The nature of quantum description is clarified. It is shown that complex-valued probability amplitudes are admissible within classical Hamiltonian mechanics. According to standard probability theory, such a description is always possible. The case of a spherical phase space is considered. It is shown that, in such a classical theory, there appears a universal constant that has dimensions of action ( h ), as well as Fock space and all attributes of quantum mechanics. Excitations of a chain of such systems are described by the equations of quantum mechanics with a correct normalization condition. It is shown that an answer to the question of what a particle and its wave function are is provided by quantum field theory (these are a single-particle field excitation and a function that describes it). Experiments are proposed that would make it possible to solve the problem of the physical nature of the wave function.
The nature of quantum description is clarified. It is shown that complex-valued probability amplitudes are admissible within classical Hamiltonian mechanics. According to standard probability theory, such a description is always possible. The case of a spherical phase space is considered. It is shown that, in such a classical theory, there appears a universal constant that has dimensions of action ( h ), as well as Fock space and all attributes of quantum mechanics. Excitations of a chain of such systems are described by the equations of quantum mechanics with a correct normalization condition. It is shown that an answer to the question of what a particle and its wave function are is provided by quantum field theory (these are a single-particle field excitation and a function that describes it). Experiments are proposed that would make it possible to solve the problem of the physical nature of the wave function.
Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic systems). The Planck constant and the Fock space appear after putting a dynamical system in a thermal bath. For harmonic oscillator in a thermal bath, the probability amplitudes can be identified with the complex valued phase functions f(q+ip) describing small deviations from the equilibrium state, when the time of relaxation is large. A chain of such oscillators models both the one-dimensional space (or string), and one-dimensional quantum field theory.
A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.
This book examines a number of problems of quantum mechanics, most of which are not usually discussed. What is the origin of probabilities in the mechanics of the microworld? What is the nature of Planck's constant h? What is the nature of probability amplitudes? What is the wave function? A system of axioms for quantum theory is formulated. A model is studied according to which a classical oscillator in a thermostat can be interpreted as a quantum one. The measurement problem is discussed in detail.
For advanced undergraduate students, graduate students, and specialists interested in the foundations of quantum theory.
It is the matter of fact that quantum mechanics operates with notions that are not determined in the frame of the mechanics' formalism. Among them we can call the notion of "wave-particle" (that, however, does not appear in both classical and high energy physics), the probabilistic interpretation of the Schroedinger wave ψ-function and hence the probability amplitude and its phase, long-range action, Heisenberg's uncertainty principle, the passage to the so-called operators of physical values, etc. Orthodox quantum mechanics was constructed as a physical theory developed in the phase space of the mentioned notions. That is why the formalism of quantum mechanics is aimed only at detailed calculations of the stationary states of the energy of the quantum system studied and is not able to describe a real path running by the system in the real space; instead, the formalism gives an averaged probabilistic prediction. Thus, if we are able to develop quantum mechanics in the real space, an option to clarify all the difficulties associated with the above notions would appear. Such a theory of quantum mechanics developed in the real space in fact has recently been constructed by the author. The theory started from deeper first principles, namely, from the consideration of the notion of a 4D space-time. So, the notion of fundamental particle, the principles of the motion of a particle and other characteristics have been made clear. The theory, rather a submicroscopic one, is characterized by short-range action that automatically means the introduction of a new kind of carriers, i.e. carriers of the quantum mechanical force. The existence of the carriers called "inertons" (because they carry inert properties of matter) has indeed been verified in a number of experiments.
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.
The famous 1927 Solvay Conference was considered a turning point in the world of Physics. The scientific realists like Albert Einstein had lost and the instrumentalists like Niels Bohr had won the fundamental conflict. Since then Physics has followed the path of the instrumentalists in which Quantum Physics has been determined by the concept of Elementary Particles and Probability Waves. Until May 2013, when for the first time a photo had been made of the Probability Waves Inside the Hydrogen Atom . It should be impossible to make a photo of a non-existing complex probability wave with mass zero But the fact could not be denied anymore that scientists had succeeded to make the “complex” and the “non-existing” quantum mechanical “Probability Wave” visible. New discoveries like “Photonic Molecules” and “Photon-Photon Interaction” require a new interpretation of the Copenhagen Interpretation . Since the Copenhagen Interpretation there is still no clarity about the physical interpretation of the Quantum Mechanical “Probability Wave Function”This Manuscript describes a third unknown interpretation of the Quantum Mechanical “Probability Wave Function”. A fundamental physical interpretation that will change the generally accepted Copenhagen Interpretation. This Manuscript will answer the fundamental question: Does an alternative theory exist in which the existence of this particular photo, of the Quantum Mechanical Probability Wave within the Hydrogen Atom, could be explained. And that the famous photo of the Hydrogen Atom can be explained in a logical and a simple way. To find the answer, we have to go back in time. Back 300 years in time to the moment when Isaac Newton discovered the beauty and one of the many secrets of the light. The moment when science walked away from light. And we return to the fundamental question: “Is it possible to build Matter out of Light (confined electromagnetic waves)?” To answer this question if it is possible to build Matter out of Light, we first have to answer the question how we have come to this world of Probability Waves and Elementary Particles and Quantum Physics.Louis de Broglie described in 1924 in his PhD thesis the wave properties of matter. Erwin Schrödinger published in 1926 the well-known Schrödinger wave equation with the characteristic spherical and elliptical wave solutions. To describe these material waves, the immediately and first explanation was: “There is only one possibility. These are “Electromagnetic Waves”. De Broglie Waves are ” Light Waves”. Because these were the only waves, known in the world of Physics, that could travel through vacuum. But there was a fundamental problem. These material waves, discovered by Louis de Broglie and mathematically described by Erwin Schrödinger could never be solutions of the well-known 4 linear equations of James Clerk Maxwell. The material waves were spherical and elliptical solutions of confinement and that was impossible for the solutions of the linear Maxwell Equations. According to the linear Maxwell Equations, superposition is possible but Electromagnetic Interaction and Electromagnetic Confinement can only be described by a set of non-linear differential equations. To solve this fundamental problem a new set of 3 non-linear Electromagnetic Equations (Equation 5 in this manuscript) has been developed which replaces the well-known 4 linear Maxwell Equations and the possible solutions of Equation (5) are also Spherical and Elliptical solutions of confinement. Table 1 in this manuscript presents a number of Electromagnetic Spherical Confinements as solutions of Equation 5a .A second problem was that the model of the atom of Bohr could not hold. How could an electron be everywhere at the same time in a kind of a spherical shell surrounding the nucleus. And be a particle at the same time. And Bohr found a compromise to declare the “De Broglie” waves to be the probability waves. Two problems were solved. Bohr’s planetary model could hold. And there was a name for the unknown material waves, discovered by Louis de Broglie and mathematically described by Erwin Schrödinger. The theory of “Probability Waves, Elementary Particles and Quantum Physics” had been created.