scholarly journals Spin 3/2 particle: Puali – Fierz theory, non-relativistic approximation

2020 ◽  
Vol 56 (3) ◽  
pp. 335-349
Author(s):  
A. V. Ivashkevich ◽  
Ya. A. Voynova ◽  
E. M. Оvsiyuk ◽  
V. V. Kisel ◽  
V. M. Red’kov
Keyword(s):  
Wave Function ◽  
Relativistic Particle ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Nonrelativistic Approximation ◽  
Matrix Algebras ◽  
First Order ◽  
Relativistic Approximation ◽  
Component Wave ◽  
Complete Matrix

The relativistic wave equation is well-known for a spin 3/2 particle proposed by W. E. Pauli and M. E. Fierz and based on the 16-component wave function with the transformation properties of the vector-bispinor. In this paper, we investigated the nonrelativistic approximation in this theory. Starting with the first-order equation formalism and representation of Pauli – Fierz equation in the Petras basis, also applying the method of generalized Kronecker symbols and elements of the complete matrix algebras, and decomposing the wave function into large and small nonrelativistic constituents with the help of projective operators, we have derived a Pauli-like equation for the 4-component wave function describing the non-relativistic particle with a 3/2 spin.

Bhabha’s equation for a particle of two mass states in Rarita—Schwinger form

1954 ◽  
Vol 222 (1148) ◽  
pp. 118-127 ◽  
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Positive Charge ◽  
Explicit Representation ◽  
Wave Functions ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Pauli Equation ◽  
Component Wave ◽  
Tensor Index

The irreducible relativistic wave equation for a particle having two different mass states and positive charge, given by Bhabha, has been written in a form similar to that given by Rarita & Schwinger for the Dirac-Fierz-Pauli equation for a particle of spin 3/2 . The components of the wave function are written as Dirac four-component wave functions, having in addition a tensor index, and one ordinary Dirac four-component wave function. The only matrices which enter into the formulation are the Dirac matrices. An explicit representation of Bhabha’s matrices in terms of the Dirac matrices is obtained. The solutions for spin 3/2 are just those given by the Dirac-Fierz-Pauli equation, but the solutions for spin ½ differ from the Dirac solutions in having additional non-vanishing components.

scholarly journals Quaternion treatment of the relativistic wave equation

1937 ◽  
Vol 162 (909) ◽  
pp. 145-154 ◽  
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Linear Functions ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Formal Representation ◽  
The Matrix ◽  
Scalar Products ◽  
Relationship Of ◽  
The Relationship

A set of matrices can be found which is isomorphic with any linear associative algebra. For the case of quaternions this was first shown by Cayley (1858), but the first formal representation was made by Peirce (1875, 1881). These were two-matrices, and the introduction of the four-row matrices of Dirac and Eddington necessitated the treatment of a wave function as a matrix of one row (as columns). Quaternions have been used by Lanczos (1929) to discuss a different form of wave equation, but here the Dirac form is discussed, the wave function being taken as a quaternion and the four-row matrices being linear functions of a quaternion. Certain advantages are claimed for quaternion methods. The absence of the distinction between outer and scalar products in the matrix notation necessitates special expedients (Eddington 1936). Every matrix is a very simple function of the fundamental Hamiltonian vectors α, β, γ , so that the result of combination is at once evident and depends only on the rules of combination of these vectors. At all stages the relationship of the different quantities to four-space is at once visible. The Dirac-Eddington matrices, the wave equation and its exact solution by Darwin, angular momentum operators, the general and Lorentz transformation, spinors and six-vectors, the current-density four-vector are treated in order to exhibit the working of this method. S and V for scalar and vector products are used. Quaternions are denoted by Clarendon type, and all vectors are in Greek letters.

scholarly journals Klein-Gordon Equation and Wave Function for Free Particle in Rindler Space-Time

2021 ◽  
Author(s):  
Sangwha Yi
Keyword(s):  
Wave Function ◽  
Gordon Equation ◽  
Free Particle ◽  
Space Time ◽  
Spinless Particle ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Rindler Space ◽  
Klein Gordon ◽  
Klein Gordon Equation

Klein-Gordon equation is a relativistic wave equation. It treats spinless particle. The wave functioncannot use as a probability amplitude. We made Klein-Gordon equation in Rindler space-time. In this paper,we make free particle’s wave function as the solution of Klein-Gordon equation in Rindler space-time.

