Based on the Bohr's correspondence principle it is shown that relativistic mechanics and quantum mechanics may be considered as generalizations of classical mechanics. A comparative description of relativistic and classical mechanics is given using three main principles: the definition of momentum, the main law of dynamics (Newton's second law) and the law of conservation of energy. The difference of relativistic mechanics from classical mechanics is due to the new definition of momentum that is proportional to energy and velocity. It is shown that new relativistic laws of dynamics make it necessary to change kinematical relations of classical mechanics such as the law of velocity composition, coordinate transformation etc. It is demonstrated that quantum mechanics may be considered as a rational statistical generalization of classical mechanics. Statistical conformities in quantum mechanics are fundamental and are not due to the incomplete information about the system. From all possible multi-parametric statistical models the root model is most remarkable. Constructing of a root multi-parametric statistical model leads to obtaining such frequencies and base functions in Fourier decomposition so that classical laws of motion are fulfilled in average. Root model leads to a consistent condition that connects eigenvectors and eigenvalues of a mechanical system that is described by a matrix equation of Heisenberg. Matrix equation of Heisenberg leads to an operator equation. The solution of the operator equation may be considered as the construction of Hamiltonian of the system and a move to the Schrodinger representation. The considered approach naturally leads to the notion of momentum operator, fundamental commutation relations, density matrix and Liouville equation construction etc.
Based on the Bohr's correspondence principle it is shown that relativistic mechanics and quantum mechanics may be considered as generalizations of classical mechanics. A comparative description of relativistic and classical mechanics is given using three main principles: the definition of momentum, the main law of dynamics (Newton's second law) and the law of conservation of energy. The difference of relativistic mechanics from classical mechanics is due to the new definition of momentum that is proportional to energy and velocity. It is shown that new relativistic laws of dynamics make it necessary to change kinematical relations of classical mechanics such as the law of velocity composition, coordinate transformation etc. It is demonstrated that quantum mechanics may be considered as a rational statistical generalization of classical mechanics. Statistical conformities in quantum mechanics are fundamental and are not due to the incomplete information about the system. From all possible multi-parametric statistical models the root model is most remarkable. Constructing of a root multi-parametric statistical model leads to obtaining such frequencies and base functions in Fourier decomposition so that classical laws of motion are fulfilled in average. Root model leads to a consistent condition that connects eigenvectors and eigenvalues of a mechanical system that is described by a matrix equation of Heisenberg. Matrix equation of Heisenberg leads to an operator equation. The solution of the operator equation may be considered as the construction of Hamiltonian of the system and a move to the Schrodinger representation. The considered approach naturally leads to the notion of momentum operator, fundamental commutation relations, density matrix and Liouville equation construction etc.
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.
Abstract The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approximation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.
Background: Recently, we obtained a unitary theory of quantum mechanics and general relativity, where a quantum particle is a continuous distribution of matter in the two conjugate spaces of the coordinates and momentum, quantized by the equality of the mass parameter describing the relativistic dynamics of the matter, with the mass as an integral of the matter density. However, in this framework, we have not explicitly revealed the connection between our new theory and the fundamental laws of quantum mechanics. Methods: We analyzed in detail the three fundamental laws of quantum mechanics, explicitly describing experimental data: 1) Planck’s law of the blackbody electromagnetic radiation of a system of electrically charged harmonic oscillators, 2) Einstein’s law of the photon energy proportionality with the photon frequency, and 3) de Broglie’s law of the quantum particle as an oscillator in space. Results: We reobtained the two dynamical equations, in the conjugate spaces of the coordinates and momentum, as functions of the Lagrangian system, unlike the Schrödinger equation, depending on the Hamiltonian. Conclusion: According to the fundamental laws of quantum mechanics, a quantum particle is a continuous distribution of matter with an intrinsic mass, unlike the conventional quantum mechanics for the state occupation probabilities of punctual entities moving with the light velocity and getting an apparent mass only by collisions with some bosons pervading the whole universe. According to these laws, we obtained a quantum theory in agreement with common sense, classical logic, and general relativity.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dtdP, P=mdtdV, and [X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
According to the correspondence principle, as formulated by Bohr, both in the old and the modern quantum theory, the classical limit should be recovered for large values of the quantum numbers in any quantum system. However, this classical limit of quantum theory is not so straightforward as in the interface of other generalizations of classical mechanics and other domains. In particular, relativistic kinematics and mechanics reduce to Newtonian equations by simple algebra in the case of bodies moving with small velocities compared to the speed of light in vacuum. In this paper we consider the correspondence limit to the two-body problem in gravitational physics, the limit in which both the principal and the angular quantum numbers, N, L are very large. In this limit, we compare with the classical elliptical orbits and we find that the macroscopic coherent quantum states correspond to the statistical average of every classical state compatible with conservation laws for the total energy and angular momentum. We also consider the perturbed Kepler problem with a central perturbation force proportional to the inverse of the cube of the distance to the central body. The exact solution for the quantum eigenstates shows that the first order perturbation to the energy eigenvalues are obtained classically as the temporal orbital average of the perturbation potential.
Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting conditions. Nevertheless, this remains a challenging objective. In this work, we explore the potential for unifying the dynamics of classical and quantum physics. This discussion does not suggest that classical behavior emerges from quantum mechanics; rather, it demonstrates the exact equivalence between the dynamics of quantum observables and their classical counterparts. It is shown that the Heisenberg equations of motion can be cast in a form that is structurally equivalent to Newton's equations of motion, with ℏ dropping out from the equations. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same form of equations, with the Heisenberg operators substituting the classical observables.