Pauli Spinor Research Articles

The spatial wavefunction of half-integer spins

Within a nonrelativistic framework, spin is generally included as an intrinsic angular momentum. It is proposed here that consistent results can be obtained with a spatial wavefunction of an oriented complex exponential , where the bivector is written in terms of the pseudoscalar i = e x e y e z and the axis of rotation given by the unit vector , which is generally an (effective) magnetic field direction (this can also be written as in terms of the Pauli vector σ ). The wavefunction is a multivector and not a complex scalar. The signs in the exponential correspond to the two directions of rotation around the axis. The transformation properties of these wavefunctions are given by the Pauli spinors. Spin can be viewed as a zero-point rotation arising from the noncommutativity of the momentum and the vector potential.

A simple linear algebra approach to Pauli spinors

The transformation law of the Pauli spinors under rotations, linked to 1/2-spin particles, leads to a mathematical problem that can be rephrased and solved in purely linear algebraic terms. We present it as a possible approach to be incorporated in a first course of quantum mechanics when introducing the concept of the spin. This concept was very elusive at early times and had an intricate development commonly overlooked in modern texts. We also take the opportunity to make some historical comments.

Efficient evaluation of the Breit operator in the Pauli spinor basis.

The frequency-independent Coulomb-Breit operator gives rise to the most accurate treatment of two-electron interaction in the non-quantum-electrodynamics regime. The Breit interaction in the Coulomb gauge consists of magnetic and gauge contributions. The high computational cost of the gauge term limits the application of the Breit interaction in relativistic molecular calculations. In this work, we apply the Pauli component integral-density matrix contraction scheme for gauge interaction with a maximum spin- and component separation scheme. We also present two different computational algorithms for evaluating gauge integrals. One is the generalized Obara-Saika algorithm, where the Laplace transformation is used to transform the gauge operator into Gaussian functions and the Obara-Saika recursion is used for reducing the angular momentum. The other algorithm is the second derivative of Coulomb interaction evaluated with Rys-quadrature. This work improves the efficiency of performing Dirac-Hartree-Fock with the variational treatment of Breit interaction for molecular systems. We use this formalism to examine relativistic trends in the Periodic Table and analyze the relativistic two-electron interaction contributions in heavy-element complexes.

Adiabatic connection in spin-current density functional theory

The spin-current density functional theory (SCDFT), when formulated in a basis of Pauli spinors, provides a proper theoretical framework for the study of materials in an arbitrarily oriented external magnetic field and/or upon inclusion of spin-dependent relativistic effects, such as spin-orbit coupling. The SCDFT is formulated in terms of the particle-number density $n$, the Cartesian components of the magnetization ${m}_{x},$ ${m}_{y},$ and ${m}_{z}$, the orbital-current density $\mathbit{j}$, and the three spin-current densities ${\mathbit{J}}^{x},$ ${\mathbit{J}}^{y},$ and ${\mathbit{J}}^{z}$, where each of these density variables depends on specific blocks of the density matrix. Exchange-correlation (xc) functionals within the SCDFT should therefore depend on all of these eight fundamental density variables: ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z},\mathbit{j},{\mathbit{J}}^{x},{\mathbit{J}}^{y},{\mathbit{J}}^{z}]$, which makes their parametrization a formidable task. Here, we formulate the adiabatic connection of the SCDFT for a treatment of exact Fock exchange in the theory. We show how the inclusion of a fraction of Fock exchange in standard functionals of the (spin) DFT (either in their collinear or noncollinear versions: ${F}_{xc}[n],$ ${F}_{xc}[n,{m}_{z}]$ and ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z}]$) allows for the two-electron potential to depend on all those blocks of the density matrix that correspond to the eight density variables of the SCDFT, in a sensible and yet practical way. In particular, in the local-density and generalized-gradient approximations of the SCDFT, the treatment of the current densities solely from the Fock exchange term is formally justified by the short-range behavior of the exchange hole. We discuss that the adiabatic coupling strength parameter modulates the two-electron coupling of the orbital- and spin-current densities with the particle-number density and magnetization. Formal considerations are complemented by numerical tests on a periodic model system in the presence of spin-orbit coupling and in the absence of an external magnetic field.

