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About the Mathematical Foundation of Quantum Mechanics M. Victoria Velasco Collado Departamento de Análisis Matemático Universidad de Granada (Spain) Operator Theory and The Principles of Quantum Mechanics CIMPA-MOROCCO research school, Meknès, September 8-17, 2014 Lecture nº 1 11-09-2014

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Page 1: About the Mathematical Foundation of Quantum Mechanics€¦ · Grupo de Investigación the Mathematical Foundation of Quantum Mechanics ... About the Mathematical Foundation of Quantum

Grupo de Investigación About the Mathematical Foundation of Quantum Mechanics

M. Victoria Velasco Collado

Departamento de Análisis Matemático

Universidad de Granada (Spain)

Operator Theory and The Principles of Quantum Mechanics

CIMPA-MOROCCO research school,

Meknès, September 8-17, 2014

Lecture nº 1

11-09-2014

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Grupo de Investigación About the Mathematical Foundation of Quantum Mechanics

Lecture nº 1: About the origins of the Quantum Mechanics

- What is the Quantum Mechanics?

- Why the Quantum Mechanics is relevant?

- The origins of Quantum Mechanics in the Physics

- The four main papers of Einstein in 1905

- The mathematical foundation of the Quantum Mechanics

- Postulates of Quantum Mechanics

- From Physics to Mathematics via Quantum Mechanics

- From Quantum Mechanics to Functional Analysis

- Hilbert spaces

Operator Theory and The Principles of Quantum Mechanics

CIMPA-MOROCCO MEKNÈS, September2014

Lecture 1: About the origins of the Quantum Mechanics

Lecture 2 : The mathematical foundations of Quantum Mechanics

Lecture 3 : About the future of Quantum Mechanics. Some problems and challenges

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What is the Quantum Mechanics?

The Quantum Mechanics (or Quantum Physics) is the branch of Physics that

studies systems with relevant quantum effects. Therefore Quantum Mechanics

deals with physical phenomena at microscopic scales. But its scope is intended

to be universal.

The Quantum Mechanics was developed in the early XX century and it marked

the beginning of Modern Physics. It arose because failure of the Gravitation

Universal Law and the classical Electromagnetic Theory to explain phenomena

such as the black body radiation, the photoelectric effect, or the Compton

effect, among others.

The discovering of the wave-particle duality of the light was essential. Thus,

depending on the circumstances, the light behaves as a particle or as a

electromagnetic wave. The idea was to generalize this duality to all known

particles.

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Why the Quantum Mechanics is relevant?

The Quantum Mechanics is the unique framework to describe the atomic world currently.

Consequently the Quantum Mechanics is essential to understand phenomena such as the

Physics of solids, lasers, semiconductor devices, superconductors, plasmas etc

In Chemical-Physics, the plasma is the fourth state of matter. It is similar to the gaseous-

fluid state but, there, many particles are electrically charged and have no electromagnetic

balance. Therefore these particles are good electrical conductors and react strongly to

electromagnetic long-range interactions.

The laser (Light Amplification by Stimulated Emission of Radiation) is a device that

uses an effect of Quantum Mechanics (the induced emission) to generate a powerful

light with shape and purity under control.

For example the plasma screen contains many tiny

cells, located between two panels of glass, which

contain a mixture of noble gases (neon and xenon).

After electricity, the gas in the cells becomes

plasma. As a consequence, a certain quantity of light

is emitted by a phosphorescent substance.

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Why the Quantum Mechanics is relevant?

Nevertheless, a photon with a certain energy can cause that an electron falls to a lower level

of energy, by emitting another photon identical to the original one. This is the called induced

or stimulated emission. Because the stimulated emission produces two identical photons from

the original one, the light is amplified.

According to Quantum Mechanics, if an electron is in a high level of

energy then, it falls spontaneously to a lower level of energy, with a

subsequent light emission. This phenomena is called spontaneous emission

and is responsible for most of the light that we see.

For many things of this type, the Quantum Mechanics is the essence of the Modern Physics

(and this includes the Physics of the Solid State, the Molecular Physics, the Atomic Physics,

the Nuclear Physics, the Optics Physics, etc.)

Quantum Mechanics is also essential for the

Chemistry and the Molecular Biology. Indeed, it

allows a precise description of the chemical bond.

Therefore the base of the called Quantum

Chemistry.

The last generation’s drugs are based on this.

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Why the Quantum Mechanics is relevant?

The Quantum Computing was created in 1981 by Paul

Benioff. He developed a theory to take advantage of the

quantum laws in the computing environment. In digital

computing, a bit can only have two values (0 or 1). In

contrast, in quantum computing, a particle can be in coherent

superposition. This means that it can be 0, 1, and also 0 and

1 simultaneously.

