# Quantum mechanics

Girl of old the Quantum theory, the quantum mechanical constitutes the pillar of a whole of Théorie S Physique S which one gathers under the general name of Quantum physics. This denomination is opposed to that traditional Physique, this one failing in its description of the microscopic world - Atome S and particles - like in that of certain properties of the electromagnetic Rayonnement.

The basic principles of quantum mechanics were established primarily between 1922 and 1927 by Bohr, Dirac, of Broglie, Heisenberg, Jordan, Pauli and Schrödinger. They allow a complete description of the dynamics of a nonrelativistic massive particle. Bohr proposed an interpretation of the formalism, called Interprétation of Copenhagen, based on the principle of correspondence.

The basic principles were supplemented by Bose and Fermi in order to authorize the description of a together of identical particles , opening the way with the development of a Physique quantum statistics. Lastly, in 1930, the mathematician Von Neumann specified the rigorous mathematical framework of the theory.

The Quantum theory of the fields is one of its relativistic extensions the most used to XXIe century.

### General panorama

#### Introduction

Girl of old the Quantum theory, quantum mechanics fixes a coherent mathematical framework which made it possible to cure all the dissensions between certain experimental results highlighted at the end of the 19th century and the corresponding theoretical predictions of the traditional Physique.

Quantum mechanics began again and developed the idea of Dualité wave-particle introduced by of Broglie in 1924 consisting in regarding the matter particles not only specific corpuscles, but also as Onde S, having a certain space extent (see Wave mechanics). Bohr introduced the concept of “complementarity” to solve this apparent paradox: any physical object is well at the same time a wave and a Corpuscule, but these two aspects, mutually exclusive, cannot be observed simultaneously. If an undulatory property is observed, the corpuscular aspect disappears. Reciprocally, if a corpuscular property is observed, the undulatory aspect disappears.

In 2007, no contradiction could be detected between the predictions of quantum mechanics and the experimental tests associated. This success has a price alas: the theory rests on an abstracted mathematical formalism, which makes its access difficult enough for the layman.

#### Some examples of success

Historically, the theory initially made it possible to correctly describe the electronic structures of the Atome S and the Molécule S, like their interactions with an electromagnetic field. It also makes it possible to explain the behavior of the condensed matter, in particular:

Another great success of quantum mechanics was to solve the Paradoxe of Gibbs: in Physical statistics traditional, identical particles are regarded as being discernible, and the Entropie is then not a extensive Grandeur. The agreement between the theory and the experiment was restored by taking account of the fact that identical particles are indistinguishable in quantum mechanics.

The Quantum theory of the fields, generalization relativistic of quantum mechanics, makes it possible as for it to describe the phenomena where the full number of particles is not preserved: Radioactivity, Nuclear fission (i.e. the disintegration of the Atomic nucleus) and nuclear Fusion.

## Canonical quantification

In traditional physics, a Onde planes progressive monochromatic pulsation $\ omega$ being propagated in the direction of X positive is written:

$\ Psi \left(X, T\right) = \ Psi_ \left\{0\right\} \ e^ \left\{- I \left(\ Omega T - K X\right)\right\}$

where the amplitude $\ Psi_ \left\{0\right\}$ is a certain constant, whereas $i$ is the imaginary unit which is defined so that $\ scriptstyle \left\{I\right\} ^2 = -1$. If we introduce into this expression the quantum relations of Broglie, we can reveal the sizes energy E and impulse p :

$\ Psi \left(X, T\right) \ = \ \ Psi_ \left\{0\right\} \ e^ \left\{- I \left(\left\{\ frac \left\{E\right\} \left\{\ hbar\right\} T \ frac \left\{p_x\right\} \left\{\ hbar\right\} X\right\}\right)\right\}$

where $\ hbar$ is the reduced Constante of Planck. This expression spreads easily in dimension 3:

$\ Psi \left(\ vec \left\{R\right\}, T\right) \ = \ \ Psi_ \left\{0\right\} \ e^ \left\{- \ frac \left\{I\right\} \left\{\ hbar\right\} \left(And \ vec \left\{p\right\} \ cdot \ vec \left\{R\right\}\right)\right\}$

To obtain energy, it is enough to derive compared to time:

$i \ \left\{\ hbar\right\} \ \ frac \ = \ E \ \ Psi \left(\ vec \left\{R\right\}, T\right)$

and to obtain the impulse, to take the Gradient:

$- \ I \ \left\{\ hbar\right\} \ \left\{\ nabla\right\} \ Psi \ = \ \ vec \left\{p\right\} \ \ Psi \left(\ vec \left\{R\right\}, T\right)$

