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.