- where X means the mean value of many measurements of the quantity X . - The standard deviation A represents the width of the distribution of A, i.e. - Consider the product of the standard deviations for the operators A ˆ and B, ˆ taking into account that ˆ u denotes (Postulate IV) the integral | ˆ u | according to (1.19). - where the Hermitian character of the operators A ˆ and B ˆ is used. - h, as in the case of x ˆ and p ˆ x . - Illustration of the Heisenberg uncertainty principle. - When the preci- sion with which x is measured increases, the particle’s momentum has so wide a distribution that the error in determining p x is huge, Fig. - In the 1920s and 1930s, Copenhagen for quantum mechanics was like Rome for catholics, and Bohr played the role of the president of the Quantum Faith Con- gregation. - 52 The picture of the world that emerged from quantum mechanics was. - 51 There is an apocryphal story about a police patrol stopping Professor Heisenberg for speeding. - Bohr presented a philosophical interpretation of the world, which at its founda- tion had in a sense a non-reality of the world.. - According to Bohr, before a measurement on a particle is made, nothing can be said about the value of a given mechanical quantity, unless the wave func- tion represents an eigenfunction of the operator of this mechanical quantity.. - A measurement gives a value of the mechanical property (A). - Then, according to Bohr, after the measurement is completed, the state of the system changes (the. - so called wave function collapse or, more generally, decoherence) to the state de- scribed by an eigenfunction of the corresponding operator A, and as the measured ˆ. - According to Bohr, there is no way to foresee which eigenvalue one will get as the result of the measurement. - This probability may be computed as the square of the overlap integral (cf. - 24) of the initial wave function and the eigenfunction of A. - Many scientists felt a strong imperative to prove that the principle is false. - Einstein believed in the reality of our world. - The key statement of the whole reasoning, given in the EPR paper, was the fol- lowing: “If, without in any way disturbing a system, we can predict with certainty (i.e.. - For exam- ple, obtaining the same results for a game of dice would require a perfect reproduction of the initial conditions, which is never feasible.. - The meaning of the words “well defined” is that, ac- cording to quantum mechanics, there is a possibility of the exact measurement of the two quantities (x and P), because the two operators x ˆ and P ˆ do commute. - We now come to the crux of the real controversy.. - When they are extremely far from each other (e.g., one close to us, the other one millions of light years away), we begin to sus- pect that each of the particles may be treated as free. - However, after we do it, we know with absolute certainty the momentum of the second particle p x2 = P − p x1 , and this knowledge has been acquired without any perturbation of particle 2. - If this happened, then we would know with absolute certainty the position of the second particle, x 2 = x − x 1 , without perturbing particle 2 at all. - A way to defend the Heisenberg principle was to treat the two particles as an indivisible total system and reject the supposition that the particles are indepen- dent, even if they are millions light years apart. - He said that the state of the total system in fact never fell apart into particles 1 and 2, and still is in what is known as. - entangled quantum state 56 of the system of particles 1 and 2 and entangled states. - any measurement influences the state of the system as a whole, independently of the distance of particles 1 and 2.. - This correlation between mea- surements on particles 1 and 2 has to take place immediately, regardless of the space. - One can apply it successfully and obtain an excellent agreement with experiment, but there is something strange in its foundations. - In the following, some precise experiments will be described, in which it is shown that quantum mechanics is right, however absurd it looks.. - Assume that the world (stars, Earth, Moon, you and me, table, proton, etc.) exists objectively. - 57 Instead of the Moon, let us begin with something simpler: how about electrons, protons or other elementary particles? This is an important question because the world as we know it – including the Moon – is mainly composed of protons. - Following Richard Feynman, 59 imagine two slits in a wall. - There is a screen behind the two slits, and when an electron hits the screen, there is a flash (fluorescence) at the point of collision. - But suddenly we begin to suspect that there is some regularity in the traces, Fig. - 57 This may indicate that the Moon exists independently of our observations and overcome importu- nate suspicions that the Moon ceases to exist, when we do not look at it. - Besides, there are people who claim to have seen the Moon from very close and even touched it (admittedly through a glove) and this slightly strengthens our belief in the Moon’s existence. - The example of the Moon also intrigued others, cf. - 58 In the darkest communist times a colleague of mine came to my office. - Conspiratorially, very excited, he whispered: “The proton decays. - He just read in a government newspaper that the lifetime of proton turned out to be finite. - I said: “Why do you look so excited then and why all this conspiracy?” He answered: “The Soviet Union is built of protons, and therefore is bound to decay as well!”