arXiv:quant-ph/0507008

Explicit Spin Coordinates

Geoffrey Hunter, I. Schlifer

Published Jul 1, 2005• 2 citations

Abstract

The recently established existence of spherical harmonic functions, $Y_\ell^{m}(θ,ϕ)$ for half-odd-integer values of $\ell$ and $m$, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles $θ_s, ϕ_s$, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, $S^2, S_z$, are represented by the usual differential operators in spherical polar coordinates (commonly used for $L^2, L_z$), and their electron-spin eigenfunctions are $\sqrt{\sinθ_s} \exp(\pmϕ_s/2)$. This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, $α, β,$ that ``integration over spin coordinates'' is a true integration (over the angles $θ_s, ϕ_s$). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.

DOI: 10.48550/arxiv.quant-ph/0507008

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Properties of Fermion Spherical Harmonics

Geoffrey Hunter, Mohsen Emami-Razavi

arXiv:quant-ph/0507006Jul 1, 20054 citations

Explicit Expression for Spherical Harmonics

L. A. Schmid

American Journal of PhysicsOct 1, 19580 citations

The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta

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arXiv:2305.11456May 19, 20230 citations

N-Dimensional Total Orbital Angular-Momentum Operator. II. Explicit Representations

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A method of generating orthogonal polar coordinate systems in N-dimensional space is given. Commuting angular-momentum operators are easily found in the coordinate systems generated; these operators are of a single form which depends on two parameters. A short table of coordinate systems and the resulting structure of quantum numbers and eigenfunctions is given.

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Addition theorems for spin spherical harmonics. I Preliminaries

Antonio O. Bouzas

Jan 1, 20134 citations

Addition theorems for spin spherical harmonics: I. Preliminaries

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arXiv:quant-ph/0507008

Explicit Spin Coordinates

Geoffrey Hunter, I. Schlifer

Published Jul 1, 2005• 2 citations

arXiv HTML PDF

Abstract

The recently established existence of spherical harmonic functions,

Y_\ell^{m}(θ,ϕ)

for half-odd-integer values of

\ell

and

m

, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles

θ_s, ϕ_s

, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators,

S^2, S_z

, are represented by the usual differential operators in spherical polar coordinates (commonly used for

L^2, L_z

), and their electron-spin eigenfunctions are

\sqrt{\sinθ_s} \exp(\pmϕ_s/2)

. This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions,

α, β,

that ``integration over spin coordinates'' is a true integration (over the angles

θ_s, ϕ_s

). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.

DOI: 10.48550/arxiv.quant-ph/0507008

Related papers

Properties of Fermion Spherical Harmonics

Geoffrey Hunter, Mohsen Emami-Razavi

The Fermion Spherical harmonics [ $Y_\ell^{m}(θ,ϕ)$ 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 [ $Y_\ell^{m}(θ,ϕ)$ 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, $θ$ and $ϕ$ .The Fermion Spherical harmonics [$Y_\ell^{m}(θ,ϕ)$ 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 [$Y_\ell^{m}(θ,ϕ)$ 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, $θ$ and $ϕ$.

arXiv:quant-ph/0507006Jul 1, 20054 citations

Explicit Expression for Spherical Harmonics

L. A. Schmid

A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.

American Journal of PhysicsOct 1, 19580 citations

The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta

Theodore Peter Rakitzis, Michail E. Koutrakis, George E. Katsoprinakis

In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$ . In contrast, here we show that a spatial wavefunction, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$ , which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $ϕ$ , $θ$ , $χ$ are the Euler angles, and $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$ ) which predicts the effective wavepacket angular uncertainty relations for $Δm Δϕ$ , $Δj Δχ$ , and $ΔϕΔθ$ , and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$ -rotation measurements. We also use the $j_m (ϕ,θ,χ)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$ , and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to $j=1/2$ , these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. In contrast, here we show that a spatial wavefunction, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$, which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $ϕ$, $θ$, $χ$ are the Euler angles, and $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$) which predicts the effective wavepacket angular uncertainty relations for $Δm Δϕ$, $Δj Δχ$, and $ΔϕΔθ$, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$-rotation measurements. We also use the $j_m (ϕ,θ,χ)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to $j=1/2$, these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.

arXiv:2305.11456May 19, 20230 citations

N-Dimensional Total Orbital Angular-Momentum Operator. II. Explicit Representations

Kenneth D. Granzow

Journal of Mathematical PhysicsOct 1, 19647 citations

Addition theorems for spin spherical harmonics. I Preliminaries

Antonio O. Bouzas

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.

Jan 1, 20134 citations

Addition theorems for spin spherical harmonics: I. Preliminaries

Antonio O Bouzas

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).

arXiv:1103.2982Journal of Physics A Mathematical and TheoreticalMar 24, 20115 citations

The recently established existence of spherical harmonic functions,

Y_\ell^{m}(θ,ϕ)

for half-odd-integer values of

\ell

and

m

, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles

θ_s, ϕ_s

, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators,

S^2, S_z

, are represented by the usual differential operators in spherical polar coordinates (commonly used for

L^2, L_z

), and their electron-spin eigenfunctions are

\sqrt{\sinθ_s} \exp(\pmϕ_s/2)

. This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions,

α, β,

that ``integration over spin coordinates'' is a true integration (over the angles

θ_s, ϕ_s

The Fermion Spherical harmonics [

Y_\ell^{m}(θ,ϕ)

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}(θ,ϕ)

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}

\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,

θ

and

ϕ

In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum

j

. In contrast, here we show that a spatial wavefunction,

j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}

, which treats

j

in the

|jm>

state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators;

ϕ

θ

χ

are the Euler angles, and

cos θ_m=(m/|j|)

is the Vector-Model polar angle. The

j_m (ϕ,θ,χ)

gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in

j

and

m

) which predicts the effective wavepacket angular uncertainty relations for

Δm Δϕ

Δj Δχ

, and

ΔϕΔθ

, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive

j

-rotation measurements. We also use the

j_m (ϕ,θ,χ)

to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles,

g=2

, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to

j=1/2

, these expressions are either exact (the last two) or excellent approximations (the first two), showing that

j_m (ϕ,θ,χ)

gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.