y', z'\rangle
\vert+\rangle$ by assuming that it operates only on \( \vert x', y', z'\rangle\). This implies that every position space operator commutes with every spin operator. In this manner, we can give meaning to the equation

    \( \vert x', y', z'\rangle\) and \( \vert x',y', z'\rangle\vert\pm\rangle\) lies in a third vector space. In mathematics, the latter space is termed the product space of the former spaces, which are termed factor spaces. The number of dimensions of a product space is equal to the product of the number of dimensions of each of the factor spaces. A general ket of the product space is not of the form \ref{435}, but is instead a sum or integral of kets of this form.

    A general state \( A\) of a spin one-half particle is represented as a ket \( \psi_+(x', y', z')\)

    \( \psi_-(x', y', z')\) \( x'\) to \( y'\) to \( z'\) to \( s_z = +1/2\) is \( x'\) to \( y'\) to \( z'\) to \( s_z = -1/2\) is \( \int\!\int\!\int dx'dy'dz' \left(\vert\psi_+\vert^{\,2} + \vert\psi_-\vert^{\,2}\right)= 1.\) \ref{438}

    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 5.3: Wavefunction of Spin One-Half Particle is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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