It is proved that in a double slit diffraction experiment, superposition principle of probability amplitude means (the probability amplitude of the event) that an electron passes through a certain slit and arrives somewhere on the screen under the condition that two slits open simultaneously, is equal to the sum of two probability amplitudes of the same event under the condition that two slits open in turn. From this thesis it is concluded that the probabilities do not obey superposition principle, which is just the reason that classical probability theory is inapplicable for the very experiment. Kolmogorov's probability theory is based on two foundations: frequency definition of probabilities and Boolean algebra of event operations. In micro processes, the former holds true while the latter, especially its multiplicative permutation law is violated. So long as we do not deal with event operations, joint probabilities holds good in micro processes. But, the event operation formulae provided are applied and the wrong conclusions present probably. The famous Bell's inequality is a wrong conclusion. (Nature and Science. 2005;3(2):82-91).
It is proved that in a double slit diffraction experiment, superposition principle of probability amplitude means (the probability amplitude of the event) that an electron passes through a certain slit and arrives somewhere on the screen under the condition that two slits open simultaneously, is equal to the sum of two probability amplitudes of the same event under the condition that two slits open in turn. From this thesis it is concluded that the probabilities do not obey superposition principle, which is just the reason that classical probability theory is inapplicable for the very experiment. Kolmogorov's probability theory is based on two foundations: frequency definition of probabilities and Boolean algebra of event operations. In micro processes, the former holds true while the latter, especially its multiplicative permutation law is violated. So long as we do not deal with event operations, joint probabilities holds good in micro processes. But, the event operation formulae provided are applied and the wrong conclusions present probably. The famous Bell's inequality is a wrong conclusion. (Nature and Science. 2005;3(2):82-91).
It is proved that for a double slit diffraction experiment,superposition principle of probability amplitudes means the probability amplitude of the event that an electron passes through a certain slit and falls on somewhere of the screen has nothing to do with the condition whether or not the other slit is open,from which it is concluded that the probability itself does not obey superposition principle.Consequently,the total probability formula as a relation among the measurable probabilities is invalid herein.As a result,it is unnecessary to give up the classical idea that an electron moves along an orbit,and thereby it is unnecessary to distinguish between classical physics and quantum physics.
It is proved that: What the two-slit experiments have refuted is the preconceived idea that of the event that a micro particle passing through one slit arrives somewhere on the screen is irrelative to the condition that whether or not the other slit is open. Consideringaddition theorem of probabilities, the above idea can be expressed as follows:
(a)Assum that a beam passes through a partition with two slit and arrives a screen. The of the event that an indivedual particle falls on somewhere of the screen under the condition that two slit open simultaneously, is equal to the sum of two of the same event under that two slit open in turn.
Defining the experimental condition such as two slit open simultaneously or in turn as passage the proposition (a)is manifested as probabilities is independent of passage conditions.
On the other hand, in terms of the above, superposition principle of in quantum mechanics ought to be indicated as probability amplitudes is independent of passage conditions. Correspondingly, (a)is named hereof as superposition presumption of probability.
For lack of the conception of passage conditions, (a)is confused with theorem of probabilities. As a result, classical physics tacitly approve (a), and thereby it fails to comprehend the two-slit experiments. In reverse, to explain two-slit experiments, quantum mechanics spurns addition theorem of so that it meets with various difficults about explanation.
Any alternative of quantum mechanics must be able to explain two-slit experiments, therefore it must contain passage conditions or amplitudes. But hidden variable theory Bell used dissatisfies such a prerequisite. It comes thus into conflict with experiment facts inevitably. As a result, Lochak's argument that Bell's inequality is irrelative to locality is rediscovered.
