Quantum mechanics

Quantum mechanics was formulated in its modern versions in 1925-1926 by E. Schrödinger on the one hand (wave mechanics) and by W. Heisenberg, P. Jordan and M. Born on the other (matrix mechanics). After the equivalence between these two formulations was demonstrated, first by Schrödinger himself and then by J. von Neumann, formalism posed no more problems, even if his interpretation would continue to raise endless discussions.
The most controversial point was the probabilistic nature of the predictions of quantum theory highlighted by Born as early as 1926. Consider, for example, the experience schematized in fig. 1: a beam of light polarized in the u direction (of intensity 0 ) affects the polarizer P oriented along the direction at which makes an angle? with the direction of polarization. You get two beams emerging: a direct, ae polarization intensity 2 cos 2 ?, The other deflected from one side, orthogonal polarization and intensity 0 sen 2?. In quantum mechanics the light beam is described as the format of polarized photons in the u direction. A fraction cos 2 ? of photons is transmitted, while the other, without 2 ?, is reflected. If we look at the process as an interaction of a single photon, it can be said that it undergoes onemeasureof polarization according to a, measure that can give two results: one (+1) corresponding to the polarization a with a probability cos 2 ? to occur; the other (-1) corresponding to the polarization orthogonal to a with a probability sen 2 ? to be obtained. This fundamentally statistical nature of quantum predictions had troubled many physicists, including Louis de Broglie andAlbert Einstein. For the latter, a fundamental theory must be able to predict with certainty the behavior of a system whose initial state is determined. The use of probability in physics is a useful aid when it is technically impossible to complete the exact treatment of the problem. This occurs, for example, in Maxwell’s kinetic theory of gases, where the exact description of the movement of 10 23molecules of a gas is evidently excluded in practice even if it is conceptually possible. But a fundamental theory cannot be probabilistic, as Einstein’s famous phrase “God does not play dice” means, and therefore the probabilistic character of quantum theory is an indication of the existence of an underlying level that would allow a more detailed description. of the physical world. So quantum mechanics would represent nothing more than a statistical description of phenomena occurring on a smaller scale: it is this more complete description that one should try to discover.
According to the interpretation, known as the ‛interpretation ofCopenhagen’, of which Niels Bohr was the greatest architect, quantum mechanics instead gives in probabilistic form the most complete physical description that can be conceived. The inability to go further is due toquantizationof certain quantities that cannot take arbitrarily small values. The validity of this position is ensured by the existence of Heisenberg’s uncertainty relationships, on which Einstein therefore concentrated his first attacks trying to contradict them with different conceptual experiences, compatible with the fundamental laws of physics but far beyond the possibilities of check with the technologies of that era. On this first phase of the discussion we must remember the Solvay congresses of 1927 and 1930, during which Bohr easily managed to overcome Einstein’s arguments.
It seems that at the end of these discussions Einstein was convinced that the quantum theory was correct, while continuing to think that it was incomplete, and since then he has tried to directly demonstrate this point of view.

b) Einstein, Podolsky and Rosen’s argument .

In 1935 Einstein and his collaborators, Boris Podolsky and Nathan Rosen, published in the journal Physical review “(see Einstein et al., 1935) the famous topic – later often called the‛ EPR paradox ‘from the name of its discoverers – which demonstrated the incompleteness of quantum mechanics based on the predictions that it formulated itself.
The experience schematized in Fig. 2 presents a variant of the situation discovered by Einstein,David Bohm(v., 1951). Quantum mechanics predicts a very close correlation between the polarization measurements of two-photon ν 1 and ν 2 furthest but issued from a common source that has produced them in a state of polarization defined by:


in which | 1 , 2 ⟩ represents a pair of photons polarized along the axis Ox perpendicular to the propagation direction, and | 1 , 2 ⟩ represents a pair of photons polarized along the axis Oy perpendicular to Oxand to the propagation direction. Quantum mechanics foresees a total correlation between the polarizations of the two photons belonging to such a state, called E EPR state ‘. For example, for parallel measurement directions, a result + 1 for ν 1 safely implies a result +1 for ν 2 (and similarly, if we get – 1 for ν 1 , we will certainly get – 1 for ν 2 ). But according to Einstein the measurement on the photon ν 2 should not be influenced by the distant measurement on ν 1 , since the relativistic causality principle prevents an interaction from propagating at a speed greater than that of light. Therefore the photon ν 2must have a property that determines the result + 1 or – 1 before the measurement is made. According to the expression of Einstein, Podolsky and Rosen this property is an element of physical reality. ”
Now, according to quantum formalism, which does not attribute any property of this type to ν 2 , all the pairs are identical even if in 50% of cases we obtain +1 and in 50% of the other cases we obtain – 1. According to Einstein and his collaborators it must therefore be concluded that quantum formalism is incomplete since it does not explain all the elements of physical reality.

c) Quantum separability is the core of the discussion .

