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

International Education and Research Journal

ELECTRON SPIN IN THE REPRESENTATION WHICH MAKES σ _z DIAGONAL AND IN THE REPRESENTATION THAT MAKES σ _x DIAGONAL

Dr. Palash Das

Published Feb 15, 2026• 0 citations
Publisher

Abstract

We calculate the density matrix of an electron spin in the representation that makes diagonal and hence calculate the ensemble average of the expectation value of .We make the same scheme in the representation that makes diagonal and show that the ensemble average of the expectation value of , resulting from this representation is precisely the same as the initial one, where are Pauli’s spin matrices and is the z-component of the intrinsic magnetic moment of the electron. We then conclude accordingly.

DOI
10.5281/zenodo.18847085

Related papers

All Moments of the Uniform Ensemble of Quantum Density Matrices

Robert R. Tucci

Given a uniform ensemble of quantum density matrices $ρ$, it is useful to calculate the mean value over this ensemble of a product of entries of $ρ$. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions $P=(P_1, P_2, ..., P_N)$ where the $P_j$ are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of $P$ components. The answer to the classical problem follows from an integral formula due to Dirichlet.

arXiv:quant-ph/0206193•Jun 28, 2002•2 citations

From the Single Electron to Spin

Plinio Innocenzi

The nature of the electron, a point particle according to the Standard Model, presents a paradox that challenges our ability to comprehend the physical reality. This paradox lies in the apparent contradiction between the complexity represented according to mathematical models that fully describe experimental results and the world around us perceived as a representation of an intuitive process. For example, how come all electrons are the same? As we shall see in this chapter, a seemingly simple question stimulated original reflections that later led to significant theoretical developments. Stern and Gerlach’s experiments demonstrated that the electron possesses an intrinsic property, the spin, whose generalization led to the distinction between fermions and bosons. The spin property, however, is apparently not reconcilable with the point properties of the electron. It has, in fact, not any analogue in classical physics. This incongruence also applies to the electron’s magnetic dipole moment. The key to everything is perhaps the spin property, which has Pauli’s Exclusion Principle as its consequence, to which we owe the existence of matter as we perceive it.

Springer eBooks•Jan 1, 2025•0 citations

Solving the Problem of Source of Electron Spin Magnetic Moment

Runsheng Tu

The concept of electron spin has a broad application prospect. However, after a century of efforts, the essence and specific form of electron spin have not been revealed. The reason is probably that it is constrained by the concept of point particle material structure. In order to solve the problem, one way to initiate the quantum mechanics revolution on the concept of material structure is to establish the concept of wave element material structure. Assuming that the composition of electrons is waves, the inherent motion of electrons (the spin of free electrons) is "the rotation of mass equivalent to the energy of waves". This rotation is not the rotation of a sphere, but similar to the rotation of a ring-shaped substance along a ring. This assumption breaks the concept of point particle structure. According to it, the angular kinetic energy and spin magnetic moment of electrons can be accurately calculated by classical force electrodynamics method. Other applications are also very successful, which has led to the revolution of quantum mechanics and material structure theory. The electron spin magnetic moment and electron spin angular momentum both originate from the motion of the waves that make up the electrons along the phase trajectory, providing a highly unified explanation for the fact that "discrete waves make up localized electrons and electrons have spin magnetic moments".

Physical Science International Journal•Nov 29, 2024•1 citation

Solving one-body ensemble N-representability problems with spin

Julia Liebert, Federico Castillo, Jean‐Philippe Labbé +2 more

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies n i of orbitals φ i according to 0 ≤ n i ≤ 2 . In this work, we first refine the underlying one-body N -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness w of the N -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope Σ N , S ( w ) ⊂ [ 0 , 2 ] d . These constraints are independent of M and the number d of orbitals, while their dependence on N , S is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

arXiv:2412.01805•Quantum•Dec 2, 2025•0 citations

Expectation Values of Many-Fermion Spin Eigenstates

Klaus Ruedenberg

Many-particle antisymmetric spin eigenstates are formulated in a manner that leads to simple expressions for expectation values of spin-independent many-particle operators. Explicit matrix elements are given for one- and two-particle operators.

