in Synthese ( The Mathematical Foundations of Quantum Mechanics by. David A. Edwards. Department of Mathematics. University of Georgia. Mathematical Foundations of Quantum Mechanics: An Advanced Short. Course. Valter Moretti. Department of Mathematics of the University of. THE MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS BY HILBERT AND VON NEUMANN. Article (PDF Available) with 2,
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basic quantum mechanics – of the abstract mathematics. Where physical In this chapter, we will set up the mathematical foundations of the day-by-day. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced. advent of Quantum Mechanics that the fusion of physics and mathematics has mathematical foundation to the concept of probability and it is this theme that I.
Planck introduced an intrinsic discontinuity: the quantum discontinuity. The hypothesis of light-quanta led Einstein to his well-known theory of the photoelectric effect, which was well supported by R. Madrid- Casado The knowledge of atomic structure was reached through the discovery of the electron, due to J.
Thomson, and through the discovery of the atomic nucleus, which we owe to E. Towards , experimental evidence existed that atoms were made up of electrons. Given that atoms were neutral, they had to contain a positive charge equal in magnitude to the negative charge provided by their electrons.
Thomson proposed a tentative model, whereby the negatively charged electrons where found inside a positively charged distribution. In , Rutherford demonstrated that the positive charge was not distributed throughout the atom, but on the contrary, was concentrated in a very small area that could be considered the atomic nucleus.
An atom was built up of a nucleus that had a positive electrical charge, together with a number of electrons which had a negative charge and move around the nucleus. This picture had a resemblance to a TE SS planetary system. S However, this conception did not provide a better explanation for the spectra of the R atoms. It was impossible to understand why atomic spectra consisted of sharp lines at AP L all. Moreover, according to classical electrodynamic theory, electrons had to fall onto H O the nucleus because their motion would emit a continuous radiation of energy from the atom.
Who could explain the data of the spectroscopy and the amazing stability of C E atoms? He avoided these difficulties by introducing concepts borrowed E O— from quantum theory. Bohr exploited the quantum discontinuity in his first atomic theory. Not discouraged by this conflict, he proposed a quantum notion of stability that was embodied in his concept of U stationary state.
In the first part of his trilogy, Bohr introduced this concept. The stability of atoms transcended classical mechanical explanation. The atomic model of Bohr solved the riddle by means of two postulates.
The first one accounted for the stability of the atom and it stated that an atomic system cannot exist in all mechanically possible states, forming a continuum, but in series of discrete stationary states.
The second postulate accounted for the line-spectra.
The Mathematical Foundations of Quantum Mechanics
By definition the stationary states were subject to the following assumptions, which were mostly suggested by the quantum theory of Planck and Einstein, and the simple regularities of the hydrogen spectrum: I. Madrid- Casado states corresponding to a discontinuous series of values for its energy, and that consequently any change of the energy of the system, including emission and absorption of electromagnetic radiation, must take place by a complete transition between two such states.
Van der Waerden , TE SS The strange conception of atoms as systems which were only able to assume discrete S energy changes was a masterpiece because it gave an explanation of the Balmer formula R and the Rydberg constant. Moreover, it sparked enormous interest in developing and extending the old quantum theory. During the war Arnold Sommerfeld made progress on the implications of quantization. He extended the circular orbits of Bohr to elliptical orbits, and he refined his atomic model by introducing several quantum numbers in order to explain the fine structure shown by the hydrogen spectrum when it was observed with a spectroscope of PL C high resolving power.
M ES With the coming of the armistice in , work in quantum mechanics expanded rapidly. Many theories were suggested and many experiments performed. To cite just one example, in O. Stern and his graduate student W.
Gerlach performed their SA N important experiment on the deflection of particles, often used to illustrate the basic principles of quantum physics. They demonstrated the space quantization rule, that is, U the magnetic moment of the silver atoms could take only two positions, not a continuum one.
At the turn of the year from to , physicists looked forward with enormous enthusiasm towards detailed solutions of the outstanding problems, such as the helium problem and the anomalous Zeeman Effect the split lines in a magnetic field.
Nevertheless, there was a great difficulty: it was not possible to use the Bohr-Sommerfeld quantization rules for the anomalous Zeeman Effect and for the helium atom, whose electrons rotate around the nucleus, because the three-body problem, of difficult mathematical treatment, is encountered.
The anomalous Zeeman Effect and the helium spectrum were the two stumbling blocks in the old quantum theory. Madrid- Casado 3. Early Quantum Mechanics Old quantum physics was a house built on sand. Each problem had to be solved first within the classical physics realm, and only then the solution could be translated by means of diverse computation rules — for instance, the correspondence principle of Bohr, i.
