Journal of Zhejiang Normal University

A simple method to determine the coupling vectors of orbit-spin angular momentums

Zhiwei Wang

Published Jan 1, 2009• 0 citations

Abstract

In most quantum mechanics textbooks,the determination of coupling vectors of multiple angular momentums was usually by looking up tables because of its complex calculation.It was introduced another method for the determination of coupling vectors,with the coupling of orbit-spin angular momentum of electrons as an example.

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Journal of Zhejiang Normal University

A simple method to determine the coupling vectors of orbit-spin angular momentums

Zhiwei Wang

Published Jan 1, 2009• 0 citations

OpenAlex

Abstract

Related papers

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M. H. Johnson

The vector model is examined in the light of the quantum theory of wave fields. First the distributions of electrons among the electronic states are limited to correspond to the atomic states arising from one electronic configuration. A set of independent angular momentum vectors ${1}_{1}$ , ${1}_{2}$ ,... ${\mathrm{S}}_{1}$ , ${\mathrm{S}}_{2}$ ... is found, in which one orbital and one spin vector are associated with each group of electronic states, rather than individual electrons, having given values of the quantum numbers $n$ and $l$ . Any other vector T may be decomposed into a sum of independent vectors ${\mathrm{t}}_{i}$ , one associated with each of the same group of electronic states. The usual commutation relations obtain among the vectors ${\mathrm{t}}_{i}$ , ${\mathrm{l}}_{i}$ and ${\mathrm{s}}_{i}$ . On further limiting the distributions to correspond to atomic states arising from electronic configurations containing no equivalent electrons, it is shown that the matrix of ${\ensuremath{\Sigma}}_{k}f({{\mathrm{l}}_{k}}^{0}, {{\mathrm{s}}_{k}}^{0})$ (an operator in coordinate space in which $f$ is any algebraic function of the components of 1 and s) can be constructed from the vectors ${1}_{1}$ , ${1}_{2}$ ... ${\mathrm{S}}_{1}$ , ${\mathrm{S}}_{2}$ ,... according to the usual laws of matrix mechanics.The vector model is examined in the light of the quantum theory of wave fields. First the distributions of electrons among the electronic states are limited to correspond to the atomic states arising from one electronic configuration. A set of independent angular momentum vectors ${1}_{1}$, ${1}_{2}$,...${\mathrm{S}}_{1}$, ${\mathrm{S}}_{2}$... is found, in which one orbital and one spin vector are associated with each group of electronic states, rather than individual electrons, having given values of the quantum numbers $n$ and $l$. Any other vector T may be decomposed into a sum of independent vectors ${\mathrm{t}}_{i}$, one associated with each of the same group of electronic states. The usual commutation relations obtain among the vectors ${\mathrm{t}}_{i}$, ${\mathrm{l}}_{i}$ and ${\mathrm{s}}_{i}$. On further limiting the distributions to correspond to atomic states arising from electronic configurations containing no equivalent electrons, it is shown that the matrix of ${\ensuremath{\Sigma}}_{k}f({{\mathrm{l}}_{k}}^{0}, {{\mathrm{s}}_{k}}^{0})$ (an operator in coordinate space in which $f$ is any algebraic function of the components of 1 and s) can be constructed from the vectors ${1}_{1}$, ${1}_{2}$...${\mathrm{S}}_{1}$, ${\mathrm{S}}_{2}$,... according to the usual laws of matrix mechanics.

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{1}_{1}

{1}_{2}

,...

{\mathrm{S}}_{1}

{\mathrm{S}}_{2}

... is found, in which one orbital and one spin vector are associated with each group of electronic states, rather than individual electrons, having given values of the quantum numbers

n

and

l

. Any other vector T may be decomposed into a sum of independent vectors

{\mathrm{t}}_{i}

, one associated with each of the same group of electronic states. The usual commutation relations obtain among the vectors

{\mathrm{t}}_{i}

{\mathrm{l}}_{i}

and

{\mathrm{s}}_{i}

. On further limiting the distributions to correspond to atomic states arising from electronic configurations containing no equivalent electrons, it is shown that the matrix of ${\ensuremath{\Sigma}}_{k}f({{\mathrm{l}}_{k}}^{0}, {{\mathrm{s}}_{k}}^{0})$ (an operator in coordinate space in which

f

is any algebraic function of the components of 1 and s) can be constructed from the vectors

{1}_{1}

{1}_{2}

...

{\mathrm{S}}_{1}

{\mathrm{S}}_{2}

,... according to the usual laws of matrix mechanics.