domenica 18 settembre 2011

"Shut up and calculate !"

The Copenhagen interpretation of quantum mechanics is fundamentally inspired by the work done by Niels Bohr and Werner Karl Heisenberg around 1927, when they were together in Copenhagen. In the classic experiment in which light passes through a screen on which is carried out a double slit are obtained, on a plate placed in front of the screen, alternating bands of light and dark color, which can be interpreted as areas where light waves interfere constructively or destructively. By reducing the intensity of the beam, so as to have only one photon at a time, in spite of the photons hit the screen one by one, on the whole it regains the interference. pattern is typical of the waves. The questions posed by this experiment are :
1) quantum mechanics provides only a statistical point where each particle hit the screen and identifies lighter and dark areas such as those for which the probability of being hit by a particle, respectively, high or low ; can not to predict exactly where a given particle will hit;
2) what happens to the particles in the path that leads from the source to the screen? Each particle is described by a wave function not localized: it appears that it interacts with both slits, but if it is viewed as a point it can cross one only.

In classical logic the principle of not contradiction asserts the falsity of every proposition which implies that a certain proposition A and its negation, the proposition that not-A are both true at the same time and in the same way. In the words of Aristotle: "It is impossible that the same attribute at the same time belong and not belong to the same subject and under the same respect"
More simply, the proposition "A is also not-A" is false.
Well, the dual nature of particle - wave of elementary particles like the photon (and the electron that has analogous behaviour), suggested by experimental evidence, is contrary to this principle. Many physicists have signed the "zero-order interpretation" of quantum mechanics, summarized in the famous saying: "Shut up and calculate!", usually (but perhaps incorrectly) attributed to Feynman. The Copenhagen interpretation confronts these issues as follows:
a) the probabilistic statements of quantum mechanics are irreducible, meaning they do not reflect our limited knowledge of some hidden variable. In classical physics, the probability is used even if the process is deterministic (for example, the roll of a dice), so as to overcome our incomplete knowledge of the initial data. On the other hand, the Copenhagen interpretation of quantum mechanics argues that the results of measurements of conjugate variables are basically non-deterministic, that is even knowing all the initial data it is impossible to predict the outcome of a single experiment, because the experiment itself affects the result ;
b) they are meaningless questions like "Where was the first particle that I have had neasured its position?", as quantum mechanics studies only observable quantities, obtained by measurement processes. The act of measurement causes the "collapse of the wave function" in the sense that it is constrained by the measurement process to take on the values ​​of one case of possible states allowed.
Many physicists and philosophers have objected to the Copenhagen and the famous words of Albert Einstein: "God does not play dice with the Universe" and "do you really think the moon is not there unless you look?" are examples of this.
The completeness of quantum mechanics has been attacked by the experiment of mind of Einstein-Podolsky-Rosen, intended to show that there must be some hidden variables, if you want to avoid instantaneous effects at a distance and not local.
In the atomic absorption spectra are observed splitting of spectral lines that, "singletons", are arranged symmetrically with respect to the middle line. In Bohr's atomic model the condition for an electron moving around the atomic nucleus along its orbit does not emit energy is that that value of angular momentum is an integer multiple of the size h/2π (quantization of angular momentum). In practice, an atom can move from the ground state to an "excited" state only absorbing a certain amount of energy that it brings along an electron to orbit in outer periphery more wide, with a radius r, called m the electron mass, and v its tangential speed, is such that nh/2π = mvr, where n is integer. In turn, de-energizing, the atom will return the same amount of energy previously absorbed, that is E = nhv. Hence the characteristic atomic spectra of absorption and/or emission lines. Each line corresponds with a certain frequency v and, therefore, a certain quantum of energy hv.
The "quantums" can be absorbed or emitted only in packages defined by the amount nhv where, precisely, n must be an integer which was defined as the principal quantum number.
Through the use of optical instruments with high resolving power is known then, that in fact these lines were, in turn, consist of discrete lines which suggested that the electrons as well as circular orbits such as those provided by Bohr , could also follow elliptical orbits in numbers equal t e so called
secondary quantum number.
Further doubling of the observed spectral lines in "singletons" that, in odd numbers, were arranged around the central line, suggested that considering the elliptical orbit of an electron around the nucleus of a loop like the plane flown by an electrical charge, this effect could be attributed to the difference in the inclination of the loop, as was observed following interaction with an external magnetic field and, therefore, the third quantum number was called magnetic quantum number. Finally, again because of other observed splitting, was introduced the quantum number of "spin" that defined the direction of rotation of each electron around its axis and, hence, its angular magnetic moment (or spin) +1/2 ; - 1/ 2, according to the "rule of the corkscrew," also influenced by an external magnetic field.
In the stable configurations the spins electrons on the same orbital must be equal and opposite to the Pauli exclusion principle. The paradox of Einstein-Podolsky-Rosen (EPR paradox) is a thought experiment that shows how a measurement of a part of a quantum system can propagate an instant effect on the outcome of another measurement, performed later on another part of the same system, regardless of the distance that separates the two sides.
This effect, known as quantum entanglement and resulting from the Copenhagen interpretation of quantum mechanics, was considered a paradox in that, not only counter intuitive but also deemed inconsistent with a postulate of special relativity (which considers that the speed of light limits the speed at which one can travel any type of information) and, more generally, with the principle of locality.
It is due to David Bohm, in 1951, a reformulation of the paradox in terms more easily verifiable experimentally.
The EPR paradox describes a physical effect that, as mentioned, is paradoxical in the following sense: if in a quantum system we assume some weak and general conditions, such as realism, locality and completeness, which are considered reasonably true for any theory that describes physical reality without contradict relativity, we arrive at a contradiction. However, note that quantum mechanics is not inherently contradictory, or in conflict with relativity.
Although originally proposed to highlight the incompleteness of quantum mechanics, additional theoretical and experimental developments followed the original article (like Bell's theorem and quantum correlation experiment of Aspect) led a large part of physicists to consider the EPR paradox only an illustrious example of how quantum mechanics contrasts sharply with the everyday experiences of the macroscopic world (although the issue is not completely closed).
Consider the simplified version of the ideal EPR experiment formulated by David Bohm.

