dorsal/arxiv
View SchemaProperties of Fermion Spherical Harmonics
| Authors | Geoffrey Hunter, Mohsen Emami-Razavi |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0507006 |
| URL | https://arxiv.org/abs/quant-ph/0507006 |
Abstract
The Fermion Spherical harmonics [$Y_\ell^{m}(\theta,\phi)$ for half-odd-integer $\ell$ and $m$ - presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [$Y_\ell^{m}(\theta,\phi)$ for integer $\ell$ and $m$]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators $M_+$ ($M_-$) that ascend (descend) the sequence of harmonics over the values of $m$ for a given value of $\ell$, do not produce the expected result {\em in just one case}: when the value of $m$ changes from $\pm{1/2}$ to $\mp{1/2}$; i.e. when $m$ changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take $m$ outside the range: $-\ell\le m\le +\ell$. The unexpected result in the one case does not invalidate this scalar coordinate representation of spin angular momentum, because the eigenfunction property is essential for a valid quantum mechanical state, whereas ladder operators relating states with different eigenvalues are not essential, and are in fact known only for a few physical systems; that this coordinate representation of spin angular momentum differs from the abstract theory of angular momentum in this respect, is simply an interesting curiosity. This new representation of spin angular momentum is expected to find application in the theoretical description of physical systems and experiments in which the spin-angular momentum (and associated magnetic moment) of a particle is oriented in space, since the orientation is specifiable by the spherical polar angles, $\theta$ and $\phi$.
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"abstract": "The Fermion Spherical harmonics [$Y_\\ell^{m}(\\theta,\\phi)$ for\nhalf-odd-integer $\\ell$ and $m$ - presented in a previous paper] are shown to\nhave the same eigenfunction properties as the well-known Boson Spherical\nHarmonics [$Y_\\ell^{m}(\\theta,\\phi)$ for integer $\\ell$ and $m$]. The Fermion\nfunctions are shown to differ from the Boson functions in so far as the ladder\noperators $M_+$ ($M_-$) that ascend (descend) the sequence of harmonics over\nthe values of $m$ for a given value of $\\ell$, do not produce the expected\nresult {\\em in just one case}: when the value of $m$ changes from $\\pm{1/2}$ to\n$\\mp{1/2}$; i.e. when $m$ changes sign; in all other cases the ladder operators\nproduce the usually expected result including anihilation when a ladder\noperator attempts to take $m$ outside the range: $-\\ell\\le m\\le +\\ell$.\n The unexpected result in the one case does not invalidate this scalar\ncoordinate representation of spin angular momentum, because the eigenfunction\nproperty is essential for a valid quantum mechanical state, whereas ladder\noperators relating states with different eigenvalues are not essential, and are\nin fact known only for a few physical systems; that this coordinate\nrepresentation of spin angular momentum differs from the abstract theory of\nangular momentum in this respect, is simply an interesting curiosity. This new\nrepresentation of spin angular momentum is expected to find application in the\ntheoretical description of physical systems and experiments in which the\nspin-angular momentum (and associated magnetic moment) of a particle is\noriented in space, since the orientation is specifiable by the spherical polar\nangles, $\\theta$ and $\\phi$.",
"arxiv_id": "quant-ph/0507006",
"authors": [
"Geoffrey Hunter",
"Mohsen Emami-Razavi"
],
"categories": [
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],
"title": "Properties of Fermion Spherical Harmonics",
"url": "https://arxiv.org/abs/quant-ph/0507006"
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