In a double slit experiment, a screen position x could be endowed with two probabilities: the probability of finding particles which passed through slit A and arrived in x, and the probability of finding particles which passed through slit B and arrived in x.Analysis showed that, the probability of finding particles in x when two slits opened simultaneously was greater than when only one slit opened; as long as the time that the two slits opened simultaneously was long enough, the particle interference phenomenon in x would occur with certainty.Sur la base du point de vue de la classification des particules, une analyse supplmentaire est fournie: dans une exprience double fente, la probabilit de trouver une particule dans l'cran lorsque deux fentes s'ouvrent simultanment est deux fois suprieure celle d'une seule fente.Ds que le temps d'ouverture des deux fentes est suffisamment long, le phnomne d'interfrence de particules se produira certainement.
The science of information theory begins with the observation that there is a fundamental link between probabilities and information. As early as the mid eighteenth century, Bayes recognized that probabilities depend on what we know; if we acquire additional information then this modifies the probability. For example, the probability that it is raining when I leave for work in the morning is about 0.2, but if I look out of the window ten minutes before leaving and see that it is raining then this additional information adjusts the probability to in excess of 0.9. Information is a function of probabilities: it is the entropy associated with the probability distribution. This conclusion grew out of investigations into the physical nature of entropy by Boltzmann and his followers. The full power of entropy as the quantity of information was revealed by Shannon in his mathematical theory of communication. This work laid the foundations for the development of information and communications theory by proving two powerful theorems which limit our ability to communicate information. The link between probability and information has far wider application than just communications. Indeed, we can expect the ideas of information theory to be applicable to any statistical or probabilistic problem. Quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed. In quantum theory, probabilities are secondary quantities calculated by taking the squared modulus of probability amplitudes and this gives rise to interference effects. Consider, for example, the famous two-slit experiment depicted in Fig. 1.1. A single particle launched at the slits can arrive at a point P on the screen by passing either through slit 1 or through slit 2. In classical statistical mechanics this leads to a probability . . . P = P1 + P2, (1.1) . . . where P1 and P2 are, respectively, the probabilities that the particle passed through slit 1 or slit 2 and went on to arrive at the point P.
We start with the discussion on misapplication of classical probability theory by Feynman in his analysis of the two slit experiment (by following the critical argumentation of Koopman, Ballentine, and the author of this paper). The seed of Feynman's conclusion on the impossibility to apply the classical probabilistic description for the two slit experiment is treatment of conditional probabilities corresponding to different experimental contexts as unconditional ones. Then we move to the Bell type inequalities. Bell applied classical probability theory in the same manner as Feynman and, as can be expected, he also obtained the impossibility statement. In contrast to Feynman, he formulated his no-go statement not in the probabilistic terms, but by appealing to nonlocality. This note can be considered as a part of the author's attempts for getting rid off nonlocality from quantum physics.
Probability and interference According to Feynman, single-particle interference is ‘the only mystery’ of quantum theory (Feynman, Leighton and Sands 1965, ch. 1, p. 1). Feynman considered an experiment in which particles are fired, one at a time, towards a screen with two holes labelled 1 and 2. With both holes open, the distribution P 12 of particles at the backstop displays an oscillatory pattern of bright and dark fringes. If P 1 is the distribution with only hole 1 open, and P 2 is the distribution with only hole 2 open, then experimentally it is found that P 12 ≠ P 1 + P 2 . According to the argument given by Feynman (as well as by many other authors), this result is inexplicable by ‘classical’ reasoning. By his presentation of the two-slit experiment (as well as by his development of the path-integral formulation of quantum theory), Feynman popularised the idea that the usual probability calculus breaks down in the presence of quantum interference, where it is probability amplitudes (and not probabilities themselves) that are to be added. As pointed out by Koopman (1955), and by Ballentine (1986), this argument is mistaken: the probability distributions at the backstop – P 12 , P 1 and P 2 – are conditional probabilities with three distinct conditions (both slits open, one or other slit closed), and probability calculus does not imply any relationship between these. Feynman's argument notwithstanding, standard probability calculus is perfectly consistent with the two-slit experiment.