Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. II. Transformation theory in nonrelativistic quantum mechanics

1974 ◽  
Vol 15 (7) ◽  
pp. 917-925 ◽  
Author(s):  
O. Melsheimer
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Systems ◽  
Mathematical Formalism ◽  
Nonrelativistic Quantum ◽  
Transformation Theory ◽  
Rigged Hilbert Space ◽  
Space Formalism ◽  
Nonrelativistic Quantum Mechanics

scholarly journals The Phase Space Model of Nonrelativistic Quantum Mechanics

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 581
Author(s):  
Jaromir Tosiek ◽  
Maciej Przanowski
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Quantum Systems ◽  
Interesting Case ◽  
Nonrelativistic Quantum ◽  
Space Model ◽  
Discrete Phase ◽  
Quantum Observables ◽  
Nonrelativistic Quantum Mechanics ◽  
Phase Space Model

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.

scholarly journals FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS V: INTERPRETATION, SUMMARY AND DISCUSSION

2002 ◽  
Vol 17 (02) ◽  
pp. 245-258 ◽  
Author(s):  
BOZHIDAR Z. ILIEV
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Pure State ◽  
Fibre Bundle ◽  
Theoretical Physics ◽  
Nonrelativistic Quantum ◽  
Time Model ◽  
Hilbert Bundle ◽  
Nonrelativistic Quantum Mechanics ◽  
Space Time Model

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.

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