Quantum theory

The quantum theory is the name given to a physical theory which tries to very model the behavior of energy with small scales using the quanta, discontinuous quantities. Its introduction hustled several generally accepted ideas in physics of the time. It was used as bridge between the traditional Physique and the Quantum physics, from which the angular stone, the quantum Mécanique, was born in 1925.

It was initiated by Planck in 1900, then developed primarily by Einstein, Bohr, Sommerfeld, Kramers, Heisenberg, Pauli and of Broglie between 1905 and 1924.

History

The traditional Physique into force at the end of the 19th century included/understood the following theories:

• the Newtonian Mechanical , published by Newton in 1687 and improved by later generations physicists for the needs for the Celestial mechanics .

• the theory of the electromagnetism, developed by Maxwell in 1865 and reformulated by Lorentz in 1895. This theory includes the undulatory Optique like particular case.

• the Thermodynamic , formalized in the years 1850 by Clausius, and a first version of the Physical statistics: the kinetic Theory of the gases, developed by Maxwell and Boltzmann.

Experimental problems of the end of the XIXe century

A certain number of known experimental facts at the end of the 19th century were unexplainable within the framework of the classical theory. These unmatched experimental facts led the physicists gradually to propose a new vision of the world, the Quantum physics. The major stages of this conceptual revolution proceeded between 1900 and 1925.

The radiation of the black body

The radiation of the black Corps is the electromagnetic radiation produced by a body completely absorbing in thermodynamic balance with its medium.

To imagine a closed enclosure maintained with a Temperature $T$: a “furnace”, and pierced with a tiny hole. Walls of the furnace being supposed completely absorbing, any radiation initially outside the furnace which penetrates via the hole towards the sudden interior of the enclosure of multiple reflections, emissions and absorptions by the walls of the furnace until reaching a complete thermalization: the enclosure and its contents of radiation are in thermal balance . Besides reciprocally, a negligible part of the thermal radiation inside the furnace can escape definitively from this one, allowing its experimental study, in particular its spectral energy distribution , i.e. the density of voluminal energy present per elementary interval of frequency. Thermodynamics makes it possible to show that the characteristics of this radiation do not depend on the nature of the material of which the walls of the furnace are made up, but only of its temperature. This radiation is called radiation of the black body .

At the end of the 19th century, the classical theory was unable to explain the experimental characteristics of the radiation of the black body: the calculation of emitted energy tended theoretically towards the Infini, which was obviously in contradiction with the experiment. This dissension was called ultraviolet Catastrophe, and constitutes one of two small clouds in the serene sky of the theoretical physics , formula celebrates marked by Thomson - alias Lord Kelvin - on April 27th 1900 at the time of a conference. In the continuation of its speech, Thomson predicted a rapid explanation of the experimental results within the framework of the classical theory. The history gave him wrong: a few months only after the conference of Thomson, Planck proposed a daring assumption which will involve a radical upheaval of the landscape of the theoretical physics.

The relation of Planck-Einstein (1900-1905)

In cause of despair, Planck made the assumption that the energy exchanges between the electromagnetic Rayonnement of the black body and the matter constituting the walls of the furnace were quantified, i.e. energy is transmitted per packages. More precisely, for a monochromatic radiation of frequency $\backslash \; nu$, the energy exchanges could take place only by multiple entireties of a minimal quantity, a quantum of energy:

$|\backslash \; Delta\; E|\; =\; N\; \backslash \; H\; \backslash \; nu$

where $n\; =\; 0,1,2,3,\dots$ is a positive integer, and $h$ a new universal constant, now called Constante of Planck or quantum of action . This constant is worth:

$H\; =\; 6,62.10^\; \{-\; 34\}$ joule S

The law of Planck for the radiation of the black body is written:

$B\_\; \backslash \; lambda\; (T)\; =\; \backslash \; frac\; \{8\; \{\backslash \; pi\}\; hc\}\; \{\backslash \; lambda^\; \{5\}\; (e^\; \{\backslash \; frac\; \{hc\}\}\; -\; 1)\}$

$\backslash \; lambda$ being the wavelength, T the temperature in Kelvin, H the constant of Plank, and C speed of light in the vacuum.

The assumption of the quanta of max Planck was taken again and supplemented by Einstein in 1905 to interpret the photoelectric effect.

The photoelectric effect (1905)

See also: photoelectric Effect

At the end of the 19th century, the physicists notice that when one lights a metal with a light, this one can emit electrons.

Their kinetic energy depends on the frequency of the incidental light, and their number depends on the luminous intensity, which is not easily comprehensible within the undulatory model of the light. In particular, if the incidental light has a frequency in lower part of a certain threshold, nothing occurs, even if one waits very a long time. This result is incomprehensible classically, because the theory of Maxwell associates with the electromagnetic waves a density of energy proportional to the luminous intensity, therefore it is classically possible to accumulate as much energy than one wants in metal in the illuminant sufficiently a long time and this whatever the frequency of the incidental radiation considered . There should not be threshold!

