Principle of uncertainty - SpeedyLook encyclopedia

The principle of uncertainty was stated in spring 1927 by Heisenberg at the time of the stammerings of the quantum Mécanique.

The " term; incertitude" is the historical term for this principle. The name of principle of indetermination is sometimes preferred because the principle does not relate to ignorance by the experimenter of sizes, but of course impossibility of determining them, and of even affirming that a more precise determination exists.

Work of Planck, Einstein and De Broglie had updated that the quantum nature of the matter involved equivalence between undulatory properties (Fréquence and Vecteur of wave) and corpuscular (energy and Impulsion) according to the laws: $E= \ hbar \ omega$ and $\ vec {p} = \ hbar \ vec {K}$ .

The Dualité wave-corpuscle confirmed then by many experiments presented a basic problem with the physicists. Indeed, to have a frequency and a vector of wave, an object must have a certain extension in space and time. A quantum object can thus be neither perfectly localized, nor to have a perfectly defined energy.

In a simplified way, this principle of indetermination thus states that - in a rather against-intuitive way from the point of view traditional mechanics - for a given massive particle, one cannot simultaneously know his position and his speed. Either one can precisely know his position (by ex: with ± 1 mm) against a great uncertainty on the value its speed (by ex: with ± 100 m/s), are one can precisely know his speed (by ex: with ± 0,0001 m/s) against a great uncertainty on the value of its position (by ex: with ± 1 km).

However, if one gives up considering the particle as a corpuscular object, the statement of this principle becomes more intuitive. The quantum object having a certain extension in space and a certain lifespan in time, it is represented then, either by a whole of scalar values (position, speed), but by a function describing his spatial distribution. All the relative information with the particle is contained in this Fonction of wave. The scalar measurements taken on this particle consist in extracting only part of this information, via mathematical operators.

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History of the term

The principle of uncertainty is often called principle of indetermination. The use of these two terms to indicate the same concept results from a problem during the translation in English of the article of Heisenberg. Indeed, during the first drafting of its article, Heisenberg employs the terms Unsicherheit (uncertainty) and Ungenauigkeit (inaccuracy), then, understanding that these terms can lend to confusion, it decides to use finally the term Unbestimmtheit (indetermination). But the article is already translated and it is the term " principle of incertitude" which will be devoted.

Although denomination “ principle of incertitude ” is most used, one would have in any rigor speech of “ principle of indétermination ”. However the expression was spread so much so that it is accepted today by all the physicists. The term of “ principe ” is also inappropriate, though often still used. It would be advisable to speak about relations of uncertainty or better of relations of indetermination.

Because of these philosophical connotations, today the physicists speak about the relations of uncertainty , or about the inequalities of Heisenberg , because it is about a inequality bearing on physical sizes not-commutative .

Relations of Heisenberg

Let us consider a nonrelativistic massive particle moving on an axis.

Traditional description

The traditional Mécanique of Newton affirms that the dynamics of the particle is entirely given if &thinsp at every moment is known;: its position X and its momentum p = mv (also called: impulse ). These two physical sizes real have values belonging to $\ mathbb {R}$ , varying $- \ infty$ with $+ \ infty$ . It is said that the couple (X, p) defines the space of the phases of the particle. Any physical size is representable by a function f (X, p) real. This theory in conformity with logic aristotelician, including the concept of third is excluded : “ it is necessary that a door is opened or fermée. ” From the mathematical point of view, one describes the state of the particle by a finished number of scalar sizes.

Quantum description

In quantum mechanics, the precise value of the physical parameters such as the position or speed is not given as long as it is not measured. Only the statistical distribution of these values is perfectly at any moment given. That can lead to the point of view of (which is an abuse language) according to which a quantum object could be " at several places at the same time ". A point of view righter would be to say than the quantum object does not have localization as long as the position is not measured.

However, the paradox is only apparent. It comes owing to the fact that the traditional scalar sizes are insufficient to describe quantum reality. One must call upon functions of wave which are vectors belonging to a Espace of Hilbert of dimension infinie.
The traditional sizes are thus makes of them only sights partial of the object, potentially correlated.

Concept of observable

Very curiously, a physical size, called an Observable , is not any more one function f (X, p) real, but is represented by a operator Hermitien $\ hat {G}$ acting on a Space of Hilbert $\ mathcal {H}$ . The value of this physical size is one of the real eigenvalues of this operator :

\ hat {G} \, | g_i \ rangle = g_i \, | g_i \rangle

If the state of the system at the moment of measurement is a vector $| \ psi \ rangle$ of space $\ mathcal {H}$ , then this vector admits the décomposition :

| \ psi \ rangle = \ Sigma_i c_i \, |g_i \rangle

where the c_i are complex numbers.