scholarly journals KALB–RAMOND FIELDS IN THE PETIAU–DUFFIN–KEMMER FORMALISM AND SCALE INVARIANCE

2010 ◽  
Vol 25 (32) ◽  
pp. 2745-2751 ◽  
Author(s):  
S. I. KRUGLOV
Keyword(s):  
Wave Equation ◽  
Density Matrix ◽  
Scale Invariance ◽  
Conformal Symmetry ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Massless Particles ◽  
First Order ◽  
Dilatation Current ◽  
Massless Fields

Kalb–Ramond equations for massive and massless particles are considered in the framework of the Petiau–Duffin–Kemmer formalism. We obtain 10×10 matrices of the relativistic wave equation of the first-order and solutions in the form of density matrix. The canonical and Belinfante energy–momentum tensors are found. We investigate the scale invariance and obtain the conserved dilatation current. It was demonstrated that the conformal symmetry is broken even for massless fields.

A MODEL OF THE DEUTERON BASED ON A BREIT TYPE EQUATION

1994 ◽  
Vol 09 (14) ◽  
pp. 1327-1333
Author(s):  
P. LEAL FERREIRA ◽  
J.A. CASTILHO ALCARÁS ◽  
A.P. GALEĀO
Keyword(s):  
Wave Function ◽  
Type Equation ◽  
Deuteron Binding Energy ◽  
Radial System ◽  
Radial Functions ◽  
First Order ◽  
Component Wave ◽  
Relativistic Treatment ◽  
Breit Equation ◽  
Conditions At Infinity

A relativistic treatment of the deuteron and its observables based on a two-body Dirac (Breit) equation, with phenomenological interactions, associated to one-boson exchanges with cutoff masses, is presented. The 16-component wave function for the deuteron (Jπ=1+) solution contains four independent radial functions which obey a system of four coupled differential equations of first order. This radial system is numerically integrated, from infinity to the origin, by fixing the value of the deuteron binding energy and using appropriate boundary conditions at infinity. Specific examples of mixtures containing scalar, pseudoscalar and vector like terms are discussed in some detail and several observables of the deuteron are calculated. Our treatment differs from more conventional ones in that nonrelativistic reductions of the order c−2 are not used.

Screening constants for relativistic wave functions

1966 ◽  
Vol 62 (4) ◽  
pp. 777-782 ◽  
Author(s):  
R. H. Garstang ◽  
D. F. Mayers
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Wave Functions ◽  
Relativistic Wave Equation ◽  
Mean Square ◽  
Relativistic Wave ◽  
Coulomb Wave Function ◽  
Relativistic Coulomb ◽  
The Mean ◽  
Good Agreement

AbstractFormulae for the mean radius and mean square radius of a relativistic Coulomb wave function are obtained. Screening constants for the energy, mean radius and mean square radius are defined relative to non-relativistic wave functions and the results of numerical calculations given. It is shown that if the screening constants so determined are added to the screening constants due to the presence of other electrons as found by the s.c.f. method, good agreement is obtained in a case where both effects have been considered together. The value of solving the relativistic wave equation in a Thomas-Fermi field is also shown.

FIRST-ORDER APPROXIMATION OF THE NUMBER-PROJECTED SO(2N) TAMM-DANCOFF EQUATION AND ITS REDUCTION BY THE SCHUR FUNCTION

1999 ◽  
Vol 08 (05) ◽  
pp. 461-483
Author(s):  
SEIYA NISHIYAMA
Keyword(s):  
Wave Function ◽  
Order Approximation ◽  
Approximate Equation ◽  
Variation Principle ◽  
Schur Function ◽  
Soliton Theory ◽  
First Order ◽  
Fermion Systems ◽  
Group Manifold ◽  
First Order Approximation

First-order approximation of the number-projected (NP) SO(2N) Tamm-Dancoff (TD) equation is developed to describe ground and excited states of superconducting fermion systems. We start from an NP Hartree-Bogoliubov (HB) wave function. The NP SO(2N) TD expansion is generated by quasi-particle pair excitations from the degenerate geminals in the number-projected HB wave function. The Schrödinger equation is cast into the NP SO(2N) TD equation by the variation principle. We approximate it up to first order. This approximate equation is reduced to a simpler form by the Schur function of group characters which has a close connection with the soliton theory on the group manifold.

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