Inspired by Planck, Einstein proposed in 1905 a simple assumption explaining the phenomenon: the electromagnetic radiation itself is quantified , each “grain of light” - which will be baptized photon later on - being carrying a quantum of energy $E\; =\; H\; \backslash \; nu$. The electrons absorbing the photons acquire this energy; if it is higher than an energy of fixed threshold (which depends only on the nature of metal), the electrons can leave metal. The emitted electrons have the kinetic energy then:

$\backslash \; frac\; \{1\}\; \{2\}\; \backslash ,\; m\; \backslash ,\; v^2\; \backslash \; =\; \backslash \; H\; \backslash ,\; \backslash \; naked\; \backslash \; -\; \backslash \; E\_\; \{threshold\}$.

This article was worth in Einstein the title of doctor of theoretical physics in 1905, and the Nobel Prize of physics in 1921.

The stability of the atoms

Two serious problems arose as of the end of the 19th century concerning the Atome S, consisted of a certain number of specific electrons negatively charged, and of a quasi-specific core, charged positively:

• the stability of an atom is incomprehensible within the framework of the classical theory. Indeed, the theory of Maxwell affirms that any load accelerated rayon of energy in the form of wave electromagnetic. In a traditional planetary model, the electrons are accelerated on their orbits within the atom, and their energy must decrease: the electrons fall then on the core. A calculation of the duration characteristic of this phenomenon is about 10 years, therefore the traditional atoms are unstable, which the experiment contradicts obviously!

• Moreover, the classical theory predicts that the radiation emitted by the accelerated electron has a frequency equal to the angular frequency of the movement. The electron falling continuement on the core, its frequency angular increases continuement, and one should observe a continuous spectrum. However the light emitted by a spectral lamp with atomic vapor presents a discrete spectrum of lines !

It is the Dane Niels Bohr who will propose the first a semi-traditional Modèle allowing to circumvent these difficulties.

The model of Bohr and Sommerfeld

The model of Bohr (1913)

See also: Model of Bohr

The Modèle of Bohr of the hydrogen atom is a model which uses two very different ingredients:

1. a description of traditional mechanics nonrelativistic: the electron turns around the proton on a circular orbit.

2. two ad hoc quantum ingredients:
   1. Seules certain circular orbits are allowed (quantification). Moreover, the electron on its circular orbit does not radiate, as opposed to what not predicted the theory of Maxwell.
   2. the electron can sometimes pass from a circular orbit allowed another allowed circular orbit, condition of emitting light of a quite precise frequency, dependant on the difference in energies between the two circular orbits in accordance with the relation of Planck-Einstein.

The exotic mixture of these ingredients produces spectacular results: the agreement with the experiment is indeed excellent.

Improvements of Sommerfeld (1916)

Sommerfeld will improve the model of Bohr in two stages:

1. generalization with the elliptic orbits.

2. relativistic treatment of the model with elliptic orbits.

The inclusion of the relativistic effects will do nothing but return the still best comparison with the experimental results.

August 1st

Relations of Broglie (1923)

Whereas it was clear that the light presented a Dualité wave-particle, Louis de Broglie proposed to boldly generalize this duality with all the known particles.

In its thesis of 1923, with Broglie with each material particle energy $E$ a frequency $\backslash \; nu$ associates according to the relation of Planck-Einstein already mentioned, and, been new, the weather proposes to associate with the impulse $p=\; m\; v$ of a nonrelativistic massive particle a wavelength $\backslash \; lambda$, according to the law:

$p\; \backslash \; =\; \backslash \; \backslash \; frac\; \{H\}\; \{\backslash \; lambda\}$

This constituted a new revolutionary step. Paul Langevin made at once read the thesis of Broglie with Einstein, which declared: “It Broglie raised a corner of the large veil. ” The undulatory character of the electron will receive a direct experimental confirmation with the experiment of diffraction of the electrons by a crystal realized by Davisson and Germer in 1927.

The relations of Broglie can be also written:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; E\; \backslash \; =\; \backslash \; \backslash \; hbar\; \backslash \; \backslash \; Omega\; \backslash \; \backslash \; \backslash \; vec\; \{p\}\; \backslash \; =\; \backslash \; \backslash \; hbar\; \backslash \; \backslash \; vec\; \{K\}\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

in term of the pulsation: $\backslash \; Omega\; =\; 2\; \backslash \; pi\; \backslash \; naked$ and of the vector of wave $\backslash \; vec\; \{K\}$, whose standard is worth: $k\; =\; 2\; \backslash \; pi/\backslash \; lambda$.

The effect Compton (1923-1925)

See also: Diffusion Compton, Effect Compton

The electron S, charged particles, interact with the light, classically described by an electromagnetic field. However, traditional physics does not make it possible to explain the variation observed wavelength of the radiation according to the direction of diffusion. The correct interpretation of this experimental fact will be given by Compton and its collaborators at the conclusion of experiments carried out between 1925 and 1927.

This effect, baptized in its honor Effect Compton, is well described by considering the shock Photon - electron, like a shock between the two particles, the photon being carrying a quantum of energy $E\; =\; H\; \backslash \; nu$ and a quantum of impulse $\backslash \; vec\; \{p\}\; =\; \backslash \; hbar\; \backslash \; vec\; \{K\}$. The photons are diffused according to variable directions, and present a variation wavelength which depends on the direction of diffusion.

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