This time Bohr could not refute Einstein’s arguments on a technical level, since on this level the reasoning of Einstein, Podolsky and Rosen is unassailable. The discussion must therefore move to the level of the concepts and to try to understand Bohr’s answer (see 1935) it is necessary, somewhat schematically, to specify the hypotheses behind that reasoning. Hypothesis 1: the predictions of quantum mechanics are right; hypothesis 2: no interaction can propagate with a speed higher than that of light (relativistic causality); hypothesis 3: when two objects are very far from each other it is possible to speak separately about the elements of the physical reality of each of them.
It is on hypothesis 3 (sometimes called the principle of separability) that Bohr’s refutation concentrates that one cannot speak in the abstract of the physical reality elements of a system; each experience of physics involves a system to be studied and a measuring device, and it is only by specifying the set of the two that one can speak of physical reality. In conclusion, according to Bohr, the notion of physical reality of Einstein, Podolsky and Rosen is meaningless.
The core of the debate therefore lies in the concept of separable physical reality. According to Einsteinthe worldit can be conceived as a format of entities localizable in space-time with properties that make up their physical reality; these entities can interact locally in a relativistic sense, that is to say through interactions that do not propagate with a speed greater than that of light. Such a conception of the world is usually called “local realistic” or “separable”. Bohr, while considering himself a realistic physicist, proposed a different (and, it must be said, less clear) version of physical reality. By refusing to consider a physical reality independent (separate) from the measuring instrument, Bohr could resist the attack of Einstein, Podolsky and Rosen, but this refutation did not satisfy Einstein,

d) The reactions .

The impact of this debate on the scientific community was very limited. Some physicists were convinced by the arguments of Einstein, Podolsky and Rosen and tried to invent theories to complete quantum mechanics with the introduction of an underlying level of description (Louis de Broglie’s pilot wave theory, theories withhidden variablesDavid Bohm and Jean-Pierre Vigier; v. Jammer, 1974). However, the great majority of physicists followed Bohr, both because they had been convinced by his answer, and, more likely, because they trusted him without worrying about investigating the problem. This confidence was reinforced by the ‘von Neumann theorem’, set out in 1932 in the book in which the mathematical foundations of quantum mechanics were laid. In this book von Neumann (v., 1932) had addressed the problem of interpreting the statistical character of quantum predictions in terms of hidden variables and had ‛demonstrated ‘the incompatibility of the mathematical properties of quantum formalism with the existence of a formalism underlying in which ‛hidden variables’ intervened.
To understand the weak reaction of Einstein, Podolsky and Rosen by the majority of physicists, it is necessary to keep in mind the enormous amount of successes of quantum mechanics in all fields, from solid physics to chemistry, passing through the quantum theory of radiation. Moreover, since everyone agreed on how to use quantum theory in practice, the choice between one or the other position had no practical consequences on research, so the problem seemed to be limited to a question of interpretation or to go beyond even in philosophy.
It is therefore understandable the interest raised by the research ofJohn Bell, who in 1964 moved the debate from the terrain of ideas to that of experiment.

2. Bell’s theorem

a) A conceptual step forward .

