arXiv:quant-ph/0112096

Clebsch-Gordan coefficients and the binomial distribution

Paul O’Hara

Published Dec 17, 2001• 3 citations

Abstract

A class of Clebsch-Gordan coefficients are derived from the properties of conditional probability using the binomial distribution. In particular, in the case of $l=l_1+l_2$ it is shown that $$[^2 =\frac{(\begin{array}{c} l_1 k_1\end{array}) (\begin{array}{c}l_2 k_2\end{array})}{(\begin{array}{c}l k \end{array})}$$

DOI: 10.48550/arxiv.quant-ph/0112096

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arXiv:quant-ph/0112096

Clebsch-Gordan coefficients and the binomial distribution

Paul O’Hara

Published Dec 17, 2001• 3 citations

arXiv HTML PDF

Abstract

A class of Clebsch-Gordan coefficients are derived from the properties of conditional probability using the binomial distribution. In particular, in the case of

l=l_1+l_2

it is shown that

[^2 =\frac{(\begin{array}{c} l_1 k_1\end{array}) (\begin{array}{c}l_2 k_2\end{array})}{(\begin{array}{c}l k \end{array})}

DOI: 10.48550/arxiv.quant-ph/0112096

Related papers

Clebsch-Gordan coefficients, hypergeometric functions and the binomial distribution

Jean‐Christophe Pain

A particular case of degenerate Clebsch-Gordan coefficient can be expressed with three binomial coefficients. Such a formula, which may be obtained using the standard ladder operator procedure, can also be derived from the Racah-Shimpuku formula or from expressions of Clebsch-Gordan coefficients in terms of $_3F_2$ hypergeometric functions. The O'Hara interesting interpretation of this Clebsch-Gordan coefficient by binomial random variables can also be related to hypergeometric functions ( $_2F_1$ ), in the case where one of the parameters tends to infinity. This emphasizes the links between Clebsch-Gordan coefficients, hypergeometric functions and, what has been less exploited until now, the notion of probability within the framework of the quantum theory of angular momentum.A particular case of degenerate Clebsch-Gordan coefficient can be expressed with three binomial coefficients. Such a formula, which may be obtained using the standard ladder operator procedure, can also be derived from the Racah-Shimpuku formula or from expressions of Clebsch-Gordan coefficients in terms of $_3F_2$ hypergeometric functions. The O'Hara interesting interpretation of this Clebsch-Gordan coefficient by binomial random variables can also be related to hypergeometric functions ($_2F_1$), in the case where one of the parameters tends to infinity. This emphasizes the links between Clebsch-Gordan coefficients, hypergeometric functions and, what has been less exploited until now, the notion of probability within the framework of the quantum theory of angular momentum.

arXiv:2402.11298Feb 17, 20240 citations

A simple short derivation of the Clebsch-Gordan coefficients

Michael J. Caola

Latin American journal of physics educationJan 1, 20102 citations

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K. T. Hecht

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SPIE eBooksSep 12, 20190 citations

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A. M. Perelomov

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A. L. Shelepin, Leonid A. Shelepin

Physics of Atomic NucleiOct 1, 19931 citation

A class of Clebsch-Gordan coefficients are derived from the properties of conditional probability using the binomial distribution. In particular, in the case of

l=l_1+l_2

it is shown that

[^2 =\frac{(\begin{array}{c} l_1 k_1\end{array}) (\begin{array}{c}l_2 k_2\end{array})}{(\begin{array}{c}l k \end{array})}

_3F_2

hypergeometric functions. The O'Hara interesting interpretation of this Clebsch-Gordan coefficient by binomial random variables can also be related to hypergeometric functions (

_2F_1

), in the case where one of the parameters tends to infinity. This emphasizes the links between Clebsch-Gordan coefficients, hypergeometric functions and, what has been less exploited until now, the notion of probability within the framework of the quantum theory of angular momentum.