An investigation of origins of the quantum mechanical momentum operator has shown that it corresponds to the nonrelativistic momentum of classical special relativity theory rather than the relativistic one, as has been unconditionally believed in traditional relativistic quantum mechanics until now. Taking this correspondence into account, relativistic momentum and energy operators are defined. Schrödinger equations with relativistic kinematics are introduced and investigated for a free particle and a particle trapped in the deep potential well.An investigation of origins of the quantum mechanical momentum operator has shown that it corresponds to the nonrelativistic momentum of classical special relativity theory rather than the relativistic one, as has been unconditionally believed in traditional relativistic quantum mechanics until now. Taking this correspondence into account, relativistic momentum and energy operators are defined. Schrödinger equations with relativistic kinematics are introduced and investigated for a free particle and a particle trapped in the deep potential well.

Journal of Applied Mathematics and PhysicsJan 1, 20173 citations

Relativistic Quantum Mechanics

WORLD SCIENTIFIC eBooksJan 1, 201210 citations

Relativistic Quantum Mechanics

WORLD SCIENTIFIC eBooksDec 3, 20144 citations

Quantum Mechanics and the Relativistic Hamilton-Jacobi Equation

Lloyd Motz, Adolph Selzer

In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where $S$ is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where $S$ is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.

Physical ReviewMar 23, 196413 citations

Relativistic Quantum Mechanics

Günter Ludyk

Springer eBooksOct 25, 20170 citations

Schrödinger group and non-linear generalizations of quantum mechanics

Marcel Reginatto

Abstract A free particle may be characterized abstractly in terms of its invariance group. Both the Hamilton-Jacobi and Schrödinger equations for classical and quantum free particles are invariant under the same group, the Schrödinger group. The aim of this paper is to discuss possible restrictions on non-linear generalizations of quantum mechanics. The basic idea is to search for those non-linear generalization of quantum mechanics for which a description of the non-relativistic free particle includes a representation of the Schrödinger group, as in classical and quantum mechanics. This condition imposes strong constraints on possible models of free particles. To carry out such an investigation it is necessary to select an appropriate mathematical formalism, one which can describe physical systems of non-relativistic particles and allows for the incorporation of non-linearities in a natural way. Here the formalism of ensembles on configuration space is used, which is an approach that is capable of describing both classical and quantum systems and, in addition, allows for more general theories that describe mixed classical-quantum systems. Modifications of the equations involving derivatives of up to fourth order are consideredAbstract A free particle may be characterized abstractly in terms of its invariance group. Both the Hamilton-Jacobi and Schrödinger equations for classical and quantum free particles are invariant under the same group, the Schrödinger group. The aim of this paper is to discuss possible restrictions on non-linear generalizations of quantum mechanics. The basic idea is to search for those non-linear generalization of quantum mechanics for which a description of the non-relativistic free particle includes a representation of the Schrödinger group, as in classical and quantum mechanics. This condition imposes strong constraints on possible models of free particles. To carry out such an investigation it is necessary to select an appropriate mathematical formalism, one which can describe physical systems of non-relativistic particles and allows for the incorporation of non-linearities in a natural way. Here the formalism of ensembles on configuration space is used, which is an approach that is capable of describing both classical and quantum systems and, in addition, allows for more general theories that describe mixed classical-quantum systems. Modifications of the equations involving derivatives of up to fourth order are considered

Journal of Physics Conference SeriesAug 1, 20200 citations

In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where

S

is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.