Bell’s first contribution to the conceptual foundations of quantum mechanics was the demonstration, in 1964, of the inaccuracy of von Neumann’s theorem (see Bell, 1964). He first observed that the existence of theories with hidden variables such as Bohm’s was a counterexample to the theorem. He also identified an apparently obvious, albeit untested, hypothesis that undermined the whole demonstration. At this point he was ready to resume Einstein, Podolsky and Rosen’s program, what he did in 1966 in a second article that remained famous (see Bell, 1966).
In this article, Bell extends Einstein, Podolsky and Rosen’s argument by examining the possible consequences of the conclusion that quantum mechanics is incomplete. He then completed the theory by introducing ‘hidden variables’ to interpret the correlations of Einstein, Podolsky and Rosen. These are properties that may vary from one pair (for example of photons) to the other but common to the two members of the same pair. The results of the polarization measurements depend on this common property, so that it is understood that the measurements made on the two members of the same pair can be correlated.
Faithful to the spirit of the article by Einstein, Podolsky and Rosen, Bell also imposed on his formalism a condition of locality closely linked to hypotheses 2 and 3 listed above. This fact showed that such a theory (separable theory with hidden variables) is in conflict with quantum mechanics. The great strength of Bell’s reasoning lies in its generality. It is not restricted to a particular theory with hidden variables, but applies to all separable theories with hidden variables whose predictions satisfy certain inequalities (Bell’s inequalities), while the predictions of quantum mechanics violate them.
The debate therefore changed in nature. Before, Bohr’s or Einstein’s position could have been adopted while continuing to believe in the accuracy of quantum predictions for Einstein, Podolsky and Rosen’s correlations. With Bell’s theory, however, it is established that, if Einstein’s position is correct, the correlations of Einstein, Podolsky and Rosen violate quantum predictions. The conflict then became quantitative and the debate returned to the domain of physics, since it is sufficient – in principle – to carry out the corresponding measurement to decide the question.

b) Bell’s inequalities and the conflict with quantum mechanics .

Although extremely effective, Bell’s theorem can be proven without great difficulty. The starting point is to find a formalism in which the hidden variables λ that characterize each pair appear, their probability density ρ (λ) and of the functions A (λ, a) and B (λ, b) with values ​​+ 1 or – 1 representing the result of the polarization measurement. This formalism allows to express the correlation coefficient of the polarization measurements when the polarizers are oriented along the directions a and b:

E (a, b) = ∫ A (λ, a) B (λ, b) ρ (λ) d λ. ( 2 )

The locality condition of Bell is expressed here by the property that the response A (λ, a) of the first polarizer does not depend on the orientation b of the other polarizer (and vice versa), or on the property that the probability density ρ (λ) which characterizes the source does not depend on the orientations a and b of the polarizers.
Expression ( 2 ) is sufficient to demonstrate Bell’s inequalities (in the form given to them by Clauser, Horn, Shimony and Holt: see Clauser and Shimony, 1978).
Consider the quantity S , defined by:

S = E (a, b) – E (a, b ′) + E (a ′, b) + E (a ′, b ′), ( 3 )

in which the correlation coefficients of the polarization corresponding to the orientations aea ′ for the first polarizer and b and b ′ for the second appear. It can take only the values:

-2 ≤ S ≤ 2. ( 4 )

To demonstrate this inequality we consider the quantity

s = α (β – β ′) + α ′ (β + β ′), ( 5 )

in which each of the four numbers α, β, α ′ and β ′ is ± 1. It is immediately verified that s is + 2 or – 2. Now the quantity S of equation ( 3 ) is the average – weighted with ρ (λ) – of numbers type ( 5), that is, they are worth either 2 or – 2. This average is therefore between 2 and – 2, so that Bell’s inequality is obtained ( 4 ).
Let us now consider the results of a quantum calculation for a situation of the Einstein-Podolsky-Rosen type, for example for a pair of photons produced in the radiative cascade of fig. 3. It is easy to demonstrate that the polarization correlation coefficient holds:

MQ (a, b) = cos [2 (a, b)], ( 6 )

being (a, b) the angle between the directions a and b of the two polarizers. We now use this result to calculate the quantity Sprovided by the quantum calculation in the case of the orientations of fig. 4: the result, when the angle between the polarizers is equal to and equal to 22.5 degrees, is:

MQ (22.5 o ) = 2 √2 ≅ 2.83; ( 7 )

this result clearly violates inequality ( 4 ). There are therefore situations in which the theories that can be separated from hidden variables and quantum mechanics lead to different predictions, and this fact determines the incompatibility of these theories.
When two theoretical approaches lead to different quantitative predictions, it is necessary to experimentally determine which is the most reliable.

3. The passage to experience

a)Exceptional situations .

Given the immense success of quantum mechanics it could have been believed a priorito already have numerous experimental results in 1965 that violated Bell’s inequality, but in fact it was quickly found that no result had been obtained in a ‘suitable’ situation (such that quantum predictions violated Bell’s inequalities). Indeed, situations where conflict can exist are very rare. Moreover, in a real experience the inevitable imperfections reduce the correlations predicted by quantum mechanics, and this attenuates or even makes the conflict disappear in situations where it can in principle exist. It was therefore necessary to conceive experiments designed in such a way as to be able to verify Bell’s inequalities.

b) First generation experiences .

