We study atomic and molecular (AM) stationary systems and electrodynamic (ED) systems, composed of an electron interacting with an electromagnetic field. We show that the Schr?dinger and Klein-Gordon equations written for these systems have a secondary solution, which is the wave function associated to a classical system. For AM systems, the wave surfaces (S surfaces) and their normals (C curves) are solutions of the Hamilton-Jacobi equation, written for the same system. The S surfaces have a periodical motion and the C curves are closed. The integral relation of the Schr?dinger equation on the C curve has a solution identical to the wave function associated to the classical motion. This solution leads to the generalized Bohr quantization relation and to the generalized de Broglie relations, which are valid in the space of the electron coordinates. An identical wave function verifies the Klein-Gordon equation, in the case of the EM systems. The above properties lead to a central field method for calculation of the energetic values and symmetry properties for AM systems, whose accuracy is comparable to the accuracy of the Hartree-Fock method. They also lead to an accurate method for modeling EM systems, which is verified by experimental data from the literature. The above properties are deduced without using any approximation.
We study atomic and molecular (AM) stationary systems and electrodynamic (ED) systems, composed of an electron interacting with an electromagnetic field. We show that the Schr?dinger and Klein-Gordon equations written for these systems have a secondary solution, which is the wave function associated to a classical system. For AM systems, the wave surfaces (S surfaces) and their normals (C curves) are solutions of the Hamilton-Jacobi equation, written for the same system. The S surfaces have a periodical motion and the C curves are closed. The integral relation of the Schr?dinger equation on the C curve has a solution identical to the wave function associated to the classical motion. This solution leads to the generalized Bohr quantization relation and to the generalized de Broglie relations, which are valid in the space of the electron coordinates. An identical wave function verifies the Klein-Gordon equation, in the case of the EM systems. The above properties lead to a central field method for calculation of the energetic values and symmetry properties for AM systems, whose accuracy is comparable to the accuracy of the Hartree-Fock method. They also lead to an accurate method for modeling EM systems, which is verified by experimental data from the literature. The above properties are deduced without using any approximation.
In previous papers we proved the existence of exact connections between quantum and classical equations in the cases of two different quantum systems, namely the stationary atomic and molecular systems and the systems composed by electrons and very intense electromagnetic field. We proved that the geometrical elements of the wave described by the Schrodinger equation, namely the characteristic surfaces and curves, denoted, respectively, by Σ and C, are solutions of the HamiltonJacobi equations, written for the same system, in the case of multidimensional stationary systems. The Σ surfaces, which have the significance of wave surfaces, and the C curves, which are their normals, correspond to the same constants of motion as the eigenvalues of the Schrodinger equation. In three recent papers we presented a central field method for the calculation of the C curves, and of the corresponding energetic values. We have verified this method for a large number of atoms and diatomic molecules, and we found that its accuracy is comparable to the accuracy of the Hartree-Fock method applied to the same atoms and diatomic molecules. In another papers we proved that the Klein-Gordon equation for the system electron-very intense electromagnetic field is verified exactly by the function associated to the classical motion of the electron. An exact connection follows between the solutions of the Klein-Gordon and the relativistic Hamilton-Jacobi equations, for that system. This result explains the accuracy of numerous classical models from the literature, belonging to the new field studying the interactions between very intense laser beams and electron plasmas. We present now a short review of these results.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.
This study examined the relativistic Klein–Gordon equation and highlighted the characteristics of positive and negative energies. Positive energies correspond to real particles, as they exhibit stable motion forward in time, while negative energies correspond to non-real systems due to antiparticles, highlighting the understanding of particles in relativistic quantum theory. The system’s stability is more pronounced at small screening parameters as the creation of particles (indicated by negative energies) also enlarged. The study further explores the contributions of component potentials in an interacting potential, with a focus on the Pöschl–Teller and symmetric trigonometric Rosen–Morse potentials. The duality of positive and negative energy solutions exemplifies the symmetry between matter and antimatter. The solution of the Schrödinger equation with the same potential is also studied and applied to diatomic models, specifically Cl2, Br2, I2, K2, N2 and F2, comparing analytic results with tested data for various quantum states. The analytic results demonstrate excellent agreement with experimental results, with deviations attributed to experimental methods or theoretical model limitations. The study underscores the reliability of the proposed analytic methods for predicting molecular properties.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.
We have demonstrated in previous papers the following property of closed, conservative and bounded systems: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. In the present article, we show that this property is connected to the intrinsic wave properties of the system. This results from the equivalence between the Schrödinger equation and the wave equation, valid for conservative systems. As a consequence of the wave properties of the system, we show that the Hamilton-Jacobi equation has always periodical solutions, whose constants of motion are identical to the eigenvalues of the Schrödinger equation, written for the same system. It results that the calculation model presented in previous papers is generally valid in the case of closed, conservative and bounded systems. We present the applications of the model to the nitrogen and oxygen atoms, to the ions with the same structure, and to the He 2 , Be 2 and B 2 molecules.
The Hamilton-Jacobi method is generalized, both, in classical and relativistic mechanics. The implications in quantum mechanics are considered in the case of Klein-Gordon equation. We find that the wave functions of Klein-Gordon theory can be considered as describing the motion of an ensemble of particles that move under the action of the electromagnetic field alone, without quantum potentials, hidden uninterpreted variables, or zero point fields. The number of particles is not locally conserved.