This allows to carry out several operations at the same time,

depending on the number of qubits (quantum bits). This

“quantum” computer was purchased by NASA (15,000,000

$) in 2012 and works since the end of 2013.

It is 50,000 times faster than a conventional computer.

(A revolution for the Cryptography).

The optical fiber is a thin strand of glass, or of melt silicon, that conducts

the light far away at high speed, without using electrical signals. Fiber

optics and lasers have been a revolution for communications. This also is

a "quantum" phenomena.

Procesador D.wave 2

The spectroscopy studies the interaction between electromagnetic

radiation and matter, with absorption or emission of radiant energy. The

nuclear magnetic resonance is based on this. The Quantum Mechanics

provides the theoretical basis for their understanding.

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The origins of the Quantum Mechanics in the Physics

At the end of the XIX century the Physics seemed to be a consistent theory with

many well-established disciplines: the Thermodynamics (study of the macroscopic

equilibrium states), Classical Mechanics (motion study) and Electromagnetism

(electric and magnetic phenomena).

However the Classical Mechanics and the Electromagnetism could not explain

certain phenomena related to the exchange of energy and matter, such as the

following ones:

The black body radiation problem, enunciated by Gustav Kirchhoff

in 1859.

A black body is a theoretical object (reproducible experimentally

to a certain extent ) which absorbs all the light and the radiant

energy that falls on it.

Every body emits energy in the form of electromagnetic waves. This radiation is

more intense as highest is the temperature of the transmitter (consequently the

color of a body changes when the body is heated).

According with the classical Electromagnetism, a black body at thermal

equilibrium should emit energy in all ranges of frequency. It follows that it must

radiate an infinite amount of energy. This is called the ultraviolet catastrophe.

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The origins of the Quantum Mechanics in the Physics

The classical Physics cannot explain this phenomenon because its theorem of

equipartition of energy (a formula that relates the temperature of a system with its

average energies) is not valid when the thermal energy is much lower than the energy

associated to the frequency of the radiation.

The photoelectric effect was discovered by Heinrich Hertz in

1887. It consists in the emission of electrons by a material under

the influence of a electromagnetic radiation. It was noted that

the energy of the photons increased with the frequency of the

𝐼 𝜈, 𝑇 =2ℎ𝜈3

𝑐21

𝑒ℎ𝜈𝑘𝑇 −1

The intensity of the medium (or spectral) radiation emitted by a

black body at temperature T and frequency 𝜈

The solution to the problem of black body's radiation is the named Planck's law.

It was given in 1900 by Max Planck. Today it is considered a principle of Quantum

Mechanics.

light falling on it. (According to the Maxwel’s laws of Electromagnetism, the energy and the

frecuency of light are independent). Indeed, it was shown that under a especific level of

frequency there was no emission of electrons, independently of the intensity of the light and

the time of emission (a contradiction with the classic laws of Physics). This was established

in:

A. Einstein, Ueber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen

Gesichtspunkt'. Annalen der Physik, 17 (1905), 132-141

(On a heuristic point of view about the creation and conversion of light)

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The origins of the Quantum Mechanics in the Physics

The Compton scattering (or Compont effect) was showed in 1922 by Arthur H.

Compton. It consists in the increase of the wavelength of a photon when it crashes

into a free electron and loses some of its energy. When the incoming photon gives

part of its energy to the electron, then the scattered photon has lower energy

according to the Planck relationship. Indeed, the variation of wavelength of the

scattered photons, can be calculated through the relation of Compton:

Because of this, A. Compton won the

Nobel Prize in Physics in 1927.

(A. Einstein won it in 1921 for his

explanation of the photoelectric effect

and their contributions to Physics).

The inverse Compton scattering also exists, where

the photon gains energy (decreasing in wavelength)

upon interaction with matter.

This effect cannot be explained using a wave nature

of light, where the wavelength does not change.

This is another clear proof of the quantum nature of

light.

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The four main papers of Einstein in 1905

Einstein A., Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen

Gesichtspunkt, Annalen der Physik, 17 (1905), 132-148 (17 de marzo).

On a Heuristic Point of View about the Creatidn and Conversion of Light

http://www.casanchi.freeiz.com/fis/einstein1905/uno/uno_i.pdf

Here, the concept of photon (quantum or corpuscle

of light) is introduced. Moreover, the problem of

the photoelectric effect is solved by using the

works of Planck, and showing the quantum nature

of light.

Many applications came in later publications, about

the photoelectric cells, the laser rays, etc.

