Abstract A wave function can be written in the form ofReiS/ℏ. We put this form of wave function into quantum mechanics equations and take hydrodynamic limit, i. e., let Planck constant be zero. Then equations of motion (EOM) describing the movement of macroscopic bodies are retrieved. From Schrödinger equation, we obtain Newtonian mechanics, including Newton’s three laws of motion; from decouple Klein–Gordon equation with positive kinetic energy (PKE), we obtain EOM of special relativity in classical mechanics. These are for PKE systems. From negative kinetic energy (NKE) Schrödinger equation and decoupled Klein–Gordon equation, the EOM describing low momentum and relativistic motions of macroscopic dark bodies are derived. These are for NKE systems, i. e., dark systems. In all cases scalar and vector potentials are also taken into account. The formalism obtained is collectively called macromechanics. For an isolated system containing PKE and NKE bodies, both total momentum and total kinetic energy are conserved. A dark ideal gas produces a negative pressure, and its microscopic mechanism is disclosed. Two-body problems, where at least one is of NKE, are investigated for both macroscopic bodies and microscopic particles. A NKE proton and a PKE electron can compose a stable PKE atom, and its spectral lines have blue shifts compared to a hydrogen atom. The author suggests to seek for these spectral lines in celestial spectra. This provides a way to seek for dark particles in space. Elastic collisions between a body and a dark body are researched.
Abstract A wave function can be written in the form ofReiS/ℏ. We put this form of wave function into quantum mechanics equations and take hydrodynamic limit, i. e., let Planck constant be zero. Then equations of motion (EOM) describing the movement of macroscopic bodies are retrieved. From Schrödinger equation, we obtain Newtonian mechanics, including Newton’s three laws of motion; from decouple Klein–Gordon equation with positive kinetic energy (PKE), we obtain EOM of special relativity in classical mechanics. These are for PKE systems. From negative kinetic energy (NKE) Schrödinger equation and decoupled Klein–Gordon equation, the EOM describing low momentum and relativistic motions of macroscopic dark bodies are derived. These are for NKE systems, i. e., dark systems. In all cases scalar and vector potentials are also taken into account. The formalism obtained is collectively called macromechanics. For an isolated system containing PKE and NKE bodies, both total momentum and total kinetic energy are conserved. A dark ideal gas produces a negative pressure, and its microscopic mechanism is disclosed. Two-body problems, where at least one is of NKE, are investigated for both macroscopic bodies and microscopic particles. A NKE proton and a PKE electron can compose a stable PKE atom, and its spectral lines have blue shifts compared to a hydrogen atom. The author suggests to seek for these spectral lines in celestial spectra. This provides a way to seek for dark particles in space. Elastic collisions between a body and a dark body are researched.
The wave function Ψ satisfies the Schrödinger equation for a free particle. Formally, the Schrödinger equation generates the magnitude of mechanical motion of the zero order 0 p = mv 0 (in the sense that it is contained in the Schrödinger equation. Comparison of the wave function Ψ and its gradient implies a formal analogue of the Schrödinger equation, which generates the magnitude of mechanical motion of the first order 1 p = mv 1. From the comparison of the wave function and its time derivative yields a formal analogue of the Schrödinger equation, which generates the magnitude of mechanical motion of the second order 2 p = mv 2/2!. The values of mechanical motion of the zero, first, and second orders are known.Obviously, other formal analogs of the Schrödinger equation can generate mechanical motion of other orders.The aim of the work is to establish such quantities and related regularities that may be of interest, which makes the study relevant. hovering. Then the spatial derivatives will be one-dimensional. The magnitude of the mechanical movement of the third order is 3 p = mv 3/3!. This value is Umov's integral vector for kinetic energy. The magnitude of the mechanical movement minus the first order -1 p = mv -1 is a reverse impulse. The meaning of this quantity and its relevance is established by the theorem: in a hy- drogen-like atom, the quantity m v-1 is quantized. A fixed (unchanged) quantum is a quantity m v-1 correspond- e e 0 ing to the basic energy level. Almost all of the results obtained were a consequence of the use of quantum mechanical differential equations, however, the results themselves are predominantly macromechanical. The quantities of mechanical motion of various orders are generated by formal analogs of the Schrödinger equation. In all formal analogs of the Schrödinger equation, the orders of the partial derivatives differ by one. For quantities of motion with a positive degree of velocity, the order of the temporal derivatives is higher than that of the spatial ones. For quantities with a negative degree, the order of spatial derivatives is higher.
Abstract A fundamental belief is that the formulism of relativistic quantum mechanics equations (RQMEs) should remain in low momentum motion. However, it is found that some formulas from RQMEs were lost in Schrödinger equation. For example, a free relativistic particle has positive and negative energy branches. The former includes positive kinetic energy (PKE) and the latter negative kinetic energy (NKE). The latter should be treated on an equal footing as the former. Nevertheless, from Schrödinger equation, a free particle can have only PKE. Starting from RQMEs and taking low momentum approximation, we derive NKE Schrödinger equation which is for the cases that free particles have NKE. Thus negative energy branch of RQMEs can be retained in low momentum motion. We point out a fact that whether Schrödinger equation is applicable in a region where a particle’s energy E is less than potential V , E < V , has never been quantitatively verified. In such a region NKE Schrödinger equation should be employed. With the help of NKE Schrödinger equation, the lost formulas are recovered. The so-called difficulty of negative probability of Klein–Gordon equation for free particles is solved. A PKE (NKE) particle can have stationary motion only when it is subject to an attractive (repulsive) potential, which is determined by Virial theorem. Two NKE electrons in a potential can constitute a stable system, a new kind of possible mechanism for electron paring. The whole discussion stems from RQMEs with no any new postulation. Experiments are suggested, which may confirm that there are indeed NKE electrons.
The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.
Macromechanical quantities are obtained using quantum mechanical differential equations. The magnitudes of mechanical motion of different orders are generated by formal analogues of the Schrödinger equation. These quantities include both known (mass, momentum, kinetic energy) and unknown (Umov’s integral vector for kinetic energy, reverse impulse, etc.) properties. The proper Schrödinger equation formally generates the zeroth-order mechanical motion mv0 (in the sense that it is in the Schrödinger equation). The magnitude of the first-order mechanical motion mv1 is generated by a formal analogue of the Schrödinger equation, which is a combination of the proper wave function and its gradient.
Abstract Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.
Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.