arXiv:2308.02884

Path distributions for describing eigenstates of orbital angular momentum

Randall M. Feenstra

Published Aug 5, 2023• 0 citations

Abstract

The manner in which probability amplitudes of paths sum up to form wave functions of orbital angular momentum eigenstates is described. Using a generalization of stationary-phase analysis, distributions are derived that provide a measure of how paths contribute towards any given eigenstate. In the limit of long travel-time, these distributions turn out to be real-valued, non-negative functions of a momentum variable that describes classical travel between the endpoints of a path (with the paths explicitly including nonclassical ones, described in terms of elastica). The distributions are functions of both this characteristic momentum as well as a polar angle that provides a tilt, relative to the z-axis of the chosen coordinate system, of the geodesic that connects the endpoints. The resulting description provides a replacement for the well-known "vector model" for describing orbital angular momentum, and importantly, it includes treatment of the case when the quantum number $\ell$ is zero (i.e., s-states).

DOI: 10.48550/arxiv.2308.02884

Related papers

Path distributions for describing eigenstates of the harmonic oscillator and other 1-dimensional problems

Randall M. Feenstra

arXiv:2306.11155Jun 19, 20230 citations

Quantum formulation of angle and orbital angular momentum

Jörg B. Götte, Stephen Mark Barnett

Cambridge University Press eBooksNov 8, 20122 citations

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

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

arXiv:2305.11456May 19, 20230 citations

A note on angular momentum and sums over classical paths

A Norcliffe

The spherical top propagator and spectral operator kernel for all spins are evaluated in terms of new rotational coordinates that are related to the Euler angles in a simple way. Both operator kernels are expressible exactly as sums over classical paths. The angular momentum wavefunctions are single-valued functions of the new rotational coordinates for all values of j.

Journal of Physics A Mathematical Nuclear and GeneralSep 1, 19731 citation

Orbital angular momentum from marginals of quadrature distributions

Luis Lorenzo Sánchez-Soto, A. B. Klimov, Pablo de la Hoz +4 more

arXiv:1312.5433Physical Review ANov 25, 20135 citations

The Vector Model and the Pauli Principle

M. H. Johnson

Physical ReviewApr 15, 19333 citations

arXiv:2308.02884

Path distributions for describing eigenstates of orbital angular momentum

Randall M. Feenstra

Published Aug 5, 2023• 0 citations

arXiv HTML PDF

Abstract

\ell

is zero (i.e., s-states).

DOI: 10.48550/arxiv.2308.02884

Related papers

Path distributions for describing eigenstates of the harmonic oscillator and other 1-dimensional problems

Randall M. Feenstra

The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.

arXiv:2306.11155Jun 19, 20230 citations

Quantum formulation of angle and orbital angular momentum

Jörg B. Götte, Stephen Mark Barnett

The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.

Cambridge University Press eBooksNov 8, 20122 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

A note on angular momentum and sums over classical paths

A Norcliffe

Journal of Physics A Mathematical Nuclear and GeneralSep 1, 19731 citation

Orbital angular momentum from marginals of quadrature distributions

Luis Lorenzo Sánchez-Soto, A. B. Klimov, Pablo de la Hoz +4 more

We set forth a method to analyze the orbital angular momentum of a light field. Instead of using the canonical formalism for the conjugate pair angle-angular momentum, we model this latter variable by the superposition of two independent harmonic oscillators along two orthogonal axes. By describing each oscillator by a standard Wigner function, we derive, via a consistent change of variables, a comprehensive picture of the orbital angular momentum. We compare this with previous approaches and show how this method works in some relevant examples.We set forth a method to analyze the orbital angular momentum of a light field. Instead of using the canonical formalism for the conjugate pair angle-angular momentum, we model this latter variable by the superposition of two independent harmonic oscillators along two orthogonal axes. By describing each oscillator by a standard Wigner function, we derive, via a consistent change of variables, a comprehensive picture of the orbital angular momentum. We compare this with previous approaches and show how this method works in some relevant examples.

arXiv:1312.5433Physical Review ANov 25, 20135 citations

The Vector Model and the Pauli Principle

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.

Physical ReviewApr 15, 19333 citations

\ell

is zero (i.e., s-states).

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.

{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.