Investigations of spin-eigenfunction correlated wavefunctions

The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.

Journal of Mathematical PhysicsFeb 1, 196836 citations

Correlated Ground-State Wavefunctions

WORLD SCIENTIFIC eBooksOct 1, 20120 citations

Explicit Spin Coordinates

Geoffrey Hunter, I. Schlifer

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.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.

arXiv:quant-ph/0507008Jul 1, 20052 citations

The Construction of Spin Eigenfunctions

Ruben Pauncz

Springer eBooksJan 1, 200055 citations

Eigenfunctions for SU( nu ) particles with 1/r2interaction in harmonic confinement

K Vacek, A Okiji, N Kawakami

We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.

arXiv:cond-mat/9402012Journal of Physics A Mathematical and GeneralApr 7, 199427 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

). 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.