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The four main papers of Einstein in 1905

Einstein A., Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in

ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik, 17 (1905), 549-560 (11 de mayo)

On the movement of small particles suspended in a stationary liquid demanded by the molecular theory of heat

http://www.casanchi.freeiz.com/fis/einstein1905/dos/dos_i.pdf

In 1827 a Scottish botanist, Robert Brown, had discovered the movement of the pollen

grains that were floating in a totally quiet liquid. This movement was continuous and

unpredictable.

Einstein provided a complete and accurate mathematical description of Brownian motion

that could be verified experimentally. Therefore he gave a experimental evidence for the

existence of atoms (a disputed fact at that time). This paper becomes one of the

foundations of Statistical Mechanics and the Kinetic Theory of fluids.

His formula applies to molecular collisions as well as to any random movement.

Imagine a drunk person walking down the street, randomly changing direction when it

hits the mailboxes, lampposts or other bystanders.

The average distance gotten by the drunk from the beginning is the product of the

length of each step by the square root of the number of steps taken.

For instance if the drunk has taken 49 steps of 1 meter each, then the drunk has

covered 7 meters from its initial position. Nevertheless, walking in a straight line, the

distance would be 49 meters.

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The four main papers of Einstein in 1905

Einstein A., Zur Elektrodynamik bewegter Körper , Annalen der Physik 17 (1905), 891-920 (June 30 th)

On the Electrodynamics of moving Bodies) http://www.casanchi.freeiz.com/fis/einstein1905/tres/tres_i.pdf

Here, the bases of the "Special (or Restricted) Theory of the Relativity" are established in

order to describe the motion of bodies (even at high speeds) in the absence of gravitational

forces. (Therefore, this theory is not applicable to astrophysical problems in which the

gravitational field plays an important role).

(In 1915 Einstein developed the General Theory of Relativity where the effects of gravity

and acceleration were considered).

Einstein used the Lorentz’s equations to describe this movement. Indeed, H. Poincare y

Heindrik Lorentz are considered prerunners of this theory.

Einstein proves in this paper Simultaneity Principle of Galileo: the laws of Physics are

invariant for all observers moving at relatively constant speed.

He proves also that the speed of light is constant for any observer with independence of the

movement of the emitting source.

The Slowing of Clocks and the Twin Paradox

An example of clocks changing their rates with changes in motion is the so called Twin Paradox, where

one twin travels at very high speed to a star and back, and returns younger than the twin that stayed

home.

Experimentally tested with clocks.

The location of the physical events in space and time

are relative to the state of motion of the observer.

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The four main papers of Einstein in 1905

Einstein A., Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik 17

(1905), 639-641 (September 27th)

Does the inertia of a body depend upon its energy-content? http://www.casanchi.freeiz.com/fis/einstein1905/cuatro/cuatro_i.pdf

¿Depende la masa inercial de la Energía?) http://www.casanchi.freeiz.com/fis/einstein1905/cuatro/cuatro_e.pdf

Until this paper, mass and energy were two separate things. Here, Einstein demonstrated that

neither mass nor energy were conserved separately. Indeed, he proved that the energy E of

a physical system is numerically equal to the product of its mass m and the speed of

light c squared. This result lies at the core of modern physics.

(Equivalence of mass and energy)

Therefore: the matter can be converted into energy and, conversely, the energy into matter. (Indeed very small amounts of mass may be converted into a very large amount of energy and conversely)

This was demonstrated by J. D. Cockcroft and E. Walton in 1932, experimentally.

This fact is essential, for instance, to understand the nuclear fission and the nuclear

fusion.

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The origins of the Quantum Mechanics in the Physics

The nuclear fusion is a nuclear reaction in which two or

more atomic nuclei atomic collide at a very high speed and

join to form a new type of atomic nucleus. During this

process, matter is not conserved because some of the

matter of the fusing nuclei is converted to photons (energy).

The nuclear fission is either a nuclear reaction or a radioactive decay process in which the

nucleus of an atom splits into smaller parts (lighter nuclei). The fission process often

produces free neutrons and photons (in the form of gamma rays), and releases a very large

amount of energy even by the energetic standards of radioactive decay.

There are a number of elements that can be used

in nuclear fission, but the most common is uranium.

For instance, the Sun generates its

energy by nuclear fusion of

hydrogen nuclei into helium.

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The origins of the Quantum Mechanics in the Physics

1907. Ernest Rutherford by shooting alpha particles (positively charged) on a gold foil

showed that some atoms were returned. This was an empirical proof that atoms have a

small atomic nucleus at its center is positively charged.

1913. Niels Bohr explains the Rydberg’s formula (1888) that models the spectrum of light

emission of the hydrogen atom.

1917. Pieter Zeeman (with the so-called Zeeman effect) showed experimentally the

conjecture of H. A. Lorentz (1895) about the splitting of the energy levels of the atom.

(He showed the splitting of a spectral line into many others under the influence of a

magnetic field).