Probabilistic interpretation

The complex number c_i makes it possible to calculate the probability p_i of obtaining the value g_i :

p_i = | c_i |^2 = c_i \, c_i^*

The measurement of the size is thus a Random variable (v.a.) with a Espérance E (G) and a standard deviation σ (G) . Measurement is thus of nature probabilistic, which implies many apparent paradoxes in logic aristotelician. One of them was immediately noticed by Heisenberg : like the operator position $\ hat {X}$ and the operator momentum $\ hat {p}$ does not commutate :

\ left \ hat {Q}, \ hat {p} \ right = I \, \ hbar \, \ hat {\ Bbb {I}}

one cannot measure simultaneously these two grandeurs : the concept of space of the phases disappears in quantum mechanics. The quantum object in fact is completely described by its function of wave. The scalar sizes used in traditional physics are insufficient and inadequate.

The deterministic evolution of Newton is replaced by a deterministic equation of evolution of Schrödinger, making it possible to predict in an unquestionable way the temporal evolution of the functions of wave (of which the square module is the probability, the phase not being known a priori).

Inequality of Heisenberg

Repeated measurements of the position and impulse will give results in general different to each mesure : each sample of values will be characterized by a standard deviation : σ_x for the position, and σ _p for the impulse. The theorem of Heisenberg shows que :

\ sigma_x \ cdot \ sigma_p \ Ge \ frac {\ hbar} {2}

where $\ hbar$ is the Quantum of action. This notion is frequently popularized by sentences of the type : “ It is impossible to know at the same time the position and the momentum of an object in manner précise ”. Indeed, if for example the position of a particle is known exactly, dispersion in position is identically nulle : σ_x = 0 . The inequality of Heisenberg implies whereas $\ sigma_p = \ infty$ : dispersion in impulse must be maximum.

General principle of Heisenberg

The theorem of Heisenberg does not apply only to the couple of values position and momentum. In its general form, it applies to each couple of operators $\ hat {has}$ and $\ hat {B}$ not commutating pas :

Statement of the principle of Heisenberg

For a state $| \ psi \ rangle$ given, one a:

\langle \psi |\ sigma_A \ \ cdot \ \ sigma_B \ psi \ rangle \ Ge \ frac {\ langle \ psi | \ hat {C} | \ psi \ rangle} {2}

where the median value of the switch $\ langle \ psi | \ hat {C} | \ psi \ rangle$ depends of course on the state $|\ psi \ rangle$ selected.

This general theorem, consequence of the Inequality of Cauchy-Schwarz, was highlighted in 1930 by Robertson and (independently) by Schrödinger ; the inequality is thus also known like the relation of Robertson-Schrödinger .

Principle or theorem?

The purists reserve sometimes the name of principle if one undervaluing not no one of $| \langle \psi | \ hat {C} | \psi\rangle |$ exists whatever the state $| \psi \rangle$ . That is not possible that if the space of Hilbert is of infinite size. Indeed, in the case of a space of dimension finished , one a:

\ mathrm {Tr} \, (\ hat {has} \ hat {B}) = \ mathrm {Tr} \, (\ hat {B} \ hat {has}) \ quad \ Longrightarrow \ quad \ mathrm {Tr} \, (\ hat {C}) = 0

There is not whereas theorem of Heisenberg, and not principe ; it is for example the case of a spin 1/2.

Another formulation of the principle of Heisenberg

The inequality of Heisenberg is often written:

\ Delta {has} \ cdot \ Delta {B} \ Ge \ frac {1} {2} \ left|  \ left \ langle \ left \ hat {has}, \ hat {B} \ \ right \ right \ rangle_ \ gamma \ right|

where:

has and B is two observable,

$\ hat {has}$ and $\ hat {B}$ corresponding operators,

represents the switch $\ hat {has}$ and $\ hat {B}$ ,

$\ left \ langle \, \ right \ rangle_ \ gamma$ is the average on the state Notation bra-ket : $| \ gamma \ rangle$ , and

Δ X is the standard deviation of X : $\ sqrt$ .

Relation time-energy

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

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

However, the deduction 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 (“ théorème ” of Pauli), i.e. one cannot build of operator $\ hat {T}$ which would obey a relation between canonical commutation and the operator Hamiltonien $\ hat {H}$ :

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

this for a reason very fondamentale : quantum mechanics was indeed invented so that each stable physical system has a fundamental state of energy mininum .

Historical prospect

It is clear that the abandonment of the logic of Aristote because of the probabilistic nature of measurement caused a sharp agitation in the community scientifique : John von Neumann is one of very first to write on quantum logic, followed by Mackey.

