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Compt. rend.

THE INTERPRETATION OF QUANTUM MECHANICS. RELATIVISTIC CASE OF A BODY WITH SPIN

A.B. Datzeff

Published Jan 22, 1962• 0 citations
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Abstract

The quantum problem of a body with spin, in the case of relativistic velocities, is treated with the aid of wellknown Dirac equations containing four wave functions. In the present work it is shown that the development of the ideas expressed previously (Compt. rend., 250: 67(1960) and Compt. rend., 254: 434(1962)) leads to the mathematical appliance of the Dirac theory, that is, the interpretation of quantum mechanics given can be transported to Dirac equations. (J.S.R.)

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THE INTERPRETATION OF QUANTUM MECHANICS. THE CASE OF THE RELATIVISTIC VELOCITY OF A PARTICLE WITHOUT SPIN

A.B. Datzeff

In previous works on probabilistic description of the quantum motion of a micro-particle mu the non-relativistic Schroedinger equation was derived. The idea developed is now used to show that one can find this probabilistic description for mu also for relativistic velocities. (T.R.H.)

Compt. rend.•Jan 1, 1960•0 citations

Sur l'interprétation de la mécanique quantique. V. Spin et relativité

A.B. Datzeff

The results of the present work are a summary of those obtained in the previous publications [1]-[4]. In (A) is considered a corpuscule with spin introduced according to the hypothesis of Uhlenbeck and Goudsmit. The probability equations obtained here coincide with the wellknown equations of Pauli. In (B) is found a probability equation for a corpuscule without spin in a relativistic case. It proves that it coincides with the well-known equation of Klein-Gordon. In (C) is studied the movement of a corpuscule with spin in a relativistic case. The probability equations obtained here coincide with the well-known equations of Dirac. The matrices of Pauli as well as those of Dirac are found here to be a conséquence of the probability equations. The results obtained here lead to the conclusion that the interpretation of quantum mechanics developed in the previous works may be spread over the cases, considered here.

Journal de physique•Jan 1, 1962•0 citations

The Dirac Equation

Kenneth G. Dyall, Knut Fægri

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

Oxford University Press eBooks•Jul 5, 2007•0 citations

STUDY OF THE PHYSICAL INTERPRETATION OF OPERATIORS IN THE RELATIVISTIC QUANTUM MECHANICS OF SPIN PARTICLES

M. Baktavatsalou

It is shown that operators that are observables can be obtained in relativistic qaantum mechanics; and the operators are determined for electrons, bosons of spin 0 or 1, and particles of arbitrary spin and nonzero mass. The definition and interpretation of the position operators, specifically, are treated. (D.C.W.)

•Jul 1, 1963•0 citations

Spin - 1/2 wave equations in relativistic quantum mechanics

D S Staudte

This thesis is concerned with the development of a new spin-1/2 wave equation in relativistic quantum mechanics. This equation is the spin-1/2 analogue of the spin-0 Feshbach-Villars equation. The thesis begins in chapter 1 with a review of the subject of relativistic wave equations and its place in the development of modern physics. Chapter 2 sees the derivation of the new equation (hereafter referred to as the FV1/2 equation). This derivation is presented at three levels of understanding, and is combined with an analysis of the equation using elements of the underlying mathematics of the theories of special relativity and quantum mechanics. The analysis justifies the methods of derivation of the equation, and its suitability as a wave equation in relativistic quantum mechanics. It is shown th a t the FV1/2 equation is different to the Dirac equation and other equations in the literature. One of the striking features of the FV1/2 equation is th a t it contains a less restrictive dynamics than the conventional Dirac equation. Chapter 3 examines the solution space and proves some results using the discrete symmetries C, P and T, which enable a subset of the solution space to be used for physical applications. It is possible to use a subset of solutions which contain the usual Dirac negative energy states, or it is possible to use a subset which contains only positive energy states, but includes states of opposite signs of charge. The second subset corresponds to the physical properties of particles and antiparticles and suggests th a t the FV1/2 equation could be useful in atomic physics. Chapter 4 represents a start on the development of the FV1/2 equation for use in atomic physics. It contains a study of hydrogenic atoms. The energy spectra and wavefunctions for hydrogenic atoms are derived using the FV1/2 equation. The method of solution is of comparable difficulty to th a t using the Dirac equation. The spectra are found to be identical to th a t of the Dirac equation, but the wavefunctions differ. A calculation of the expectation value of the Coulomb energy is performed and the results obtained differ from the Dirac equation results by 6 x 10-3% for Z — 1, and 40% for Z = 70. This suggests th a t physically measureable quantities should be calculated and compared using the FV1/2 and Dirac equations. A calculation is presented to show the energy spectra and wavefunctions obtained are consistent with the literature. The results obtained in this chapter (again) suggest th a t the FV1/2 equation could be useful for calculations in atomic physics. Chapter 5 discusses important further work which should be undertaken if the FV1/2 equation is to become useful to physics. The development in the previous chapters concentrated on the FV1/2 equation in relativistic quantum mechanics. It should be also considered as a quantised field equation. The two electron problem and the calculation of physical quantities are crucial tests of the FV1/2 equation’s applicability in atomic physics. The mathematical analysis in chapter 2 can be extended. Suggestions for gaining a deeper mathematical understanding of the less restrictive dynamics of the FV1/2 equation are given. Finally some conclusions on what has been achieved in the thesis are presented in chapter 6.