From this experiment Arnold Sommerfeld suggests the

existence of elliptical orbits (besides the spherical ones)

in the atom.

To do this he postulates that the negatively charged

electrons turn around a nucleus positively charged,

in quantum orbits to a certain distances. These

orbits are associated with a specific level of energy. The

movement of the electrons between orbits requires

emission or absorption of quantum energy.

1915. A. Einstein. General theory of the relativity

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The origins of the Quantum Mechanics in the Physics

1923. Louis-Victor de Broglie postulates that the moving of an electron has a wavelength

associated that is given by 𝜆 =ℎ

𝑚𝑣 (where 𝑕 denotes the Planck’s constant).

1926. Erwin Schrödinger (by using the postulate of de Broglie) developed a wave equation

that mathematically represents the distribution of the charge of an electron through

space. With this model, the spectrum of the atom of hydrogen was properly explained.

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The mathematical foundation of the Quantum Mechanics

1900. In the 2nd International Congress of Mathematicians

(ICM) held in Paris, David Hilbert delivered a famous

lecture, in which the Mathematical treatment of the axioms

of Physics was formulated as one of the 23 more important

problems in Mathematics (the sixth problem).

D. Hilbert 1862-1943 Gottingen University

In 1895, David Hilbert obtained the position of

Professor of Math. at the University of Göttingen,

(and remained there for the rest of his life).

At that time Göttingen was the best research center

for mathematics in the world. The leadership of his

president, Felix Klein was decisive in this respect.

He proposed “to threat in the same manner, by means of

axioms, those physical sciences in which mathematics

plays an important part.”

Carl F. Gauss taught there, in the 19th century. Bernhard Riemann, Peter G. L. Dirichlet,

Herman Minkowski, and a number of significant mathematicians made their contributions to

mathematics in Göttingen.

By 1900, David Hilbert and Felix Klein had attracted mathematicians from around the world to

Göttingen, which made out of Göttingen a world-mecca of mathematics at the beginning of the

20th century.

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The mathematical foundation of the Quantum Mechanics

The problem of the independence of the fifth

postulate (axiom of parallels) led to a critical reading of

the elements of Euclid. In this way, it arises the problem

of the foundations of the Euclidean geometry, as well as

that of all possible geometries.

Felix Klein made major discoveries in geometry. He

showed that Euclidean and non-Euclidean geometries

could be considered special cases of a projective

surface, with a specific conic section associated.

The Erlangen Program (1872) of F. Klein for classifying geometries according with their

underlying symmetry groups, caused a deep influence for the evolution of the mathematics by

this time.

The work of Hilbert about the axiomatization of geometry, was a strong motivation for the

axiomatization of Physics. Moreover the new geometries helped to consider more sofisticated

systems where the time was fully included as a fourth dimension.

The idea was to extend the rigor of the Analysis and the Arithmetic to the Geometry as well

as to the Physical Sciences.

On the other hand, around 1902, the Hilbert’s research was strongly focused to the study of

linear integral equations. Incidentally, this allowed him to give a solution to the Boltzmann’s

quation in kinetic theory of gases in 1912.

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The mathematical foundation of the Quantum Mechanics

It often happened that in a course of a semester the program in an advanced lecture was

completely changed, because I wanted to discuss issues in which I was currently involved as

a researcher and which had not yet by any means attained their definite formulation.

Since 1898, Hilbert delivered courses and seminaries in many topic of Physics: Mechanics,

the structure of matter, kinetic theory of gases etc. From 1912 he published many papers

about the mathematical foundations of these topics.

The works of Minkowski published between 1907 and 1909 (the year

in which he died prematurely) related to the mathematical foundation

of the Special Theory of Relativity, were highly discussed in Hilbert’s

seminaries.

Also were discussed in Hilbert’s seminaries those papers of Einstein

and Grossmann drawing the General Theory of Relativity.

Göttingen was perhpas the unique scientific center that brought

together a gallery first world-class researchers in Mathematics

and Physics.

The lectures in Göttingen University became into important

occasions for the free exploration of yet untried ideas.

(D. Hilbert). I always tried to illuminate the problems and difficulties

and to offer a bridge leading to currently open questions…

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The mathematical foundation of the Quantum Mechanics

From 1911 Hilbert also was interested in the atomic structure of matter influenced by Max

Born. Indeed, between 1914 and 1915, Hilbert studied these theories deeply with the idea of

promoting a unified research programme for the sixth problem.

Because his interest in the axiomatic foundation of the whole Physics, Hilbert was working in

the formulation of the gravitational field-equations of the General Theory of Relativity. He was

strongly persuaded by the ideas of Heinrich Hertz and Ludwing Boltzmann.