This consisted in the obvious requirement that ordinary classical mechanics had to hold to a high degree of approximation in the limiting case where the numbers of the stationary states, the so-called quantum numbers, were very large. The correspondence principle acted as a code book for translating a classical relation into its quantum counterpart.
It was a daring fusion of old and new.
But these rules revealed a dismaying TE SS state of affairs in C E Quantum physicists became more and more convinced that a radical change on the E O— foundations of physics was necessary, that is to say: a new kind of mechanics which they called quantum mechanics.
To tell the truth, the name was coined by Max Born in a paper.
Heisenberg aimed at constructing a quantum-mechanical formalism corresponding as closely as possible to that of classical mechanics.
The classical position q and momentum p and their operations q 2 , p 2 , pq Madrid- Casado quantities contained information about the measurable line spectrum of an atom rather than the unobservable orbit of the electron. To put it C E another way, the old picture of electronic orbits. Given that an electron trajectory inside an atom was not observable, it was necessary to drop such a concept and the concepts E O— associated to it, like those of position and velocity.
Only intensities, frequencies, and amplitudes of radiation were observable, because they could be determined by spectral lines. SA N Heisenberg did not really arrange his quantum-theoretical quantities into a table or array. He began to deal with sets of allowed physical quantities. But Born looked at U these sets of numbers and he suddenly saw that they could be interpreted as mathematical matrices.
In , matrix calculus was an advanced abstract technique, well known to Born from his student days from the lectures of Rosanes in Breslau, but Heisenberg struggled with it. That is, A times B does not necessarily equal B times A in quantum mechanics.
This S was particularly important when Born and Jordan obtained the quantum mechanical R expression corresponding to the quantum conditions in the old quantum theory. This rejection failed to demoralize Born, who immediately set out to work with a more benevolent collaborator, his pupil Pascual Jordan, who overheard Born discussing matrix theory with Pauli on the train.
The mathematical method of PL C treatment inherent in the new quantum mechanics was characterized by the use of matrix calculus in place of the usual number analysis. Consequently, the basic matrix-mechanical problem was merely that of integrating these motion equations 13 , i. Born, Jordan and Heisenberg applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory.
However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Madrid- Casado way of matrices; on the other hand, the satisfactory explanation of the hydrogen spectrum created the expectation that finally it would be possible to explain multielectronic atoms. U MM3. Otherwise it MM4. The classical moment problem. Quantum Mechanics and Experience. Harvard: Harvard University Press.
Chicago: University of Chicago Press. Quantum Theory.
New York: Dover. On the constitution of atoms and molecules, Philosophical Magazine 26, , and Physics , pp. Amsterdam: Elsevier Publishing Company. The quantum theory of radiation, Philosophical Magazine 47, Reprinted in B. Physik 37, The statistical interpretation of quantum mechanics Nobel Lecture, December 11, , in Nobel Lectures. Madrid- Casado Born, M. Zur Quantenmechanik, Z. Physik 34, Zur Quantenmechanik II, Z. Physik 35, A new formulation of the laws of quantization of periodic and aperiodic phenomena, Journal of Mathematics and Physics M.
The Quantum Physicists. Oxford: Oxford University Press. Berkeley: University of California Press. History of Functional Analysis. Amsterdam: North-Holland. A, On Quantum Algebra, Cam.
On the Theory of Quantum Mechanics, Proc. The physical interpretation of the quantum dynamics, Proc. The Principles of Quantum Mechanics. Oxford: Clarendon Press. A new notation for quantum mechanics, Cam. Madrid- Casado Einstein, A. Reviews of Modern Physics 20, The Shaky Game. Einstein Realism and the Quantum Theory. Chicago: Chicago University Press.
Weimar culture, causality, and quantum theory: adaptation by German physicists and mathematicians to a hostile environment, Historical Studies in the Physical Sciences 3, Image and Logic: A material culture of microphysics. Are wave mechanics and matrix mechanics equivalent theories? Feigl and G. C E Maxwell eds.
APM421H1: Mathematical Foundations of Quantum Mechanics
Wheeler, was published in by Princeton University Press. The book mainly summarizes results that von Neumann had published in earlier papers. From Wikipedia, the free encyclopedia.
Nicholas A. Wheeler, ed. Mathematical Foundations of Quantum Mechanics. New Edition. Translated by Robert T. Princeton University Press.
Mathematische Annalen: Mathematische Annalen. Retrieved from " https: Hidden categories:Schilpp ed. Harper and Bros. Google Scholar  Velo, G.
Harvard: Harvard University Press. U MM3. Google Scholar  Colodny, R. C E Quantum physicists became more and more convinced that a radical change on the E O— foundations of physics was necessary, that is to say: a new kind of mechanics which they called quantum mechanics. Chicago: University of Chicago Press.