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Annalen der Physik

MAXWELL‐Theorie (in Medien) und Quantentheorie in einem rotierenden Bezugssystem

Ernst Schmützer

Published Jan 1, 1973• 4 citations
Publisher

Abstract

Abstract On the basis of our 3‐dimensional conceptions of the electromagnetic quantities [1] the MAXWELL theory is represented in a rotating frame of reference. For such a frame the MAXWELL equations obtain additional terms depending on the angular velocity (analogous to the CORIOLIS term etc. in NEWTONian mechanics). Using these results with the help of the DIRAC equation we investigate quantum mechanics in a rotating frame of reference. Also in this case we obtain interesting additional terms depending on the angular velocity. The need for such a theory is obvious for many reasons: 1. Our Earth is such a rotating laboratory. Later the rotation effects must not be neglected. 2. In magnetic resonance physics theoretical questions with respect to rotating systems are important. 3. Interesting statements result for a rotational JOSEPHSON effect of superconductors. — All our calculations are carried out up to the first order in the angular velocity.

DOI
10.1002/andp.19734840109

Related papers

MAXWELL-Theorie (in Medien) und Quantentheorie in einem rotierenden Bezugssystem

Ernst Schmützer

On the basis of our 3-dimensional conceptions of the electromagnetic quantities [1] the MAXWELL theory is represented in a rotating frame of reference. For such a frame the MAXWELL equations obtain additional terms depending on the angular velocity (analogous to the CORIOLIS term etc. in NEWTONian mechanics). Using these results with the help of the DIRAC equation we investigate quantum mechanics in a rotating frame of reference. Also in this case we obtain interesting additional terms depending on the angular velocity. The need for such a theory is obvious for many reasons: 1. Our Earth is such a rotating laboratory. Later the rotation effects must not be neglected. 2. In magnetic resonance physics theoretical questions with respect to rotating systems are important. 3. Interesting statements result for a rotational JOSEPHSON effect of superconductors. — All our calculations are carried out up to the first order in the angular velocity.

Annalen der Physik•Jan 1, 1973•19 citations

The first and second order equations of the quantum theory

H. T. Flint

Abstract The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator (h/2πi ∂/∂xα - eϕα) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc. Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.

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Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.

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The concept of invariance relates to both the intrinsic symmetries of physical systems and the symmetry of the set of equivalent reference frames used to observe them. Standard algebraic expressions for electrostatic potentials and crystal-field effective operators display both types of invariance. The concept of a reference frame is generalized to that of an ‘observing system’, which can, for example, be the basis states of a quantum system. This idea is related to Racah’s mathematical machinery for evaluating the matrix elements of many-electron 4f open-shell states in lanthanide ions. It is argued, on the basis of computational flexibility and ease of interpretation, that all equations that represent physical processes be expressible in terms of invariants of the set of observing systems. This ‘Principle of Invariance’ is then applied to special relativity, leading to a simple geometrical interpretation of Maxwell’s electromagnetic field equations. The close relationship between Dirac’s relativistic wave equation and Maxwell’s equations is then exposed. This leads to the concept of an inner structure of space-time and the reinterpretation of particle spin. Finally, it is shown that the use of invariants in relativity theory identifies a set of observing systems with a higher symmetry than that of Minkowski space-time.

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arXiv:1104.2473•Chinese Physics C•Feb 1, 2011•0 citations
Abstract On the basis of our 3‐dimensional conceptions of the electromagnetic quantities [1] the MAXWELL theory is represented in a rotating frame of reference. For such a frame the MAXWELL equations obtain additional terms depending on the angular velocity (analogous to the CORIOLIS term etc. in NEWTONian mechanics). Using these results with the help of the DIRAC equation we investigate quantum mechanics in a rotating frame of reference. Also in this case we obtain interesting additional terms depending on the angular velocity. The need for such a theory is obvious for many reasons: 1. Our Earth is such a rotating laboratory. Later the rotation effects must not be neglected. 2. In magnetic resonance physics theoretical questions with respect to rotating systems are important. 3. Interesting statements result for a rotational JOSEPHSON effect of superconductors. — All our calculations are carried out up to the first order in the angular velocity.
On the basis of our 3-dimensional conceptions of the electromagnetic quantities [1] the MAXWELL theory is represented in a rotating frame of reference. For such a frame the MAXWELL equations obtain additional terms depending on the angular velocity (analogous to the CORIOLIS term etc. in NEWTONian mechanics). Using these results with the help of the DIRAC equation we investigate quantum mechanics in a rotating frame of reference. Also in this case we obtain interesting additional terms depending on the angular velocity. The need for such a theory is obvious for many reasons: 1. Our Earth is such a rotating laboratory. Later the rotation effects must not be neglected. 2. In magnetic resonance physics theoretical questions with respect to rotating systems are important. 3. Interesting statements result for a rotational JOSEPHSON effect of superconductors. — All our calculations are carried out up to the first order in the angular velocity.
Abstract The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator (h/2πi ∂/∂xα - eϕα) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc. Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
The concept of invariance relates to both the intrinsic symmetries of physical systems and the symmetry of the set of equivalent reference frames used to observe them. Standard algebraic expressions for electrostatic potentials and crystal-field effective operators display both types of invariance. The concept of a reference frame is generalized to that of an ‘observing system’, which can, for example, be the basis states of a quantum system. This idea is related to Racah’s mathematical machinery for evaluating the matrix elements of many-electron 4f open-shell states in lanthanide ions. It is argued, on the basis of computational flexibility and ease of interpretation, that all equations that represent physical processes be expressible in terms of invariants of the set of observing systems. This ‘Principle of Invariance’ is then applied to special relativity, leading to a simple geometrical interpretation of Maxwell’s electromagnetic field equations. The close relationship between Dirac’s relativistic wave equation and Maxwell’s equations is then exposed. This leads to the concept of an inner structure of space-time and the reinterpretation of particle spin. Finally, it is shown that the use of invariants in relativity theory identifies a set of observing systems with a higher symmetry than that of Minkowski space-time.
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum jjj and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3)\text{SO}(3)SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large jjj, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.