Non surpresvely, in the summer of 1915 (June and July) Einstein was

in Göttingen invited by Hilbert to give some lectures about the state of

their research (the six Wolfskehl lectures). Both exchanged many

ideas and were impressed each other. After this summer, the

correspondence among them was almost daily.

Also in 1915, on November 20th, Hilbert provides his version of the

gravitational field-equations of the General Theory of Relativity in

Göttingen. Five days later (on November 25th) Einstein provided his own formulation in

Berlin. This fact caused some controversy (the so-called “nostrification”).

It seems that Einstein developed the theory, and Hilbert was probably pionner in getting the

right formulation of the essential equations. The way of working in Göttingen was so particular

that is not easy to clarify it. Anyway, in spite of the controversy, Hilbert always recognized the

authority of Einstein about the Relativity Theory.

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The mathematical foundation of the Quantum Mechanics

Max Born (1882-1970)

Assistants of David Hilbert were Max Born, Lothar W. Nordheim (also assistant of Max Born),

and a very young John von Neumman who just joined to the team.

Lothar Nordheim (1889-1985) John von Neumann (1903-1957)

In 1923, Werner Karl Heisenberg was an assistant of Max Born. From 1924 to

1927 he got a grant to work with Niels Bohr in Copenhague. For his Uncertainty

Principle, in 1932, he got the Nobel Price of Physic. From 1941 he was the

President of the Max Planck Institute.

In the winter of 1925, Werner Heisenberg exposed his ideas in the Hilbert’s

Seminary in Göttingen. And after this, Hilbert was even more interested in the

foundation of the new Physics.

Werner Karl

Heisenberg

(1901-19769

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The mathematical foundation of the Quantum Mechanics

The term "Quantum Mechanics" was coined by Max Born to denote a canonical theory of

motion of the atom and the electron, with the same level of consistency and generality

than the classical mechanics.

A first essential paper in the theory was the following:

W. Heisenberg, On a quantum theoretical interpretation of kinematical

and Mechanical relations, Z. Phys. 33 (1925), 879-893.

The idea of Heisenberg was to retain the classical equations of Newton, but to replace the

classical position coordinate with a quantum theoretical quantity.

M. Born was realized that the rule for multiplying kinematic quantities related to the quantum

position was very similar to that of the matrix product. Therefore the next step was the

formulation of Heisenberg's theory in terms of matrices.

In this way it arises the called Matrix Quantum Mechanics with the works:

M. Born and P. Jordan, “Zur Quantenmechanik,” Z. Phys. 34, (1925), 858–888.

M. Born, W. Heisenberg, and P. Jordan, “Zur Quantenmechanik II,” Z. Phys. 35, (1926), 557–615.

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The mathematical foundation of the Quantum Mechanics

P. Jordan (1902-1980)

He uses the delta “function” to define the derivative of a non-continuous function. He imagine 𝛿(𝑥) as «a

function having an infinite value at x=0, and a zero value in some other point, in such a way that its integral

is equal to zero».

A mathematical fiction, in words of von Neumann. However experimentally it gets very good predictions,

and it is easy to work with, so nowadays many people still use it.

Independently, Paul Dirac discovered the general equations of Quantum

Mechanics without the use of the matrices.

These works of Born, Jordan, and Heisenberg, mark the beginning of a new

era in Physics, in which matrices, commutators, and eigenvalues become

mathematical milestones of the atomic age.

Pascual Jordan was an assistant of Max Born. Some mathematicians have

speculated that P. Jordan could have shared with Max Born the Nobel Prize,

in 1954, in case of not being joined to the Nazi Party (in 1933).

Paul Dirac (1902-1984)

P. A. M. Dirac, The fundamental equations of quantum mechanics, Proc. R. Soc.

London, Ser. A 109 (1925), 642–653.

In this work Dirac developes a hamiltonian Mechanics for the atom. This is a

quantum non-commutative theory based in the Diract’s delta.

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The mathematical foundation of the Quantum Mechanics

E. Schrödinger 1887-1961

In these works, he established the so-called Schrödinger equations (he got Nobel Prize of

Physic in 1933 for this).

He pioneered to relate the Matrix Mechanics with the Wave Mechanics. He gave the key of

the equivalence of both theories (no in a rigorous way) in the paper:

E. Schrödinger, On the relation of the Heisenbergg-Born-Jordan Quantum Mechanics and Mine, Annalen

der Physick 79 (1926), 734-756.

Just after this, Dirac and Jordan proved, independently, the equivalence of both theories

without the use of the Dirac’s delta functions.

During the course 1926-1927, Hilbert delivered a lecture entitled "Mathematische Methoden

der Quantentheorie" which gave rise to the first publication of von Neumann about Quantum

Mechanics.