### Rules of the canonical quantification

The canonical quantification consists in replacing the traditional dynamic variables of position and impulse, of the real numbers, by operators, according to the following rules of substitution:

• with the coordinate of position $x^i$ is associated an operator with position $\ hat \left\{X\right\} ^i$ such as:

$\ hat \left\{X\right\} ^i \ F \left(\ vec \left\{R\right\}\right) \ = \ x^i \ F \left(\ vec \left\{R\right\}\right)$

• with the variable of impulse $p_i$ is associated an operator impulse $\ hat \left\{p\right\} _i$ such as:

$\ hat \left\{p\right\} _i \ F \left(\ vec \left\{R\right\}\right) \ = \ - \ I \ \left\{\ hbar\right\} \ \ frac \left\{\ partial F \left(\ vec \left\{R\right\}\right)\right\}\left\{\ partial x^i\right\}$, is: $\ hat \left\{p\right\} _i \ = \ - \ I \ \left\{\ hbar\right\} \ \ frac \left\{\ partial\right\} \left\{\ partial x^i\right\}$

• with the variable energy is associated the operator with temporal derivation:

$E \ F \left(\ vec \left\{R\right\}, T\right) \ = \ I \ \left\{\ hbar\right\} \ \ frac \left\{\ partial F \left(\ vec \left\{R\right\}, T\right)\right\}\left\{\ partial T\right\}$, is: $E \ \ to \ I \ \left\{\ hbar\right\} \ \ frac$

### Equation of Schrödinger

#### Heuristic derivation of the equation

The Hamiltonien giving the total mechanical energy of a nonrelativistic massive particle subjected to a force deriving from a potential is given by the traditional expression:

$H \left(\ vec \left\{R\right\}, \, \ vec \left\{p\right\}\right) \ = \ \ frac \left\{p^2\right\} \left\{2m\right\} \ + \ V \left(\ vec \left\{R\right\}\right) \ = \ E$

This size contains all the necessary information being studied traditional of the dynamic evolution of the system via the canonical equations of Hamilton, with the help of the data of an initial condition. With this traditional particle a wave $\ Psi is associated \left(\ vec \left\{R\right\}, T\right)$, which one seeks the equation of evolution. According to the rules of the canonical quantification, the traditional Hamiltonian becomes an operator:

$\ hat \left\{H\right\} \ = \ \ frac \left\{\ hat \left\{p\right\} ^2\right\} \left\{2m\right\} \ + \ V \left(\ hat \left\{R\right\}\right) \ = \$

- \ \ frac {\ hbar^2} {2m} \ \ vec {\ nabla} ^2 \ + \ V (\ vec {R})

The operator $\ vec \left\{\ nabla\right\} ^2$ is the Laplacien $\ Delta = \ vec \left\{\ nabla\right\} ^2$. The traditional equation of conservation of energy:

$H \left(\ vec \left\{R\right\}, \, \ vec \left\{p\right\}\right) \ = \ E$

give, while multiplying each dimensioned by the function of wave, the equation of Schrödinger dependant on time:

$- \ \ frac \left\{\ hbar^2\right\} \left\{2m\right\} \ \ Delta \ \ Psi \left(\ vec \left\{R\right\}, T\right) \ + \ V \left(\ vec \left\{R\right\}\right) \ \ Psi \left(\ vec \left\{R\right\}, T\right) \ = \ I \ \left\{\ hbar\right\} \ \ frac$

It is valid only for small traditional speeds in front of the Speed of light in the vacuum.

#### Physical interpretation of the function of wave

The physical interpretation of the function of wave $\ Psi$ will be given by Born in 1926: the module with the square of this function of wave $\ left| \Psi \right|^2 = \ overline \left\{\ Psi\right\} \ Psi$ represents the density of probability of presence of the particle considered, i.e.:

be interpreted as being the probability of finding the particle in a small $dV$ volume located in the vicinity of the point $\ vec \left\{R\right\}$ of space at the moment $t$. In particular, the particle being necessarily located some share in whole space, one in the condition of standardization:

This statistical interpretation poses a problem when the studied quantum system is the whole Universe, as in quantum Cosmologie. In this case, the physicists theorists preferentially use the interpretation known as of the “multiple worlds” of Everett.

#### Methods of resolution

Apart from some particular cases where one can integrate it exactly, the equation of Schrödinger does not lend itself in general to an exact analytical resolution. It is necessary then:

• is to develop techniques of approximations like the Théorie of the disturbances.

• is to solve it numerically. This numerical resolution in particular makes it possible to visualize the provision curious about the orbital S electronic.

