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Quamtuum MechanicsQuamtuum MechanicsQuantum mechanics replaces classical mechanics and classical electromagnetism at the atomic and subatomic levels. It is a important part of theoretical physics and has many applications in experimental physics. Quantum (quantification) is connected to the discrete units that the theory assigns to certain physical quantities, such as the energy of an atom at rest.In the first half of the twentieth century the physicists Werner Heisenberg, , Louis de Broglie, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Albert Einstein and Wolfgang Pauli established the foundations of quantum mechanics..
Werner Heisenberg
The Heisenberg uncertainty principle discovered in 1927 He invented matrix mechanics, the first formalization of quantum
mechanics in 1925Together with Bohr, he formulated the Copenhagen interpretation of
quantum mechanicsHe received the Nobel Prize in physics in 1932 “for the creation of
quantum mechanics”He collaborated with Wolfgang Pauli, and along with Paul Dirac,
developed an early version of quantum electrodynamics.
Max Planck
The founder of quantum theory, the problem of black-body radiation (Planck’s law), Nobel Prize in Physics 1918 (awarded 1919)
Louis de Broglie
De Broglie hypothesis which states that any moving particle or object has an associated wave. For this hypothesis he won the Nobel Prize in Physics in 1929. Among the applications of this work has been the development of electron microscopes to get much better image resolution than optical ones, because of shorter wavelengths of electrons compared with photons
Niels Bohr
Bohr's model; quantization of electron's orbital angular momentum; electrons travel in discrete orbits around the atom's nucleus, with the chemical properties of the element being largely determined by the number of electrons in each of the outer orbits; an electron could drop from a higher-energy orbit to a lower one, emitting a photon of discrete energy; the principle of complementarity which states that items could be separately analyzed as having several contradictory properties. He received the Nobel Prize for Physics for this work in 1922.
Erwin Schrödinger
the Schrödinger equation. He received the Nobel Prize in 1933 for his contributions to quantum mechanics, especially the Schrödinger equation
Max Born
formulated together with Heisenberg the matrix mechanics representation of quantum mechanics. He formulated the now-standard interpretation of the probability density function for ψ*ψ in the Schrödinger equation of quantum mechanics. He won the 1954 Nobel Prize in Physics
John von Neumann
was a pioneer of the modern digital computer and the application of operator theory to quantum mechanics
Paul Dirac
the Dirac equation which describes the behavior of fermions and which led to the prediction of the existence of antimatter; the founder of quantum electrodynamics . He shared the Nobel Prize in physics for 1933 “for the discovery of new productive forms of atomic theory” with Erwin Schrödinger
Albert Einstein
the probabilistic interpretation of quantum theory, the quantum theory of a monatomic gas, the thermal properties of light with a low radiation density which laid the foundation of the photon theory of light, the theory of radiation. He was awarded the 1921 Nobel Prize for Physics for his 1905 explanation of the photoelectric effect and for his services to theoretical physics
Wolfgang Pauli
the theory of nonrelativistic spin, and in particular the discovery of the exclusion principle which explains why matter occupies space exclusively for itself and does not allow other material objects to pass through it, at the same time allowing light and radiation to pass. It states that no two identical fermions may occupy the same quantum state simultaneously
Mathematical Statements
1. De Broglie hypothesisDe Broglie hypothesis (1924) states that any moving particle or
object has an associated wave. These waves are called de Broglie waves.
If a particle behaves as a wave, the wave-particle duality, we have to determine the wavelength and frequency.
The de Broglie hypothesis states that the expressions of energy and momentum of photons can be generalized at the electromagnetic radiation and also at particles.
The wavelength of a particle which has the mass and the velocity is given by
vm
hph
The monentum and the energy are given by
Experimental confirmation of de Broglie hypothesis
The Davisson-Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. Since the original Davisson-Germer experiment for electrons, the de Broglie hypothesis has been confirmed for other elementary particles
Ekp ,
If a particle behaves as a wave this allows to describe it with a wave function
),( tr . Its quantum state can be represented as a wave, of arbitrary shape and
extending over all of space, called a wavefunction. The position and momentum of
the particle are observables. The quantity 2 represents the density of
probability in localizing the particle in space. The quantity rd 32 is the
probability, at time t , of finding the particle in the infinitesimal region of volume
rd 3 surrounding the position r. This statistic interpretation of the wave function
of a particle was given by Max Born. The modification in time o the wave function
is connected to the modification of the probability of finding it in different points
in space.
2. Heisenberg's uncertainty principle
In the formalism of quantum mechanics, the state of a system at a given time
is described by a complex wave function . This abstract mathematical object
allows one to compute the probability of finding an electron in a particular region
around the nucleus at a particular time. One cannot ever make simultaneous
predictions of conjugate variables, such as position and momentum, with arbitrary
accuracy. For instance, electrons may be considered to be located somewhere
within a region of space, but with their exact positions being unknown.
Heisenberg's uncertainty principle it applies to the position and momentum
of a single particle. According to it, if we continue increasing the accuracy with
which one of these is measured, there will come a point at which the other must be
measured with less accuracy. If x and p are the uncertainties in the
measurements of the position and momentum we have
2
px , (1)
where 2h
is the reduced Planck's constant with sJh 341062,6 is
Planck’s constant.
The fundamental postulate of quantum mechanics: measurements of position
and momentum taken in several identical copies of a system in a given state will
vary according to known probability distributions.
Furthermore, for the x , y and z we have
2
xpx ,
2
ypy , (2)
2
zpz .
Every measured particle in quantum mechanics exhibits wavelike behaviour so there is an exact, quantitative analogy between the Heisenberg uncertainty relations and properties of waves or signals. In a time-varying signal such as a sound wave, it is meaningless to ask about the frequency spectrum at a single moment in time because the measure of frequency is the measure of a repetition recurring over a period of time. In order to determine the frequencies accurately, the signal needs to be sampled for a finite (non-zero) time. This necessarily implies that time precision is lost in favor of a more accurate measurement of the frequency spectrum of a signal. This is analogous to the relationship between momentum and position, and there is an equivalent formulation of the uncertainty principle which states that the uncertainty of energy of a wave (directly proportional to the frequency) is inversely proportional to the uncertainty in time with a constant of proportionality identical to that for position and momentum.
Very important is the connection to the measurement of energy and t
moment when a particle is moving at a point. For an electron which is moving
along the x axis the energy is given by
0
20
22v
mpmE xx , (3)
where kgm 310 101,9 is the rest mass of the electron. Differentiating and
replacing Ed by E and xpd by xp one gets
xxxxx pp
mpmE v
22
2v
0
20 . (4)
If the time needed or observing the electron is t , we have tx x v and
we have
x
xtv
. (5)
By multiplying (4) and (5) it results
xpxtE (6)
and using first equation of (2) we obtain
2 tE . (7)
This is the uncertainty in determination of the energy and moment for a
particle or uncertainty relation between energy and time. The relation has an
important implication for spectroscopy. As excited states have a short lifetime their
energy uncertainty is not negligible. For this reason sharp lines cannot be obtained
even under ideal conditions. This relation helps also to give an idea of the
“chaotic” behavior of the space-time, wherein very small time steps authorize huge
energy variations.