Mathematical surprises and Dirac's formalism in quantum mechanics
@article{Gieres1999MathematicalSA, title={Mathematical surprises and Dirac's formalism in quantum mechanics}, author={François Gieres}, journal={Reports on Progress in Physics}, year={1999}, volume={63}, pages={1893-1931} }
By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and…
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References
SHOWING 1-10 OF 119 REFERENCES
Dirac Formalism and Symmetry Problems in Quantum Mechanics. I. General Dirac Formalism
- Physics
- 1969
Dirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory. First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of…
Elementary Quantum Mechanics
- Physics, EducationNature
- 1935
THERE is now a number of volumes of an introductory character on the new quantum theory, and in reading them one is struck by the diversity in the methods adopted for introducing the beginner to this…
Mathematical Physics.
- EducationNature
- 1930
THE scope and quality of the two works under review are very different, but they are both concerned with branches of theoretical physics, and attack them rather from the point of view of the pure…
Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory
- Mathematics
- 1974
Roberts' proposal of a rigged Hilbert space Φ⊂G⊂Φ× for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces…
Quantum mechanics : a modern development
- Physics
- 1998
Although there are many textbooks that deal with the formal apparatus of quantum mechanics (QM) and its application to standard problems, none take into account the developments in the foundations of…
The Principles of Quantum Mechanics
- PhysicsNature
- 1947
IT is now twenty years since the theory of quantum mechanics was founded, and not much less since the first edition of Dirac's book was published. Ever since, it has been a classic of scientific…
Linear Transformations in Hilbert Space: and their Applications to Analysis
- MathematicsNature
- 1933
THIS most welcome treatise fills a serious gap in English mathematical literature. It provides for the first time a comprehensive account of the general transformation theory which steadily dominates…
Conceptual Foundations Of Quantum Mechanics
- Physics, Education
- 1971
A detailed view of the conceptual foundations and problems of quantum physics by a highly regarded French theoretical physicist.. Conceptual Foundations of Quantum Mechanics provides a detailed view…
Phase and Angle Variables in Quantum Mechanics
- Physics
- 1968
The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state…