Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Expansions of spin-1/2 expectation values

Richard Clinton Fernow

Published Jun 1, 1976• 3 citations
Publisher

Abstract

We review some general properties of density and scattering matrices and composite spin spaces of spin-1/2 particles. We give a derivation of a useful expansion of spin expectation values. The expansion is used in a number of simple examples. Spin tensors and pure spin cross sections are introduced.
DOI
10.1119/1.10362

Related papers

Calculation of Expectation Values of S$_{z}$ and S$^{2}$ operators for spin-1 and spin-3/2 particles

Selahattin GÖNEN, Ali Havare

In the present article we calculate the expectation values of of S$_{z}$ and S$^{2}$ operators for spin-1 and spin-3/2 particles by expanding a general wave function which includes all spin values. The results are same as in the stantard quantum mechanics.

arXiv:hep-th/0210106•Oct 11, 2002•0 citations

The classification of three-parameter density matrices for a qutrit

Istok P. Mendaš

The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.

Journal of Physics A Mathematical and General•Aug 18, 2006•66 citations

Exact Relations for a Strongly Interacting Fermi Gas from the Operator Product Expansion

Eric Braaten, Lucas Platter

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 at large momentum k, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.

arXiv:0803.1125•Physical Review Letters•May 21, 2008•298 citations

Generalized Probability Amplitudes for Spin Projection Measurements on Spin 2 Systems

Habatwa Vincent Mweene

In previous work, we have shown how the description of spin may be generalized and we have worked out this generalization for the cases spin 1/2 and spin 1. In this paper, we deal with the case of spin 2 and give the generalized probability amplitudes.

arXiv:quant-ph/0502005•Feb 1, 2005•1 citation

Classification and time evolution of density matrices for a N-state system

Istok P. Mendaš

A possible method of classifying all density matrices for a N-state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the SU(N) generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined time-dependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a N-state system. The general discussion is illustrated throughout with the special case of a N=4-state quantum system. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.

Journal of Mathematical Physics•Sep 1, 2008•5 citations

Expansion of the spin-spin interaction

Keh‐Ning Huang

The space part of a general spin-spin interaction is expanded in a complete orthonormal set of irreducible tensors constructed from spherical harmonics of the interacting particles. With this expansions, the $m$-scheme matrix element of the spin-spin interaction is represented as a sum of weighted radial integrals along with its graphical representation. The graphical form of the $m$-scheme matrix element enables one to obtain graphically matrix elements between many-particle configurations in any coupling scheme.

Physical Review A•Nov 1, 1980•0 citations
Calculation of Expectation Values of Sz_{z}z​ and S2^{2}2 operators for spin-1 and spin-3/2 particles
In the present article we calculate the expectation values of of Sz_{z}z​ and S2^{2}2 operators for spin-1 and spin-3/2 particles by expanding a general wave function which includes all spin values. The results are same as in the stantard quantum mechanics.
The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.
The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 at large momentum k, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.
A possible method of classifying all density matrices for a N-state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the SU(N) generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined time-dependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a N-state system. The general discussion is illustrated throughout with the special case of a N=4-state quantum system. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.
The space part of a general spin-spin interaction is expanded in a complete orthonormal set of irreducible tensors constructed from spherical harmonics of the interacting particles. With this expansions, the mmm-scheme matrix element of the spin-spin interaction is represented as a sum of weighted radial integrals along with its graphical representation. The graphical form of the mmm-scheme matrix element enables one to obtain graphically matrix elements between many-particle configurations in any coupling scheme.