It is shown that the wave function describes the state of the statistical ensemble E[S] of individual particles, or the statistical average particle . This result follows from the fact that in the classical limit h=0 the Schroedinger equation turns to the dynamic equations for the statistical ensemble of classical particles. The statement that the wave function describes the state of an individual particle is incompatible with the quantum mechanics formalism. It is shown that the statistical interpretation of quantum mechanics is a corollary of the fact, that the QM formalism is the technique of the statistical ensemble description, restricted by constraints of the dynamic equation linearity.
It is shown that the wave function describes the state of the statistical ensemble E[S] of individual particles, or the statistical average particle . This result follows from the fact that in the classical limit h=0 the Schroedinger equation turns to the dynamic equations for the statistical ensemble of classical particles. The statement that the wave function describes the state of an individual particle is incompatible with the quantum mechanics formalism. It is shown that the statistical interpretation of quantum mechanics is a corollary of the fact, that the QM formalism is the technique of the statistical ensemble description, restricted by constraints of the dynamic equation linearity.
It is proved the mathematical theorem, that the wave function describes the statistical ensemble of particles, but not a single particle. Supposition, that the wave function describes a single particle appears to be incompatible with formalism of quantum mechanics. One discusses the reasons, why this very simple statement has not been proved mathematically for many years. The reason lies in application of the trial and error methods for construction of the quantum mechanics. Application of this method as the main tool of investigation during eighty years generated ”fitting mentality ” of all microwold researchers. In this paper we prove a very important theorem, which claims that the wave function may not describe an individual quantum particle. It describes always a statistical ensemble of quantum particles. We shall show, that the action AS for the Schrödinger particle SS (the dynamic system described by the Schrödinger equation) turns into the action AE[Scl] for the statistical ensemble E [Scl] of free classical particles Scl, when the quantum constant � → 0. Such a transition is possible only in the case, when the wave function ψ describes a statistical ensemble of quantum particles, but not a single particle. For the free Schrödinger particle SS the action has the form
One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions. The wave-function plays a central role because it is a good parametrization that allows us to represent a group of transformations using linear transformations of the hypersphere. It is precisely the fact that the hypersphere is not the phase-space of the theory that implies the collapse of the wave-function. Without collapse, the wave-function parametrization would be inconsistent.
One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. On the other hand, Classical Statistical Field Theory suffers from severe mathematical inconsistencies (specially for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical field theory). We claim that both problems are related to each other and we propose a solution to both. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions; 4) the wave-function parametrization is key in the mathematical definition of Classical Statistical Field Theory we propose here. The same formalism is applied to Quantum Yang-Mills theory and Quantum Gravity in another article.
In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.
It is shown that the statistical conception of quantum mechanics is dynamical but not probabilistic, i.e. the statistical description in quantum mechanics is founded on dynamics. A use of the probability theory, when it takes place, is auxiliary. Attention is drawn to the fact that in the quantum mechanics there are two different objects: an individual object to be statistically described and a statistical average object, which is a result of the statistical description. Identification of the two different objects (a use of the same term for both) is an origin of many known quantum mechanics paradoxes.
Because of the use of different limiting procedures, there are two conflict conclusions on wave function. One is that a wave function can describe a single particle and the other is that it can only describe an ensemble in the classical limit. In this paper the limiting procedures have been compared. We put the synthesized limit n→∞, →0 but keep E(n, ) to be the actual measured value of energy. Under this limit condition, we can not only prove that a wave function can only describe an ensemble, but also can make clear the dynamical behavior of the particles of the ensemble. Calculations show that the reason why quantum mechanics cannot describe a particle is not because of the motion equation but because of the definition of state for the particle.