At this time, Erwin Rudolf Schrödinger worked in his Wave Mechanics

in the University of Zurich (after to participate in the First World War).

E. Schrödinger, Quantisierung als Eigenwertproblem, (The quantification as an

eigenvalue problem) Ann. Phys. (1926)

1ª communication: vol. 79, p. 361-376, 2ª communication vol. 79, p. 489-527,

3ª communication vol. 80, p. 437-490, y 4ª communication vol. 81, p. 109-139.

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The mathematical foundation of the Quantum Mechanics D. Hilbert. J. von Neumann & L. Nordheim, Über die Grundlagen der Quatenmechanik, Math. Annalen 98

(1927), 1-30.

Nevertheless, in the same year (1927), J. von Neumann published the following 3 essential

papers about Quantum Mechanics. With them, he developed a rigorous Mathematical

formulation of Quantum Mechanics. He showed the equivalence between Matrix Mechanics

and Wave Mechanics also in a rigorous way (avoiding, of course, the Dirac’s delta).

J. von Neumann, Mathematische Begundung der Quantenmechanik, Nachr. Ges. Wiss. Göttingen (1927),

1-57.

J. von Neumann, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik, Nachr. Ges. Wiss.

Göttingen (1927), 245-272.

J. von Neumann, Thermodynamik quantenmechanischer Gesamtheiten Nachr. Ges. Wiss. Göttingen

(1927), 273-291.

Some years ago, Hilbert was working in the problem of finding linear operators whose

eigenvalues were able to represent the spectral lines.

He was not success with this, because he was not able to proof that a sequence of such

eigenvalues has to converge to zero (when its terms represent the energy of the atoms).

Because of this Hilbert gave up his research in spectroscopy. This problem was solved

finally by von Neumann.

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The mathematical foundation of the Quantum Mechanics

J. Von Neumann (1903- 1975)

The above papers constitute the essence of the famous book of von Neumann (1932). There

the mathematical foundations of Quantum Mechanics are developed in terms of separable

Hilbert spaces, and operators among them.

J. von Neumann, Mathematische Grundlagen der Quantenmechanik, J. Springer (1932).

Dover Publications, New York, 1943; Presses Universitaires de France, 1947; Instituto de Mathematicas

"Jorge Juan” Madrid, 1949; Translation from German ed. by Robert T. Beyer, Princeton Univ. Press, 1955.

In this book, von Neumann thanks the simplicity and usefulness of the formulation of Dirac,

but he considers it unacceptable (“a mathematical fiction”). He points out not only the

mathematical inconsistency of the Dirac’s delta, but also the assumption that every self-

adjoint operator is diagonalizable. (Infinite dimensional Hilbert spaces).

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The mathematical foundation of the Quantum Mechanics

J. von Neumann extended the Matrix Mechanics to the framework of the separable Hilbert

spaces. (Indeed, in the own definition of Hilbert space he assumed the hipothesis of the

separability).

As a concrete Hilbert space, he considered the given by the square integrable functions.

The third chapter is addressed to introduce the Statistic in the Quantum Mechanics.

The question if either the Quantum Mechanics is a statistical theory, or if this fact is

avoidable, is analyzed in the fourth chapter.

Finally, the problem of the measurement is studied in the chapters five and six.

In the two first chapters von Neumann developes the theory of the Hilbert

spaces. More precisely:

In the first chapter he introduced, by means several postulates, the ideas

about which the Quantum Mechanics is structured. To this aim, he

developed the basic theory of Hilbert spaces.

The second chapter is a is a purely mathematical treatise (almost funny in

a book of Physic). There he developed the topic of the continuous linear

operators on a Hilbert space, as well as the eigenvalue problem.

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Postulates of Quantum Mechanics Postulate 1 (The wavefunction): Each physical system is associated with a separable

complex Hilbert space H (the State Space). Any instantaneous state of the system

corresponds to a unit vector of H (called ket) (which encodes the probabilities of all possible

outcomes of measurements made to the system). Two vectors represent the same state if

they differ only by a phase factor, that is a complex number with module 1).

Dirac’s notation for kets: |𝜓⟩ (this is like 𝑣 ∈ 𝐻 with 𝑣 = 1).

The exact nature of the Hilbert space that defines the state space depends on the system.

For instance the state space for position and momentum is the space of square integrable

functions.

Postulate 2 (Operators and observables): The observables of a physical system are

represented by hermitian (i.e. self-adjoint) linear operators on 𝐻 (the space state). The set of

eigenvalues of an observable is called the spectrum. The values that we can obtain after a measurement of an observable 𝐴 belong to the spectrum of 𝐴. The observable's eigenvectors ⟨𝑎| form an orthonormal basis. Any quantum state can be

represented as a superposition of the eigenstates of an observable.