### Formalism of Dirac: fundamental arms, kets, and postulates

Dirac introduced in 1925 a powerful notation, derived from the mathematical theory of the linear forms on a vector Space. In this abstracted formalism, the Postulats of quantum mechanics take a concise and particularly elegant form.

## Quantum mechanics and relativity

Quantum mechanics is a theory not relativist : it does not incorporate the principles of the restricted Relativité. By applying the rules of the canonical quantification to the relativistic relation of dispersion, one obtains the equation of Klein-Gordon (1926). The solutions of this equation present however serious difficulties of interpretation within the framework of a supposed theory of describing a only particle: one cannot in particular build a density of probability of presence everywhere positive, because the equation contains a temporal derivative second. Dirac will then seek another relativistic equation of the first order in time , and will obtain the equation of Dirac, which describes very well the Fermion S of Spin a-half like the electron.

The Quantum theory of the fields makes it possible to interpret all the relativistic quantum equations without difficulty.

The equation of Dirac naturally incorporates the Invariance of Lorentz with quantum mechanics, as well as the interaction with the electromagnetic field but which is still treated in a traditional way (one speaks about semi-traditional Approximation). It constitutes the Mécanique quantum relativist. But of the fact precisely of this interaction between the particles and the field, it is then necessary, in order to obtain a coherent description of the unit, to also apply the procedure of Quantification to the electromagnetic field. The result of this procedure is the quantum electrodynamic in which the unit between field and particle is even more transparent since from now on the matter it also is described by a field. The quantum electrodynamics is a particular example of Quantum theory of the fields.

Other theories quantum of the fields were developed thereafter with the fur and as the other fundamental interactions were discovered (électrofaible Théorie, then quantum Chromodynamique).

## Inequalities of Heisenberg

The relations of uncertainty of Heisenberg translate impossibility of preparing a quantum state corresponding to precise values of certain couples of combined sizes. This is related to the fact that the quantum operators associated with these traditional sizes do not commutate not .

### Inequality position-impulse

Let us consider for example the position $x \,$ and the impulse $p_x \,$ of a particle. By using the rules of the canonical quantification, it is easy to check that the operators of position and impulse check:

$\ left \ hat \left\{X\right\} ^i, \ hat \left\{p\right\} _j \ right F \left(\ vec \left\{R\right\}\right) \ = \ \ left \left(\ hat \left\{X\right\} ^i \ hat \left\{p\right\} _j - \ hat \left\{p\right\} _j \ hat \left\{X\right\} ^i \ right\right) F \left(\ vec \left\{R\right\}\right) \ = \ I \ hbar \ \ delta^i_j \ F \left(\ vec \left\{R\right\}\right)$

The relation of uncertainty is defined starting from the average standard deviations of combined sizes. In the case of the position $x \,$ and impulse $p_x \,$ of a particle, it is written for example  :

$\left\{\ Delta\right\} X \, \ cdot \, \left\{\ Delta\right\} p_x \ \left\{\ Ge\right\} \ \ frac \left\{\ hbar\right\} \left\{2\right\}$

The more the state has a distribution tightened on the position, plus its distribution on the values of the impulse which is associated is broad for him. This property points out the case of the waves, via a result of the Transformée of Fourier, and expresses the duality wave-corpuscle here. It is clear that this leads to a questioning of the traditional concept of trajectory like differentiable continuous way.

### Inequality time-energy

There exists also a relation of uncertainty relating to the energy of a particle and the variable time. Thus, the duration $\ Delta T \,$ necessary to the detection of a particle of energy $E \,$ with $\ Delta E \,$ close checks the relation:

$\left\{\ Delta\right\} E \, \ cdot \, \left\{\ Delta\right\} T \ \left\{\ Ge\right\} \ \ frac \left\{\ hbar\right\} \left\{2\right\}$

However, the derivation of this inequality energy-time is rather different from that of the inequalities position-impulse.

Indeed, if the Hamiltonian is well the generator of the translations in time in Hamiltonian Mécanique, indicating that times and energy are combined, there does not exist operator time in quantum mechanics (“theorem” of Pauli), i.e. one cannot build of operator $\ hat \left\{T\right\} \,$ which would obey a relation between canonical commutation and the Hamiltonian operator $\ hat \left\{H\right\} \,$:

$\ left \ hat \left\{H\right\}, \ hat \left\{T\right\} \ right \ = \ I \ hbar \ \ hat \left\{1\right\}$

this for a very fundamental reason: quantum mechanics was indeed invented so that each stable physical system has a fundamental state of energy miminum . The argument of Pauli is the following: if the operator time existed, it would have a continuous spectrum. However, the operator time, obeying the canonical relation of commutation, would be also the generator of the translations in energy . This involves whereas the Hamiltonian operator would have to him also a continuous spectrum , in contradiction with the fact that the energy of any stable physical system must be limited in a lower position .