Reappraisal of P, T-odd parameters from the improved calculation of electric dipole moment of 225Ra atom

By employing the recently developed relativistic normal coupled-cluster theory, we obtained the electric dipole moment (EDM) of 225Ra due to nuclear Schiff moment (NSM) and tensor–pseudotensor (T–PT) electron–nucleus (e–N) interaction as −6.3(5) × 10−17 S |e| cm/(|e| fm3) and −13(1) × 10−20 CT⟨σN⟩ |e| cm, respectively, for the NSM S, T–PT coupling coefficient CT and nuclear Pauli spinor σN. This is corroborated by analyzing the dipole polarizability value of 225Ra, which is obtained as 257(8) . Our EDM results, in combination with the measurement (2015 Phys. Rev. Lett. 114 233002), are about 8% and 28% improvements over the limits that were reported using the relativistic coupled-cluster theory for the NSM and T–PT e–N coupling coefficient, respectively. Further combining with the nuclear calculations, we infer limits on the EDM of neutron dn < 1.15 × 10−23 |e| cm, the EDM of proton dp < 1.1 × 10−22 |e| cm, the quantum chromodynamics parameter and the combined up- and down-quark chromo-EDMs . Though these bounds are not competitive at present with the current-best limits acquired from 199Hg atom, they can be bettered when the measurement of EDM in 225Ra improves by four orders.

On the quaternionic form of the Pauli–Schrödinger equation

A thorough approach to quantum mechanics should rely on an algebraic operation of multiplication based on proper algebraic structures, i.e. division algebras. Quaternions, with their commutation rules isomorphic to SU(2), offer such a framework. In this work, a quaternion algebraic approach derived from the finite groups is used to solve the spatially invariant time-dependent Pauli–Schrödinger equation and derive the Rabi formula and the principle of nuclear magnetic resonance and its generalization. The spatial and time-dependent Pauli–Schrödinger equation is then written using left-chirality quaternions, i.e. by expressing the Pauli spinor, its time derivative, and the Hamiltonian as elements of the same division algebra. The quaternion formalism is used to derive the dispersion relation of a free electron and of an electron in a constant magnetic field. Finally, the formalism is applied to calculate the transmission probabilities through a one-dimensional delta scatterer and through a resonant tunneling structure composed of two one-dimensional quaternionic delta-scatterers in series. The results are compared to the transmission probabilities calculated using traditional (complex) quantum mechanics.

On the quaternion representation of the Pauli spinor of an electron

The solution to the time-dependent Pauli–Schrödinger equation is written using quaternion algebra, i.e. by expressing the unitary operator describing the time evolution of the Pauli spinor and the Pauli spinor itself as left-chirality quaternions. The quaternion approach is used to express the time evolution of a qubit on the Bloch sphere under the action of the commonly used quantum gates and to implement the two-qubit controlled-NOT (CNOT) gate and the three-qubit Tofolli gate. The quaternion formalism is also applied to describe: (1) a general two-qubit quaternionic gate, (2) the Larmor precession of an electron spin in an uniform magnetic field, and (3) the issue of measurements in quantum mechanics. Finally, the time-dependent Pauli–Schrödinger equation is also solved using right-chirality quaternions.

Exact solutions of Schrödinger and Pauli equations for a charged particle on a sphere and interacting with non-central potentials

We calculate exact solutions of the Schrödinger equation for a particle constrained to move along a spherical surface and interacting with non-central potentials, namely, (i) Makarov, (ii) ring-shaped pseudo-harmonic oscillatory, and (iii) Kratzer potentials. We also study exact solutions of the Pauli equation in the same geometrical setting for a charged particle in the presence of a uniform magnetic field. In this case, the two-component spinor can adhere to the surface only if the magnetic field intensity has certain special values. The solutions of Schrödinger equations allow us to obtain exact Pauli spinors and their corresponding energy eigenvalues for the same non-central potentials.