•Jan 1, 1993•0 citations

Relativistic Quantum Mechanics and Spin

Masatoshi Kajita

Spin is one of the most important characteristics of particles, which results in a fine and hyperfine energy structure and Zeeman energy shift. Spin is a property of particles, independent of the density distribution given by the potential field. The spin eigenfunction should be described using a vector, and the spin operator is given by a matrix. The property of spin is not derived from the Schroedinger equation, which treats wave functions as scalars. The Dirac equation was developed to obtain wave functions from the relationship between frequency (energy) and wavenumber (momentum) given by the theory of special relativity, for which the equation is given by the 4 x 4 matrix and the wave functions are given by four-dimensional vectors. There are four solutions for one equation, which correspond to ± 1/2 spin states and the positive and negative rest energies. The existence of negative rest energy was confirmed by the discovery of positrons (antiparticles). The Zeeman shift induced by spin is derived from the exchange of matrix production.

BENTHAM SCIENCE PUBLISHERS eBooks•Mar 7, 2022•0 citations
The quantum problem of a body with spin, in the case of relativistic velocities, is treated with the aid of wellknown Dirac equations containing four wave functions. In the present work it is shown that the development of the ideas expressed previously (Compt. rend., 250: 67(1960) and Compt. rend., 254: 434(1962)) leads to the mathematical appliance of the Dirac theory, that is, the interpretation of quantum mechanics given can be transported to Dirac equations. (J.S.R.)
The results of the present work are a summary of those obtained in the previous publications [1]-[4]. In (A) is considered a corpuscule with spin introduced according to the hypothesis of Uhlenbeck and Goudsmit. The probability equations obtained here coincide with the wellknown equations of Pauli. In (B) is found a probability equation for a corpuscule without spin in a relativistic case. It proves that it coincides with the well-known equation of Klein-Gordon. In (C) is studied the movement of a corpuscule with spin in a relativistic case. The probability equations obtained here coincide with the well-known equations of Dirac. The matrices of Pauli as well as those of Dirac are found here to be a conséquence of the probability equations. The results obtained here lead to the conclusion that the interpretation of quantum mechanics developed in the previous works may be spread over the cases, considered here.
The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.
This thesis is concerned with the development of a new spin-1/2 wave equation in relativistic quantum mechanics. This equation is the spin-1/2 analogue of the spin-0 Feshbach-Villars equation. The thesis begins in chapter 1 with a review of the subject of relativistic wave equations and its place in the development of modern physics. Chapter 2 sees the derivation of the new equation (hereafter referred to as the FV1/2 equation). This derivation is presented at three levels of understanding, and is combined with an analysis of the equation using elements of the underlying mathematics of the theories of special relativity and quantum mechanics. The analysis justifies the methods of derivation of the equation, and its suitability as a wave equation in relativistic quantum mechanics. It is shown th a t the FV1/2 equation is different to the Dirac equation and other equations in the literature. One of the striking features of the FV1/2 equation is th a t it contains a less restrictive dynamics than the conventional Dirac equation. Chapter 3 examines the solution space and proves some results using the discrete symmetries C, P and T, which enable a subset of the solution space to be used for physical applications. It is possible to use a subset of solutions which contain the usual Dirac negative energy states, or it is possible to use a subset which contains only positive energy states, but includes states of opposite signs of charge. The second subset corresponds to the physical properties of particles and antiparticles and suggests th a t the FV1/2 equation could be useful in atomic physics. Chapter 4 represents a start on the development of the FV1/2 equation for use in atomic physics. It contains a study of hydrogenic atoms. The energy spectra and wavefunctions for hydrogenic atoms are derived using the FV1/2 equation. The method of solution is of comparable difficulty to th a t using the Dirac equation. The spectra are found to be identical to th a t of the Dirac equation, but the wavefunctions differ. A calculation of the expectation value of the Coulomb energy is performed and the results obtained differ from the Dirac equation results by 6 x 10-3% for Z — 1, and 40% for Z = 70. This suggests th a t physically measureable quantities should be calculated and compared using the FV1/2 and Dirac equations. A calculation is presented to show the energy spectra and wavefunctions obtained are consistent with the literature. The results obtained in this chapter (again) suggest th a t the FV1/2 equation could be useful for calculations in atomic physics. Chapter 5 discusses important further work which should be undertaken if the FV1/2 equation is to become useful to physics. The development in the previous chapters concentrated on the FV1/2 equation in relativistic quantum mechanics. It should be also considered as a quantised field equation. The two electron problem and the calculation of physical quantities are crucial tests of the FV1/2 equation’s applicability in atomic physics. The mathematical analysis in chapter 2 can be extended. Suggestions for gaining a deeper mathematical understanding of the less restrictive dynamics of the FV1/2 equation are given. Finally some conclusions on what has been achieved in the thesis are presented in chapter 6.
Spin is one of the most important characteristics of particles, which results in a fine and hyperfine energy structure and Zeeman energy shift. Spin is a property of particles, independent of the density distribution given by the potential field. The spin eigenfunction should be described using a vector, and the spin operator is given by a matrix. The property of spin is not derived from the Schroedinger equation, which treats wave functions as scalars. The Dirac equation was developed to obtain wave functions from the relationship between frequency (energy) and wavenumber (momentum) given by the theory of special relativity, for which the equation is given by the 4 x 4 matrix and the wave functions are given by four-dimensional vectors. There are four solutions for one equation, which correspond to ± 1/2 spin states and the positive and negative rest energies. The existence of negative rest energy was confirmed by the discovery of positrons (antiparticles). The Zeeman shift induced by spin is derived from the exchange of matrix production.