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Springer eBooks

Intrinsic Non-Invariant Forms of Dirac Equation

Josep M. Parra

Published Jan 1, 1993• 1 citation
Publisher
DOI
10.1007/978-94-011-1719-7_27

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We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.

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The Invariant Form of Quantum Equations and the Schroedinger-Heisenberg Parallelism

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Some years have passed since the appearance of Dirac's striking interpretation of the Heisenberg-Schroedinger parallelism in the equations of motion, and yet it appears that there still remain obscure points which have led to misunderstanding. Although the Schroedinger method suffices to deal with many practical problems, the flexibility of the Dirac-Jordan representation theory, aside from its intrinsic unity and beauty, recommends it to general theoretical investigation. The present paper considers the invariant form of the equations of motion in relation to the Schroedinger-Heisenberg pictures of quantum mechanics insofar as the latter refer to the "pictured" intrinsic motion of states (vectors) and observables (Hermitian operators). The discussion is divided roughly into 5-sections: Operator and Vector Transforms, Contact Transformations, Heisenberg-Schroedinger Operators, Equations of Motion and Invariant Properties, and the Density Operator and Operator Spur. Where possible, quasi-geometrical diagrams are given to illustrate the relation and transition between the several "pictures of the motion."

Physical Review•Feb 1, 1947•0 citations
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.
Some years have passed since the appearance of Dirac's striking interpretation of the Heisenberg-Schroedinger parallelism in the equations of motion, and yet it appears that there still remain obscure points which have led to misunderstanding. Although the Schroedinger method suffices to deal with many practical problems, the flexibility of the Dirac-Jordan representation theory, aside from its intrinsic unity and beauty, recommends it to general theoretical investigation. The present paper considers the invariant form of the equations of motion in relation to the Schroedinger-Heisenberg pictures of quantum mechanics insofar as the latter refer to the "pictured" intrinsic motion of states (vectors) and observables (Hermitian operators). The discussion is divided roughly into 5-sections: Operator and Vector Transforms, Contact Transformations, Heisenberg-Schroedinger Operators, Equations of Motion and Invariant Properties, and the Density Operator and Operator Spur. Where possible, quasi-geometrical diagrams are given to illustrate the relation and transition between the several "pictures of the motion."