The very first experiments (see Pipkin, 1978; see Clauser and Shimony, 1978), which used pairs of γ photons produced by the disintegration of positronium, are easy to interpret. Since there is no effective polarizer for photons of this energy (0.5 MeV), it is necessary to go through very indirect measurements and reasonings. An experience using protons presents similar problems.
The pairs of visible photons emitted in well chosen atomic radiative cascades represent much more suitable candidates, since there are excellent polarizers for light. The first experiments of this type (carried out byR. Holt and F. Pipkin at Harvard, and by J. Clauser and R. Friedman a Berkeley) gave contradictory results, which is not too strange considering the extreme weakness of the signals obtained. Thanks to the use of a laser, Fry and Thompson were able to obtain a much more reliable result in 1986 in favor of quantum mechanics. The experience nevertheless remained very far from the ideal scheme of fig. 2, since the polarizers used allowed only one result of the measurement out of the two possible ones: the comparison with Bell’s inequality still remained indirect.

c) Indisputable results .

A second generation of experiments, carried out at the Institut d’Optique d’Orsay (Université Paris-sud) by Alain Aspect, Philippe Grangier, Gerard Roger and Jean Dalibard, was made possible thanks to the preparation of a new source of photon pairs related, at the same time intense and very pure (see fig. 3). In this figure three energy levels are represented, f , r , e , of an atom. For a suitable choice of the system, the atom brought to the level and de-energized by emitting in succession two photons ν 1 and ν 2which have Einstein-Podolsky-Rosen correlations. In Orsay’s experiences a pure and intense source of Einstein-Podolsky-Rosen photon pairs was obtained by directly exciting the transition f → and through a processnot lineartwo-photon ν ′ and ν ″. Such excitement was made possible by the development of very stable tunable lasers. This source made it possible to obtain, in an experience lasting one minute, a measurement precision (1%) which in previous experiences required many hours. Associated with optical components close to perfection, this source first gave a very clear confirmation of the previous results in favor of quantum mechanics, with improved statistical precision. Furthermore, it allowed to close the debate on the possibility of the disappearance of the quantum correlation ( 6) when the polarizers were far enough away from the source. In fact, even when using polarizers located 6 m away from the source, which is greater than the coherence length of the photons emitted in the radiative cascade of fig. 3, the measurements gave results compatible with quantum predictions and incompatible with Bell’s inequalities corresponding to this situation (see Aspect et al., 1981).
The use of real polarization analyzers capable of measuring the two possible results then allowed to make the first experience that exactly reproduced the ideal scheme of fig. 2, and therefore to directly verify Bell’s inequalities (see Aspect et al., Experimental realization …, 1982). For the orientation of fig. 4 we got:

S = 2.697 ± 0.015, ( 8 )

a result that unquestionably violates inequality ( 4 ). Furthermore, all the measurements are in excellent agreement with the quantum forecasts that take into account the small imperfections of the appliance. Due to its precision and the conceptual simplicity of the scheme used, this experience represents a particularly convincing confirmation of the quantum predictions in a Einstein-Podolsky-Rosen ‛sensitive ‘situation.

d) Experiments with variable polarizers .

In his initial article Bell had insisted on the need for a locality condition to arrive at a conflict with quantum mechanics. This completely natural condition excludes any direct interaction between the polarizers or between the polarizers and the source. In Bell’s formalism, explained above, this condition translates, as we have seen, into the property whereby the response A (λ, a) of the first polarizer does not depend on the orientation b of the other polarizer (and vice versa), or in that for which the probability density ρ (λ) which characterizes the emission of pairs from the source does not depend on the orientations a and b of the polarizers.
Although very natural, this condition cannot be demonstrated with the previous experimental schemes and must be accepted as a hypothesis. On the contrary, in an experience in which the directions of the polarizers are changed very quickly and in a random way, the possibility of a direct interaction between polarizers or an influence of the polarizers on the source is excluded from the principle of relativistic causality (hypothesis 3 ). The principle of separability as a whole is therefore at stake (hypotheses 2 and 3).
To create rapid oscillations, the Orsay group (see Aspect and others, Experimental tests …, 1982) replaced each polarizer represented in fig. 2 with an optical exchange followed by two polarizers with different orientations (see fig. 5). The exchange 1 alternately sends the light towards the polarizers I or I ′ with orientations a and a ′. The set is equivalent to a single polarizer oriented according to aea ′. The exchange 2 followed by the polarizers II and II′ Performs a similar function. In this experience the time interval between two exchanges was of the order of 10 ns, which is shorter than the light propagation time between the two ends of the apparatus, 12 m (40 ns) long. This experiment is less probative than could have been hoped for due to the technical difficulties associated with the introduction of optical ‘taps’. However, it provided results in good agreement with quantum mechanics that clearly violate Bell’s inequalities (for 5 standard deviations).
It can therefore be concluded that the only experience that incorporates variable polarizers is also in favor of quantum mechanics, although the level of certainty of the data thus obtained is less high than that of the data obtained from the best static experience. It would certainly be useful to check this result by means of similar improved experiments.

