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arXiv:hep-th/0210106

Calculation of Expectation Values of S$_{z}$ and S$^{2}$ operators for spin-1 and spin-3/2 particles

Selahattin GÖNEN, Ali Havare

Published Oct 11, 2002• 0 citations
arXivHTMLPDF

Abstract

In the present article we calculate the expectation values of of S$_{z}$ and S$^{2}$ operators for spin-1 and spin-3/2 particles by expanding a general wave function which includes all spin values. The results are same as in the stantard quantum mechanics.

DOI
10.48550/arxiv.hep-th/0210106

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We show that the spectral weights ${W}_{m\stackrel{P\vec}{K}}(\stackrel{P\vec}{k})$ used for the unfolding of two-component spinor eigenstates $|{\ensuremath{\psi}}_{m\stackrel{P\vec}{K}}^{\mathrm{SC}}\ensuremath{\rangle}=|\ensuremath{\alpha}\ensuremath{\rangle}|{\ensuremath{\psi}}_{m\stackrel{P\vec}{K}}^{\mathrm{SC}},\ensuremath{\alpha}\ensuremath{\rangle}+|\ensuremath{\beta}\ensuremath{\rangle}|{\ensuremath{\psi}}_{m\stackrel{P\vec}{K}}^{\mathrm{SC}},\ensuremath{\beta}\ensuremath{\rangle}$ can be decomposed as the sum of the partial spectral weights ${W}_{m\stackrel{P\vec}{K}}^{\ensuremath{\mu}}(\stackrel{P\vec}{k})$ calculated for each component $\ensuremath{\mu}=\ensuremath{\alpha},\ensuremath{\beta}$ independently, effortlessly turning a possibly complicated problem involving two coupled quantities into two independent problems of easy solution. Furthermore, we define the unfolding-density operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\stackrel{P\vec}{K}}(\stackrel{P\vec}{k};\phantom{\rule{0.16em}{0ex}}\ensuremath{\varepsilon})$, which unfolds the primitive cell expectation values ${\ensuremath{\varphi}}^{\mathrm{pc}}(\stackrel{P\vec}{k};\ensuremath{\varepsilon})$ of any arbitrary operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varphi}}$ according to ${\ensuremath{\varphi}}^{\mathrm{pc}}(\stackrel{P\vec}{k};\ensuremath{\varepsilon})=\text{Tr}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\stackrel{P\vec}{K}}(\stackrel{P\vec}{k};\phantom{\rule{0.16em}{0ex}}\ensuremath{\varepsilon})\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varphi}})$. As a proof of concept, we apply the method to obtain the unfolded band structures, as well as the expectation values of the Pauli spin matrices, for prototypical physical systems described by two-component spinor eigenfunctions.
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