Postulate 3 (Measurement and operator eigenvalues): If a physical system of

observables is in the state |𝜓⟩, then the more that we can predict about a measurement of an

observable 𝐴 (𝐴 ∈ 𝐿(𝐻)) is that the probability of obtaining as the outcome of the

measurement of 𝐴 the eigenvalue λ (with associate eigenvector ⟨𝑎|) is given by

𝑃𝐴|𝜓⟩ = 𝑎|𝜓2 .

(the transition probability).

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Postulates of Quantum Mechanics

Therefore, the expected value of 𝐴 will be ⟨𝐴⟩|𝜓⟩= 𝜆𝑖𝑖 𝑎𝑖|𝜓2 = ⟨𝜓|𝐴|𝜓⟩

The standard deviation of the measurement is ∆|𝜓⟩𝐴 = ⟨𝜓|𝐴2|𝜓⟩ −⟨𝜓|𝐴|𝜓⟩ 2

The Heisemberg’s uncertainty principle establishes that the product of the standard deviation

of two observables 𝐴 and 𝐵 over the same state |𝜓⟩ is such that

∆𝐴∆𝐵 ≥1

2⟨𝜓|,𝐴, 𝐵-|𝜓⟩

For instance, typical observables are the position 𝑋 and the linear moment 𝑃𝑋. Because

𝑋, 𝑃𝑋 = 𝑖ℏ we have that the uncertainty principle means that

∆𝐴∆𝐵 ≥1

2ℏ

where ℏ is the rationalized Planck constant or Dirac constant (ℏ=ℎ

2𝜋= 1.054589 × 10−34

joules per second)

Postulate 4 (Expectation values): The measurement of an observable 𝐴 cause an

(unpredictable) instantaneous collapse of the state vector 𝜓⟩ into an eigenstate of 𝐴. Indeed,

with the probability given before, after a measurement of 𝐴 we obtain the value 𝜆𝑖 . This collapse should be interpreted as an updating of the information contained in the

mathematical object 𝜓⟩ that represent the state of the system. .

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Postulates of Quantum Mechanics

Postulate 5 (The time-dependence Schrödinger equation)

The wave function or state function 𝜓⟩ of a system evolves in time (without perturbations)

according to the time- dependent Schrödinger equation

𝑖ℏ𝑑

𝑑𝑡 𝜓(t)⟩ = 𝐻(𝑡)|𝜓(t)⟩

The operator 𝐻 is called the Hamiltonian. This is the hermitian operator corresponding

to the total energy of the system (commonly expressed as the sum of operators

corresponding to the kinetic and potential energies). Theferore its eigenvalues are the

unique allowed values for the total energy of the system (and hence they are

quantized values).

Postulate 6 (Permutation symmetry of the wavefunction):

There are two types of particles, classified by their spin quantum numbers: Particles with

integral spin quantum numbers called bosons; and particles with half-integral spin quantum

numbers (such as electrons and protons) called fermions. The total wave function must be

antisymmetric with respect to the interchange of all coordinates (spatial and spin) of one

fermion with those of another. Bosons are symmetric under such an operation.

For instance: The operators of position and momentum satisfy the following

commutation rule: ,𝑋𝑖 , 𝑋𝑗]= 0, ,𝑃𝑖, 𝑃𝑗]= 0, ,𝑋𝑖 , 𝑃𝑗]= 𝑖ℏ𝛿𝑖𝑗𝐼.

This implies that the dimension of the Hilbert space have to be infinite dimensional.

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Seminario kkjkjkjkjk

From Physics to Mathematics via Quantum Mechanic

Phisical states

of the quantum system H Hilbert spaces

Observables 𝑇:𝐻 → 𝐻 Operator Theory

𝑻 ∈ 𝑳(𝑯)

Observables values

of a state

(new states)

Eigenvalues

(eigenvectors)

Spectral theory

𝑻 − 𝝀𝑰 𝒙 = 𝒚

Physic Modern Physics

Quantum Mechanics

Mathematics Functional Analysis

Operator Theory

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From Quantum Mechanics to Functional Analysis

The Hilbert spaces are classical examples of the named Banach spaces.

The aim of the Functional Analysis is the study of Banach spaces and the operators defined

on them.

The historical roots of the Functional Analysis are ubicated in the variational calculus

(optimization problems of continuous real valued functionals, defined on a set of functions),

the Fourier transformations, the diferential equations and the integral equations.

In 1920, Stefan Banach presented for defense his PhD Thesis,

published two years later.

S. Banach, Sur les Opérations dans les ensembles abstraits et leur applications

aux équations intégrales”, Fundamenta Mathematicae 3 (1922), 133-181.

There, the notions of normed space and Banach space were

introduced as well as the foundations of Functional Analysis.