Non-perturbative calculation of orbital and spin effects in molecules subject to non-uniform magnetic fields.

External non-uniform magnetic fields acting on molecules induce non-collinear spin densities and spin-symmetry breaking. This necessitates a general two-component Pauli spinor representation. In this paper, we report the implementation of a general Hartree-Fock method, without any spin constraints, for non-perturbative calculations with finite non-uniform fields. London atomic orbitals are used to ensure faster basis convergence as well as invariance under constant gauge shifts of the magnetic vector potential. The implementation has been applied to investigate the joint orbital and spin response to a field gradient-quantified through the anapole moments-of a set of small molecules. The relative contributions of orbital and spin-Zeeman interaction terms have been studied both theoretically and computationally. Spin effects are stronger and show a general paramagnetic behavior for closed shell molecules while orbital effects can have either direction. Basis set convergence and size effects of anapole susceptibility tensors have been reported. The relation of the mixed anapole susceptibility tensor to chirality is also demonstrated.

Universal lower bounds on the kinetic energy of electronic systems with noncollinear magnetism

The distribution of noncollinear magnetism in an electronic system provides information about the kinetic energy as well as some kinetic energy densities. Two different everywhere-positive kinetic densities related to the Schr\"odinger--Pauli Hamiltonian are considered. For one-electron systems described by a single Pauli spinor, the electron density, spin density and current density completely determines these kinetic energy densities. For many-electron systems, lower bounds on the kinetic energy densities are proved. These results generalize a lower bound due to von Weizs\"acker, which is based on the electron density alone and plays an important role in density functional theory. The results have applications in extensions of density functional theory that incorporate noncollinear spin densities and current densities.

The Stationary Dirac Equation as a Generalized Pauli Equation for Two Quasiparticles

By analyzing the Dirac equation with static electric and magnetic fields it is shown that Dirac’s theory is nothing but a generalized one-particle quantum theory compatible with the special theory of relativity. This equation describes a quantum dynamics of a single relativistic fermion, and its solution is reduced to solution of the generalized Pauli equation for two quasiparticles which move in the Euclidean space with their effective masses holding information about the Lorentzian symmetry of the four-dimensional space-time. We reveal the correspondence between the Dirac bispinor and Pauli spinor (two-component wave function), and show that all four components of the Dirac bispinor correspond to a fermion (or all of them correspond to its antiparticle). Mixing the particle and antiparticle states is prohibited. On this basis we discuss the paradoxical phenomena of Zitterbewegung and the Klein tunneling.

Measurement in the de Broglie-Bohm Interpretation: Double-Slit, Stern-Gerlach, and EPR-B

We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation, the position of a quantum particle exists and is piloted by the phase of the wave function. We show how this position explains determinism and realism in the three most important experiments of quantum measurement: double-slit, Stern-Gerlach, and EPR-B. First, we demonstrate the conditions in which the de Broglie-Bohm interpretation can be assumed to be valid through continuity with classical mechanics. Second, we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961 with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics. Third, we present an analytic expression of the wave function in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinor with a spatial extension. This solution enables us to demonstrate the decoherence of the wave function and the three postulates of quantum measurement. Finally, we study the Bohm version of the Einstein-Podolsky-Rosen experiment. Its theoretical resolution in space and time shows that a causal interpretation exists where each atom has a position and a spin.