Physical Review Letters•Oct 25, 1971•55 citations

The Density Matrix

Michael J.R. Hoch

In dealing with systems whose eigenstates are known, such as ideal spin systems in a magnetic field, it is sometimes useful to introduce a formalism involving the density matrix. Knowledge of the density matrix operator permits the expectation values of other operators to be determined in a way that is straightforward in principle. In the presence of a magnetic field or some other polarizing mechanism, the form of the density matrix will change, and the expectation value, for example, may be nonzero. Although the situation that has been considered involves an electron beam, a related but somewhat more complicated approach is used for polarized light beams.

CRC Press eBooks•Mar 30, 2021•0 citations
We calculate the density matrix of an electron spin in the representation that makes diagonal and hence calculate the ensemble average of the expectation value of .We make the same scheme in the representation that makes diagonal and show that the ensemble average of the expectation value of , resulting from this representation is precisely the same as the initial one, where are Pauli’s spin matrices and is the z-component of the intrinsic magnetic moment of the electron. We then conclude accordingly.
Given a uniform ensemble of quantum density matrices ρρρ, it is useful to calculate the mean value over this ensemble of a product of entries of ρρρ. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions P=(P1,P2,...,PN)P=(P_1, P_2, ..., P_N)P=(P1​,P2​,...,PN​) where the PjP_jPj​ are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of PPP components. The answer to the classical problem follows from an integral formula due to Dirichlet.
The nature of the electron, a point particle according to the Standard Model, presents a paradox that challenges our ability to comprehend the physical reality. This paradox lies in the apparent contradiction between the complexity represented according to mathematical models that fully describe experimental results and the world around us perceived as a representation of an intuitive process. For example, how come all electrons are the same? As we shall see in this chapter, a seemingly simple question stimulated original reflections that later led to significant theoretical developments. Stern and Gerlach’s experiments demonstrated that the electron possesses an intrinsic property, the spin, whose generalization led to the distinction between fermions and bosons. The spin property, however, is apparently not reconcilable with the point properties of the electron. It has, in fact, not any analogue in classical physics. This incongruence also applies to the electron’s magnetic dipole moment. The key to everything is perhaps the spin property, which has Pauli’s Exclusion Principle as its consequence, to which we owe the existence of matter as we perceive it.
The concept of electron spin has a broad application prospect. However, after a century of efforts, the essence and specific form of electron spin have not been revealed. The reason is probably that it is constrained by the concept of point particle material structure. In order to solve the problem, one way to initiate the quantum mechanics revolution on the concept of material structure is to establish the concept of wave element material structure. Assuming that the composition of electrons is waves, the inherent motion of electrons (the spin of free electrons) is "the rotation of mass equivalent to the energy of waves". This rotation is not the rotation of a sphere, but similar to the rotation of a ring-shaped substance along a ring. This assumption breaks the concept of point particle structure. According to it, the angular kinetic energy and spin magnetic moment of electrons can be accurately calculated by classical force electrodynamics method. Other applications are also very successful, which has led to the revolution of quantum mechanics and material structure theory. The electron spin magnetic moment and electron spin angular momentum both originate from the motion of the waves that make up the electrons along the phase trajectory, providing a highly unified explanation for the fact that "discrete waves make up localized electrons and electrons have spin magnetic moments".
The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies n i of orbitals φ i according to 0 ≤ n i ≤ 2 . In this work, we first refine the underlying one-body N -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness w of the N -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope Σ N , S ( w ) ⊂ [ 0 , 2 ] d . These constraints are independent of M and the number d of orbitals, while their dependence on N , S is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.
In dealing with systems whose eigenstates are known, such as ideal spin systems in a magnetic field, it is sometimes useful to introduce a formalism involving the density matrix. Knowledge of the density matrix operator permits the expectation values of other operators to be determined in a way that is straightforward in principle. In the presence of a magnetic field or some other polarizing mechanism, the form of the density matrix will change, and the expectation value, for example, may be nonzero. Although the situation that has been considered involves an electron beam, a related but somewhat more complicated approach is used for polarized light beams.