de Investigación
AboutGrupo
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
de Investigación
AboutGrupo
the Mathematical
Foundation of Quantum Mechanics
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
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
CIMPA-MOROCCO
MEKNÈS, September2014
Operator Theory and The Principles of Quantum Mechanics
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.
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 gaseousfluid 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.
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.
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.
Why the Quantum Mechanics is relevant?
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.
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.
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.
Why the Quantum Mechanics is relevant?
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.
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.
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).
Procesador D.wave 2
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.
The origins of the Quantum Mechanics in the Physics
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.
𝐼 𝜈, 𝑇 =
2ℎ𝜈 3 1
ℎ𝜈
𝑐2
−1
𝑒 𝑘𝑇
The intensity of the medium (or spectral) radiation emitted by a
black body at temperature T and frequency 𝜈
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
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)
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:
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.
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 four main papers of Einstein in 1905
“Do not worry about your difficulties in Mathematics. I can assure
you mine are still greater.” Albert Einstein
“
Volume 17 of 1905. Edited by Max Planck
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.
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.
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 location of the physical events in space and time
are relative to the state of motion of the observer.
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 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
fusion.
nuclear
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).
For instance, the Sun generates its
energy by nuclear fusion of
hydrogen nuclei into helium.
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.
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.
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
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.
The origins of the Quantum Mechanics in the Physics
1923. Louis-Victor de Broglie postulates that the moving of an electron has a wavelength
ℎ
associated that is given by 𝜆 =
(where 𝑕 denotes the Planck’s constant).
𝑚𝑣
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.
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).
He proposed “to threat in the same manner, by means of
axioms, those physical sciences in which mathematics
plays an important part.”
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.
D. Hilbert 1862-1943
Gottingen University
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.
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.
The mathematical foundation of the Quantum Mechanics
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…
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.
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.
The mathematical foundation of the Quantum Mechanics
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.
Max Born (1882-1970)
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.
Werner Karl
Heisenberg
(1901-19769
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.
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.
The mathematical foundation of the Quantum Mechanics
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).
P. Jordan (1902-1980)
Independently, Paul Dirac discovered the general equations of Quantum
Mechanics without the use of the matrices.
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.
Paul Dirac (1902-1984)
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.
The mathematical foundation of the 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.
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.
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.
The mathematical foundation of the Quantum Mechanics
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.
J. Von Neumann (1903- 1975)
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 selfadjoint operator is diagonalizable. (Infinite dimensional Hilbert spaces).
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.
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.
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.
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|𝜓 2 .
(the transition probability).
Postulates of Quantum Mechanics
Therefore, the expected value of 𝐴 will be
〈𝐴〉|𝜓〉 =
𝑖 𝜆𝑖
The standard deviation of the measurement is ∆|𝜓〉 𝐴 =
𝑎𝑖 |𝜓
2
= 〈𝜓|𝐴|𝜓〉
〈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
1
∆𝐴∆𝐵 ≥ 〈𝜓|,𝐴, 𝐵-|𝜓〉
2
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
1
∆𝐴∆𝐵 ≥ ℏ
2
ℎ
where ℏ is the rationalized Planck constant or Dirac constant (ℏ= = 1.054589 × 10−34
2𝜋
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.
.
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
𝑑
𝑖ℏ 𝜓(t)〉 = 𝐻(𝑡)|𝜓(t)〉
𝑑𝑡
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 ]= 𝑖ℏ𝛿𝑖𝑗 𝐼.
This implies that the dimension of the Hilbert space have to be infinite dimensional.
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
Quantum
Mechanics
Mathematics
Physic
Modern Physics
Functional Analysis
Operator Theory
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
published two years later.
for defense his PhD Thesis,
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)
From Quantum Mechanics to Functional Analysis
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).
From Quantum Mechanics to Functional Analysis
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).
Definition: A Banach space is a complete normed space.
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, 𝑏f) Spaces of integrable functions 𝐿𝑝 ,𝑎, 𝑏g) Spaces of bounded linear operators: 𝐿 𝑋, 𝑌 , 𝐿 𝑋 , 𝐿 𝐻 .
From Quantum Mechanics to Functional Analysis
To generalize the euclidean space ℝ3 .
Let 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 )∈ ℝ3 . Then
𝑥 =
That is 𝑥 =
𝑥12 + 𝑥22 + 𝑥32
𝑥, 𝑥 where 𝑥, 𝑥 = 𝑥12 + 𝑥22 + 𝑥32
Note also that 𝑥, 𝑒𝑖 = 𝑥𝑖 , where 𝐵
In fact, if 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 ) then
𝑥1 = 𝑥, 𝑒1 =
𝑥2 = 𝑥, 𝑒2 =
𝑥3 = 𝑥, 𝑒3 =
Therefore:
= *𝑒1 , 𝑒2,𝑒3 + denotes the canonical basis.
(𝑥1 , 𝑥2 , 𝑥3 ), (1,0, 0)
(𝑥1 , 𝑥2 , 𝑥3 ), (0,1, 0)
(𝑥1 , 𝑥2 , 𝑥3 ), (0,0, 1)
𝑖=3
𝑥=
(coordinate-coordinate product).
𝑥, 𝑒𝑖 𝑒𝑖
𝑖=1
The goal is to replace 𝑖 = 3 by 𝑖 = ∞.
Hilbert 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.
Examples of Hilbert spaces:
a) ℝ, ℂ
b) ℝ𝑛 , ℂ𝑛
c) 𝑙2 = * 𝛼𝑛 ∶ 𝛼𝑛 2 <∞+ with the inner product
𝛼𝑛 , 𝛽𝑛
=
Banach spaces
Hilbert spaces
𝛼𝑛 𝛽𝑛
d) The space 𝐿2 ,𝑎, 𝑏- of all square-integrable real-valued functions on an interval
,𝑎, 𝑏- with the inner product
𝑏
⟨𝑓, 𝑔⟩ =
e)
𝐿2(𝑋, 𝜇
𝑓 𝑥 𝑔 𝑥 𝑑𝑥
𝑎
) the space of those complex-valued measurable functions on a measure
space space (𝑋, 𝑀, 𝜇) with the inner product
𝑓, 𝑔 =
𝑋
𝑓 𝑡 𝑔 𝑡 𝑑𝜇 𝑡 .
.
M. Victoria Velasco