Axial Anomaly and the Triality Symmetry of Leptons and Hadrons

We apply the supersymmetric model of \'E. Cartan to the pseudoscalar meson decay into two photons, $\pi_0\to\gamma\gamma$, $\eta\to\gamma\gamma$ and $\eta'\to\gamma\gamma$. In the book of \'E. Cartan published in 1966, Dirac spinors ${^t(}A,B)$ and ${^t(}C,D)$ and vector fields $E$ and $E'$ were introduced and five supersymmetric transformations $G_{23}, G_{12}, G_{13}, G_{123}$ and $G_{132}$ were considered. The Pauli spinor is treated as a quaternion and the Dirac spinor is treated as an octonion. In the pseudoscalar meson decay, when the two final vector fields belong to the same group ($EE$ or $E'E'$), we call the diagram rescattering diagram. When they belong to different groups ($EE'$), the diagram is called twisted diagram. Assuming the triality selection rules of octonions, dark matter is interpreted as matter emitting photons in a different triality sector than that of electromagnetic probes in our world.

Field equations for massless bosons from a Dirac–Weinberg formalism

Dirac’s first-order differential equation for relativistic spin 1/2 fermions can be extended to massless bosons using a formalism developed by Steven Weinberg. Weinberg’s theory is based heavily on details of the Lorentz and Poincaré groups and may not be accessible to the average reader. Although Weinberg’s theory was formulated over 40 years ago and his equations are easy to write down, few physicists seem to be familiar with them, and they are not discussed in standard quantum texts. In this paper we demonstrate the utility of his methods. Weinberg’s equations, which are valid for arbitrary spin, can be put in a form analogous to the Dirac equation for massless particles. Due to transversality conditions imposed on the theory so that it satisfies the second-order Klein–Gordon equation, we recast the theory in terms of a two-component Dirac–Pauli spinor. The theory is presented in some detail for j=1, corresponding to the photon and the vector Maxwell equations, and for j=2, corresponding to a graviton-like particle and a tensor version of Maxwell’s equations. These latter equations determine the polarization of gravitational waves in the weak field approximation. Our results correspond to those from the traceless-transverse gauge of general relativity.

Mesoscopic rings with spin-orbit interactions

A didactic description of charge and spin equilibrium currents on mesoscopic rings in the presence of spin–orbit interaction is presented. Emphasis is made on the non-trivial construction of the correct Hamiltonian in polar coordinates, the calculation of eigenvalues and eigenfunctions and the symmetries of the ground-state properties. Spin currents are derived following an intuitive definition, and then a more thorough derivation is built upon the canonical Lagrangian formulation that emphasizes the SU(2) gauge structure of the transport problem of spin-1/2 fermions in spin–orbit active media. The quantization conditions that follow from the constraint of single-valued Pauli spinors are also discussed. The targeted students are those of a graduate condensed matter physics course.

Relativistic two-component geometric approximation of the electron-positron contribution to magnetic properties in terms of Breit–Pauli spinors

An alternative approach for the calculation of the electron-positron (e-p) contribution to magnetic properties based on two-component Breit-Pauli spinors is presented. In it, the elimination of the small component scheme is applied to the inverse propagator matrix of e-p pairs. The effect of the positronic manifold is expressed as an operator acting on Breit-Pauli spinors. The operator form thus obtained sums up the relativistic correction as a geometric series and as a result a totally different behavior in the vicinity of a nucleus is obtained as compared to the one of the linear response approximation. This feature has deep influence in numerical values of the e-p contribution to the nuclear magnetic shielding of heavy atoms. Numerical calculations carried out for Kr, Xe, and I show that with this approach, the e-p contributions to this property are in good agreement with those of four-component methods.

The relativistic energy spectrum of hydrogen

AbstractThe method originally used by Dirac to derive his equation for a single electron is here applied to the two‐particle system electron plus proton, considered as an elementary fermion, to obtain a relativistic two‐particle Dirac‐Breit‐type Hamiltonian for the hydrogen atom. This problem can be solved exactly. Thus the relativistic energy levels of the hydrogen atom as a system bound by the static Coulomb force are obtained. The radial part of the resulting Hamiltonian operates in a four‐dimensional space describing particles, respectively antiparticles, i. e. electron and positron as well as proton and antiproton. The spin of a particle is described by the normal two‐component Pauli spinor, and therefore standard theoretical tools for dealing with angular‐momentum coupling can be exploited. The classical energy states of the hydrogen atom are retained in the appropriate non‐relativistic limit, in particular the energy levels resulting from Schrödinger's equation. The exact energy spectrum shows the expected dependence on the reduced mass of the two‐particle system, and thus describes the recoil of the core properly. The fine structure of the hydrogen spectrum arises from a dependence of the energy levels upon the quantum number of the total angular momentum.