The Einstein-Bohr controversy is in addition célèbre : for Einstein, “ God does not play dés ! ”, it with what Bohr répondra : “ Einstein, cease saying to God what It owes faire ”. The Paradoxe EPR will involve Bell via its inequalities to give up the traditional concept of Localité. This assumption will be confirmed by the experiment of Aspect in 1982 ; this experiment will be still refined by Zeilinger in 1998. The paradox of the Chat of Schrödinger will lead to a major reflection on the role of the coupling to the environment and the Décohérence of the Intricat S . From where fulgurating progression of the quantum Cryptology, the quantum Teleportation, technical realities in 2005, and of the quantum Data-processing , still stammering in 2005.

Difficulty of interpretation

Examples

This correlation of uncertainties is sometimes explained in an erroneous way by affirming that the measure of location modifies obligatorily the momentum of a particle. Heisenberg even offered this explanation in 1927 initially to him. This modification does not play any part, because the theorem applies even if the position is measured in a copy of the system, and momentum in another perfectly identical copy.

A better analogy would be the following one: either a variable signal in time, like a sound wave, and or to know the exact Frequency of this signal at one moment precise $t$ . This is impossible in general, because to determine the frequency precisely, it is necessary to sample the signal for a certain length of time. In treatment of the signal, this aspect is in the middle of the approach time-frequency Spectrogramme where one uses the principle of uncertainty under the formulation of Gabor.

The theorem of Heisenberg applies in particular to the crucial experiment of the slits of Young with a single photon: all the tricks which the physicists invent to try to see passing the " particule" through one of the holes, destroy the phase and thus the interferences of the wave: there is Complémentarité of Bohr, i.e. so before any measurement, the quantum state $|\ psi >$ describes at the same time an undulatory aspect and a corpuscular aspect, after measurement, there remains an undulatory aspect or a corpuscular aspect. According to the famous sentence of Dirac, the “particlonde” interfered with itself.

This experiment is presented to the Palais of Discovered the with a single source of photon. The reason produced by million Photon S passing through the slits can be calculated using quantum mechanics, but the way of each Photon can be predicted by no known method. The Interprétation of Copenhagen says that it could be calculated by no method. In 2005, one even made a success of this experiment with Fullerène S, these large carbon molecules containing 60 atoms!

The Bohr-Einstein controversy

Einstein did not like the theorem of uncertainty. At the time of the 5th Congress Solvay (1927), it subjected to Bohr a famous experimental challenge: we fill a box with a radioactive material which emits in a random way a radiation. The box has a slit which is opened and immediately closed by a clock of precision, making it possible some radiations to leave. Thus time is known with precision. We want to always measure precisely the energy which is a combined variable. No problem, answers Einstein, it is enough to weigh the box before and afterwards. The principle of equivalence between the mass and the energy given by the restricted Relativité thus makes it possible to precisely determine the energy which left the box. Bohr answered him this: so of energy had left the system then the lighter box would be assembled on the balance. What would have modified the position of the clock. If the clock deviates of our stationary reference frame, by restricted relativity it follows that its measurement of time differs from our, which inevitably leads to a margin of error. In fact the detailed analysis shows that the inaccuracy is given correctly by the relation of Heisenberg. See for example the site of the Nobel foundation for a '' figure '' of this “clock in the box”.

In the Interpretation of Copenhagen of quantum mechanics, largely accepted but not universally, the theorem of uncertainty implies that on an elementary level, the physical universe “does not live” not in a space of the phases, but rather like a whole of potential achievements, exactly given of probability: the probabilities are, they, given with an absolute precision, in so far as the state of the system is pure (i.e. it is not itself roughly given!)

Compressed states

In fact, to circumvent the inequalities of Heisenberg, the physicists carry out states known as compressed (in Franglais: states “ squeezés ” ), where there is no uncertainty on the phase (but then the number of particles is unspecified) or, on the contrary, a well defined number of particles (in particular of photons), but one loses information on the phase. It was shown by work of Glauber ('' Nobel Prize 2005 '') that the quantum Information is not sullied by the theorem with Heisenberg. One can thus hope to extract the maximum of information quantum from a numeric photography, while respecting the second principle of the Thermodynamique.

Quantum claustrophobia

Definition

The quantum claustrophobia is the tendency which have the particles to vibrate frantically when they are confined in a very small medium. The systems become unstable on microscopic scales.

Planck's constant

Consequences

That wants to say, that if ħ were larger, these turbulent behaviors would appear no longer on scales subplanckienne but on macroscopic scales. Two dice in a box would start to circle frantically if the Planck's constant were higher.

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