## Intrication

### Definition

Intrication is a quantum state (see also Fonction of wave) describing two traditional systems (or more) not factorisables in a product of states corresponding to each traditional system.

Two systems or two particles can be intricate as soon as there exists an interaction between them. Consequently, the intricate states are the rule rather than the exception. A measurement taken on one of the particles will change its quantum state according to the quantum postulate of measurement. Because of intrication, this measurement will have a simultaneous effect on the state of the other particle. Nevertheless, it is incorrect to compare this change of state to a transmission of information faster than speed of light (and thus a violation of the theory of relativity). The reason is that the result of measurement of the first particle is always random in the case of intricate states. It is thus impossible “to transmit” some information that it either since the modification of the state of the other particle, for immediate that it is, does not remain quite as random about it. The correlations between measurements of the two particles, although very real and highlighted in many laboratories all over the world, remain undetectable as long as the results of measurements are not compared, which implies necessarily a traditional exchange of information, respectful of relativity (see also the Paradoxe EPR). The quantum Téléportation makes use of intrication to ensure the transfer of the quantum state of a physical system towards another physical system. This process is the only known means to transfer quantum information perfectly. It cannot exceed speed of light and “is also désincarné”, in the sense that there is no transfer of matter (contrary to the fictitious teleportation of Star Trek).

This state should not be confused with the state of superposition . The same quantum object can have two (or more) superimposed states . For example the same photon can be in the state " polarity longitudinale" and " polarity transversale" at the same time. The Chat of Schrödinger is simultaneously in the state " mort" and " vivant". A photon which passes a semi-reflective blade is in the state superimposed " photon transmis" and " photon réfléchi". It is only at the time of the act of measurement that the quantum object will have a given state.

In the formalism of the quantum physics, a state of intrication of several objects quantum is represented by a tensorial Produit of the vectors of state of each quantum object. A state of superposition relates to only one object quantum (which can be an intrication), and is represented by a linear Combinaison various possibilities of states from this one.

### Quantum teleportation

One can determine the state of a quantum system only by observing it, which causes to destroy the state in question. This one can on the other hand, once known, being recreated in theory elsewhere. In other words, the duplication is not possible in the quantum world, only is a rebuilding in another place , close to the concept of teleportation in the Science-fiction.

Worked out theoretically in 1993 per C.H. Bennett, G. Arm-band, C. Crépeau, R. Jozsa, A. Peres, and W. Wootters in the article Teleporting year unkown dual quantum state by classical and EPR channels , of the Physical Review Letter , this rebuilding was carried out in experiments in 1997, on photons, by the team of Anton Zeilinger with In, and more recently on atoms of Hydrogène.

These “paradoxes” question us on the interpretation of quantum mechanics, and reveal in certain cases at which point our intuition can appear misleading in this field which does not concern directly the daily experiment of our directions.

; Cat of Schrödinger

This paradox highlights the problems of interpretation of the postulate of Réduction of the package of wave.

; Paradox EPR and experiment of Alain Aspect

This paradox highlights the not-locality of the quantum physics, implied by the intricate states.

; Experiment of Marlan Scully

This experiment can be interpreted as a demonstration that the results of an experiment recorded at one moment T depend objectively on an action carried out at a later time T+t. According to this interpretation, the not-locality of the intricate states would not be only space, but also temporal.

However, the Causalité is not strictly violated because it is not possible - for fundamental reasons - to highlight, before the T+t moment, that the state recorded at the moment T depends on a later event. This phenomenon cannot thus give any information on the future.

; Contrafactuality

According to quantum mechanics, events which could have occurred, but which did not occur , influential on the results of the experiment.

## The décohérence: quantum world in the traditional world

Whereas the principles of quantum mechanics apply a priori to all the objects contained in the universe (us including), why do we continue to perceive classically the essence of the world Macroscopique? In particular, why the quantum superpositions aren't observable in the macroscopic world? The theory of the Décohérence explains their very fast disappearances because of the inevitable coupling between the quantum system considered and its environment.

This theory received an experimental confirmation with the studies relating to systems Mésoscopique S for which the time of décohérence is not too short to remain measurable, such as for example a system of some photons in a cavity (Haroche and Al , 1996)

## See too

### Related articles

#### Interpretation

There exist many interpretations of the effects of quantum mechanics, some being in total contradiction with others. For lack of observable consequences of these interpretations, it is not possible to slice in favor of one or other of these interpretations. Only exception, the school of Copenhagen whose principle is precisely to refuse any interpretation of the phenomena.