e) A new generation of experiments .

The experiments discussed above are not perfect due to the poor overall detection efficiency of photons: the photons emitted by the source, in the vast majority, escape the detectors. The measurements made are nevertheless valid if it is assumed that the subset of the photons actually detected provides a non-influenced sample, in a statistical sense, of the set of photons emitted. This hypothesis is completely natural since the processes that lead to the non-detection of photons are not related to their polarization. However, it would be logically preferable to avoidhaving make this additional hypothesis (with respect to Bell’s hypotheses): this is in principle possible provided that all pairs are revealed.
The first improvement to be made to these experiments would therefore be to use more efficient detectors, which is possible thanks to continuous technological progress in the field of detectors. We could then think of an ideal experiment on condition that the emission directions of the photons of the pair are perfectly correlated. Now this cannot happen in experiences that use a radiative cascade, since the atom in the final state is free and can recoil in any direction. There is however another process of emission of pairs of photons, called ‛parametric fluorescence ‘, which satisfies this condition of angular correlation: it is a non-linear process that occurs in a crystalline material and that generates two related photons by ‛division’ of an incident photon.
For about ten years this process has been the subject of intense study (see Rarity and Tapster, 1990) as it would make it possible to carry out checks on Bell’s inequalities using physical quantities other than polarization (for example, the energy or the impulse of the photons). It is important to note that quantum formalism applies similarly in seemingly very different situations: the violation of Bell’s inequalities always appears as an effect of ‛not local ‘interference between probability amplitudes.
Many experiments of this type have been carried out and have generally led to a good agreement with quantum mechanics and a clear violation of Bell’s inequalities in the order of a dozen typical waste in the best experiences. John Rarity and Paul Tapster, authors of this experiment in Britain, also managed to send each photon in an optical fiber, thereby managing to verify Bell’s inequalities over distances of up to 4 km.
It should however be noted that, despite the efforts concentrated in this experiment, the technical difficulties make the detection efficiency very low and the verifications carried out in the early eighties by means of radiative cascades remain, after more than 10 years, the best in terms of signal to noise ratio. This situation could change, since several groups started experiments in 1993 aimed at obtaining a detection efficiency of more than 90%: R. Chiao’s group in Berkeley proposes to use pairs of parametric photons (see Kwiat and others, 1994), while that of E. Fry inTexasworks on mercury atoms obtained by photodissociation of molecules (see Fry et al., 1995).

4. Conclusion

Apart from some reservations, the experiences carried out have given results in favor of quantum mechanics, which, thanks to Bell’s theorem, have a very broad meaning: it is impossible to interpret these experiments by means of local theories or separable to hidden variables and therefore there is no hope of completing quantum mechanics in this direction.
It is therefore necessary to renounce a certain conception of the physical world in the spirit of hypotheses 2 and 3 of the reasoning of Einstein, Podolsky and Rosen, a conception called separable or “local realistic”. It is in this sense that we must understand the statements according to which quantum mechanics would be inseparable: at the same time, locality and an almost classical description of physical reality cannot be preserved. However, it must not be concluded that it is possible to transmit usable information by violating the principle of relativistic causality (with a speed higher than that of light).
Sometimes we wanted to see in the violation of Bell’s inequalities the proof that Einstein was wrong in his debate with Bohr. This is certainly unfair, if only because Einstein, who died before the proof of Bell’s theorem, did not know the incompatibility of his position with the predictions of quantum mechanics. He had the immense merit of discovering a situation in which quantum predictions appear extraordinary if one follows the interpretations of Copenhagen. Naturally Niels Bohr should be credited with having immediately understood the scope of Einstein, Podolsky and Rosen’s reasoning and having understood that this did not in itself lead to the inconsistency of quantum theory.

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