(In 1918, Frigyes Riesz had provided for the first time the axioms for a

normed space without further development).

From this moment the development of Functional Analysis is

spectacular, because the power of their methods and their

applicability.

Stefan Banach

(1892-1945)

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From Quantum Mechanics to Functional Analysis

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From Quantum Mechanics to Functional Analysis

A golden year for the Science: 1932

J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer (1932).

As said before, the mathematical foundations of the Quantum Mechanic were

provided here.The first part of this book it is a treatise about the general theory

of Hilbert spaces.

M. Stone, Linear Transformations in Hilbert Space and their applications to Analysis,

American Mathematical Society (1932).

Here, the spectral theory of hermitian operators on Hilbert spaces is developed

with applications to classical Analysis, and to the differential and integral

equations.

S. Banach, Théorie des Opérations Linéaires, Chelsea N. Y. (1932),

Here, the most important results of the theory of Banach spaces are showed.

The book contains the celebrated 23 open problems that have been a source of

inspiration for many researches later (many of which remain still unresolved).

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From Quantum Mechanics to Functional Analysis

Definition: A Banach space is a complete normed space.

Definition: A normed space is a (real or complex) linear space 𝑋 equipped

with a norm, i. e. a function ∙ : 𝑋 → ℝ satisfying

i) 𝑥 = 0 ⇒ 𝑥 = 0 (separates points)

ii) 𝛼𝑥 = 𝛼 𝑥 (absolute homogeneity)

iii) 𝑥 + 𝑦 ≤ 𝑥 + 𝑦 triangle inequality (or subadditivity).

Examples of Banach spaces: a) ℝ, ℂ b) ℝ𝑛, ℂ𝑛 c) Matrices 𝑀𝑛×𝑛 d) Sequences spaces 𝑙𝑝

e) Spaces of continuous functions 𝐶,𝑎, 𝑏- f) Spaces of integrable functions 𝐿𝑝,𝑎, 𝑏-

g) Spaces of bounded linear operators: 𝐿 𝑋, 𝑌 , 𝐿 𝑋 , 𝐿 𝐻 .

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From Quantum Mechanics to Functional Analysis

To generalize the euclidean space ℝ3.

Let 𝑥 = (𝑥1, 𝑥2, 𝑥3)∈ ℝ3. Then

𝑥 = 𝑥12 + 𝑥2

2 + 𝑥32

That is 𝑥 = 𝑥, 𝑥 where 𝑥, 𝑥 = 𝑥12 + 𝑥2

2 + 𝑥32 (coordinate-coordinate product).

Note also that 𝑥, 𝑒𝑖 = 𝑥𝑖 , where 𝐵 = *𝑒1, 𝑒2,𝑒3+ denotes the canonical basis.

In fact, if 𝑥 = (𝑥1, 𝑥2, 𝑥3) then

𝑥1 = 𝑥, 𝑒1 = (𝑥1, 𝑥2, 𝑥3), (1,0, 0)

𝑥2 = 𝑥, 𝑒2 = (𝑥1, 𝑥2, 𝑥3), (0,1, 0)

𝑥3 = 𝑥, 𝑒3 = (𝑥1, 𝑥2, 𝑥3), (0,0, 1)

Therefore:

The goal is to replace 𝑖 = 3 by 𝑖 = ∞.

𝑥 = 𝑥, 𝑒𝑖

𝑖=3

𝑖=1

𝑒𝑖

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Hilbert spaces

Examples of Hilbert spaces:

a) ℝ, ℂ b) ℝ𝑛, ℂ𝑛 c) 𝑙2 = * 𝛼𝑛 ∶ 𝛼𝑛

2 <∞+ with the inner product

𝛼𝑛 , 𝛽𝑛 = 𝛼𝑛𝛽𝑛

d) The space 𝐿2,𝑎, 𝑏- of all square-integrable real-valued functions on an interval

,𝑎, 𝑏- with the inner product

⟨𝑓, 𝑔⟩ = 𝑓 𝑥 𝑔 𝑥 𝑑𝑥𝑏

𝑎

e) 𝐿2(𝑋, 𝜇 ) the space of those complex-valued measurable functions on a measure

space space (𝑋,𝑀, 𝜇) with the inner product 𝑓, 𝑔 = 𝑓 𝑡 𝑔 𝑡 𝑑𝜇 𝑡𝑋.

Hilbert spaces

Banach spaces

Definition: A Hilbert space is inner product Banach space 𝐻. That is a Banach

space 𝐻 whose norm is given by 𝑥 = 𝑥, 𝑥 , for every 𝑥 ∈ 𝐻 where ∙,∙ is an

inner product.

From now on all the linear spaces considered here will be either complex or real ones.