. - 59 After Richard Feynman, “The Character of Physical Law”, MIT Press, 1967.. - (a) 10 electrons (b) 100 electrons (c) 3000 electrons – one begins to suspect something (d) 20000 electrons – no doubt, we will have a surprize (e) 70000 electrons – here it is! Conclusion: there is only one possibility – each electron went through the two slits. - This resembles the interfer- ence of waves, e.g., a stone thrown into water causes interference behind two slits:. - two slits.. - The common sense tells us that nothing like this could happen with the elec- tron, because, firstly, the electron could not pass through both slits, and, sec- ondly, unlike the waves, the electron has hit a tiny spot on the screen (trans- ferring its energy). - go through the slit and make flashes on the screen here and there, but there is only a single major concentration re- gion (just facing the slit) fading away from the centre (with some minor min- ima).. - Why? You would like the Moon, a proton or an electron to be solid objects, wouldn’t you? All investigations made so far indicate that the electron is a point-like elementary particle. - If, in the exper- iments we have considered, the electrons were to be divided into two classes: those that went through slit 1 and those that passed slit 2, then the electron patterns would be different. - The pattern with the two slits had to be the sum of the patterns corresponding to only one open slit (facing slit 1 and slit 2). - The only explanation for this interference of the electron with itself is that with the two slits open it went through both.. - Clearly, the two parts of the electron united somehow and caused the flash at a single point on the screen. - Despite the fact that the wave function is delocalized, the measurement gives its single point position (decoherence). - How could an electron pass simultaneously through two slits? We do not understand this, but this is what happens.. - Maybe it is possible to pinpoint the electron passing through two slits? Indeed, one may think of the Compton effect: a photon collides with an electron, changes its direction and this can be detected (“a flash on the electron. - When one pre- pares two such ambushes at the two open slits, it turns out that the flash is always on a single slit, not on both. - There is no interference. - When we observe them, there is no interference. - 60 Somehow we perturb the electron’s momentum (the Heisenberg principle) and the interfer- ence disappears.. - We have to accept that the electron passes through two slits. - This is a blow to those who believe in the reality of the world. - And this ocean liner passed through two slits separated by thousands of Å. - about million times more massive than the electron. - There is something intriguing in this.. - John Bell proved a theorem in 1964 that pertains to the results of measurements carried out on particles and some of the inequalities they have to fulfil. - The the- orem pertains to the basic logic of the measurements and is valid independently of the kind of particles and of the nature of their interaction. - of the bars are always parallel to each other and always perpendicular to the straight line. - If the bar’s longer axis coincides with the longer dimension of the slit then. - In the 1960s Bell reconsid- ered an old controversy of lo- cality versus non-locality, hid- den variables, etc., a subject apparently exhausted after exchange of ideas between Einstein and Bohr.. - If the bar’s longer axis coincides with the shorter axis of the slit, then the bar will not go through for sure, and will be detected as “0”. - 61 To observe such phenomena the slit distance has to be of the order of the de Broglie wave length, λ = h/p, where h is the Planck constant, and p is the momentum. - Cohen-Tannoudji lowered the tem- perature to such an extent that the momentum was close to 0, and λ could be of the order of thousands of Å.. - Kołos, Proceedings of the IV Castel Gandolfo Symposium, 1986.. - In the first experiment the two slits will be parallel.. - This means that the fate of both bars in any pair will be exactly the same: if they go through, they will both do it, if they are stopped by the slits, they will both be stopped. - There are two differences (highlighted in bold) between the lists for the two detectors.. - There are two differences (bold) between the two detectors.. - Therefore, at Detector A we obtain the same results as in Experiment II, while at Detector B – the same as in Experiment III
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