The relativistic energy spectrum of hydrogen

The method originally used by Dirac to derive his equation for a single electron is here applied to the two-particle system electron plus proton, considered as an elementary fermion, to obtain a relativistic two-particle Dirac-Breit-type Hamiltonian for the hydrogen atom. This problem can be solved exactly. Thus the relativistic energy levels of the hydrogen atom as a system bound by the static Coulomb force are obtained. The radial part of the resulting Hamiltonian operates in a four-dimensional space describing particles, respectively antiparticles, i. e. electron and positron as well as proton and antiproton. The spin of a particle is described by the normal two-component Pauli spinor, and therefore standard theoretical tools for dealing with angular-momentum coupling can be exploited. The classical energy states of the hydrogen atom are retained in the appropriate non-relativistic limit, in particular the energy levels resulting from Schrodinger's equation. The exact energy spectrum shows the expected dependence on the reduced mass of the two-particle system, and thus describes the recoil of the core properly. The fine structure of the hydrogen spectrum arises from a dependence of the energy levels upon the quantum number of the total angular momentum.

Bundle Theory of Improper Spin Transformations

We give a geometrical description of the action of the parity operator (\(\hat{P}\)) on non-relativistic spin 1/2 Pauli spinors in terms of bundle theory. The relevant bundle, \(SU(2)\odot Z_2\to O(3)\), is a non-trivial extension of the universal covering group \(\hbox{\it SU}(2)\to \hbox{\it SO}(3). \hat{P}\) is the non-relativistic limit of the corresponding Dirac matrix operator \({\cal P}=i\gamma_0\) and obeys \(\hat{P}^2=-1\). From the direct product of O(3)$ by \(Z_2\), naturally induced by the structure of the Galilean group, we identify, in its double cover, the time-reversal operator (\(\hat{T}\)) acting on spinors, and its product with \(\hat{P}. \hat{P}\) and \(\hat{T}\) generate the group \(Z_4 \times Z_2\). As in the case of parity, \(\hat{T}\) is the non-relativistic limit of the corresponding Dirac matrix operator \({\cal T}=\gamma^3 \gamma^1\), and obeys \(\hat{T}^2=-1\).

Dirac–Schrödinger equation for quark–antiquark bound states and derivation of its interaction kernel

The four-dimensional Dirac–Schrödinger equation satisfied by quark–antiquark bound states is derived from quantum chromodynamics. Different from the Bethe–Salpeter equation, the equation derived is a kind of first-order differential equation of Schrödinger-type in the position space. Especially, the interaction kernel in the equation is given by two different closed expressions. One expression which contains only a few types of Green's functions is derived with the aid of the equations of motion satisfied by some kinds of Green's functions. Another expression, which is represented in terms of the quark, antiquark and gluon propagators and some kinds of proper vertices, is derived by means of the technique of irreducible decomposition of Green's functions. The kernel derived not only can easily be calculated by the perturbation method, but also provides a suitable basis for nonperturbative investigations. Furthermore, it is shown that the four-dimensional Dirac–Schrödinger equation and its kernel can be directly reduced to rigorous three-dimensional forms in the equal-time Lorentz frame and the Dirac–Schrödinger equation can be reduced to an equivalent Pauli–Schrödinger equation which is represented in the Pauli spinor space. To show the applicability of the closed expressions derived and to demonstrate the equivalence between the two different expressions of the kernel, the t-channel and s-channel one gluon exchange kernels are chosen as an example to show how they are derived from the closed expressions. In addition, the connection of the Dirac–Schrödinger equation with the Bethe–Salpeter equation is discussed.