Suppose you have a source that emits pairs of electrons, one of which is sent to destination A, where there is an observer named Alice, and the other is sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can adjust the source so that each pair of emitted electrons occupy a quantum state called a singlet spin. This can be described as a quantum superposition of two states, denoted by I and II. In the I state, the electron has spin parallel to the z-axis (+ z) and electron B has spin antiparallel to the z axis (-z). In the II state, electron A has spin-z and electron B has spin + z. It is therefore impossible to associate one of the two electrons in the spin singlet to defined spin state : the electrons are entangled so called, that is twisted.

Returning the experiment suggested by Einstein, Podolsky and Rosen, performed with electrons. A source sends electrons toward two observers, Alice (left) and Bob (right), which are capable of measuring the projection of the spin of electrons along an axis.

Alice measures the spin along the axis of getting one of two possible results: + z-z. Suppose that obtains + z ; according to quantum mechanics the wave function that describes the singlet state of two electrons collapses in the state I (the different interpretations of quantum mechanics say this in different ways, but at the end result is the same) and the quantum state determines the probability of any other measure of results done on the system. In this case, if Bob subsequently measures the spin along the z axis, would -z with a probability of 100%. Similarly, if Alice would -z, Bob would+ z with a probability of 100%.
In quantum mechanics, the projection of the spin along the x and z are along observable quantities that are incompatible, for which the associated operators do not commute, that is that a quantum state can not have definite values ​​for both variables (uncertainty principle). Suppose Alice measures the spin along z and obtains + z, so that the system collapses in the state I. Now, instead of measuring the spin along z, Bob measures the spin along x: according to quantum mechanics, there is a 50% chance that he gets + x and a 50% probability of obtaining x-. In addition, it is impossible to predict what the outcome will be as long as Bob does not fit.
It should be noted that although the spin is used as an example, we can consider many other physical quantities (observables), entangled with each other. The original paper of EPR, for example, used as the impulse observable quantities. Experiments today often use the polarization of the photons, because more easy to prepare and then measure.
Suppose that Alice decides to measure the spin along z (we will call z-spin). After Alice performs the measurement, the z-spin of the electron is known to Bob, so it's a physical element of reality. Similarly, if Alice decides to measure spin along x, the x-spin of Bob would be a physical element of reality after his measure.
A quantum state can not simultaneously have a definite value for the x-spin and z- spin. If quantum mechanics is a complete physical theory in the sense given above, the x-spin and the z-spin can not be physical elements of reality at the same time. This means that the decision of Alice to a measurement along the x or z-axis has an instant effect on the physical elements of reality in the place where Bob is at work with his measures. However, this is a violation of the principle of locality or the principle of separation.
The principle of locality states that physical processes may have no immediate effect on the physical elements of reality to another location separate from that in which they occur. At first sight this assumption seems reasonable (in fact, at the macro level it is), as a consequence of special relativity, which states that information can never be transmitted at a rate faster than light without violating causality. Generally believed that any theory which violates causality is also internally inconsistent, and therefore completely unsatisfactory.
However, the principle of locality is strongly recalls macroscopic physical intuition, and Einstein, Podolsky and Rosen did not want to leave. Einstein derided the predictions of quantum mechanics as "horrific action at a distance." The conclusion that they drew was that quantum mechanics is not a complete theory.
There are several possible ways to resolve the paradox. The one suggested by EPR is that quantum mechanics, despite the success in a broad variety of experimental scenarios, is actually an incomplete theory. In other words, there is some theory of nature not yet discovered, with respect to which quantum mechanics plays the role of statistical approximation. This more complete theory contains variables that take into account all the "elements of physical reality" and that give rise to the fact that quantum mechanics can predict only probability level. A theory with such characteristics is called the theory of hidden variables.
To illustrate this idea may make a very simple theory of hidden variables that explain the results of the experiment described above.
Suppose that the quantum states of spin singlet emitted from the source are actually approximate descriptions of "real" physical states that have defined values ​​for the z-spin and the 'x-spin. In these
"true" states, the electron going to Bob has always the opposite of the electron spin values ​​goini towards Alice, but these values ​​are completely random. For example, the first pair emitted from the source can be "(+ z,-x) to Alice and (-z + x) to Bob" ; the next pair "(-z,-x) to Alice and (+ z , + x) to Bob, "and so on. For this, if the axis of the measurement of Bob is aligned with that of Alice, he will necessarily get the opposite of whatever Alice, otherwise he will "+" and "-" with equal probability.
Assuming the measures to restrict only the axis z and the x-axis, the theory of hidden variables is experimentally indistinguishable from the theory of quantum mechanics.
In fact there is obviously an infinite number (countable) set of axes along which Alice and Bob can perform the respective measures, which means that, in theory, you might consider an infinite number of independent hidden variables. However, it should be noted that this is a very simplistic formulation of a theory of hidden variables and a more sophisticated theory would be able to solve the problem in mathematics.
There was another objection to do: once emitted, and then released from atomic structure to which they belonged, the electrons in the singlet are still bound to obey the exclusion principle?The answer is yes because the wave function describing the quantum state of the singlet is single and stationary, that is that doesn't depend on time, but only by the space and "collapses" in a solution instantly when Alice or Bob perform the measurement regardless of where you found each of the two electrons making up the singlet "entangled" of the atom from which the two electrons were "torn". After that Alice have done her measure and Bob derived his correlated measure, each electron becomes independent of the other associated with a new wave function.

In 1964, John Bell showed that the predictions of quantum mechanics thought experiment of EPR are actually slightly different from the predictions of a very large class of theories of hidden variables. Roughly speaking, quantum mechanics predicts much stronger statistical correlations between the results of measurements on different axes than do the theories of hidden variables. These differences, expressed employing unequal relations known as Bell inequalities are in terms of principle experimentally detectable.
Following the publication of the artiche, Bell began to be prepared a series of experiments to test the Bell inequalities (as mentioned above, these experiments generally deal with polarization measurements of photons). All experiments conducted so far have shown a behavior in line with the predictions of standard quantum mechanics.
Currently the majority of physicists believes that quantum mechanics is correct and that the EPR paradox is indeed only a "paradox" because classical intuitions (at the macroscopic level) does not correspond to reality. One can draw several conclusions from this lot, depending on which interpretation of quantum mechanics is used. In the old Copenhagen interpretation, produced by Niels Bohr, Werner Karl Heisenberg, Pascual Jordan and Max Born, it is concluded that the principle of locality (or separation) should not apply and that they do so the collapse of the wave function instantaneously. In the “Many-worlds interpretation” of Hugh Everett III, the locality is maintained and the effects of the measures rise from separating and branching of the "stories" or “world lines” of observers. Perhaps "shut up and calculate" is the only possible approach to understand QM, if only to try to check the tightness of theoretical models to the amount of data expecting, for example, from results of experiments such those in progress at CERN (LHC). There is a risk, however ', that this guidance "on sight" leads erroneously to identify mathematical models and physical realities increasing, so', the "spread" between form and substance which, in my opinion, is the weak link the chain of logic that should guide the path of Knowledge.

Stefano Gusman

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