By expressing the Schrödinger wave function in the form ψ = R e i S / ℏ , where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, viz., V Q = - ℏ 2 2 m ∇ 2 R R , we have obtained a spin potential V S = - S ∇ 2 S m that depends on S which is attributed to the particle spin. The spin force is found to give rise to dissipative viscous force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass m c 2 with an internal frequency of 2 m c 2 / ℏ , satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.
By expressing the Schrödinger wave function in the form ψ = R e i S / ℏ , where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, viz., V Q = - ℏ 2 2 m ∇ 2 R R , we have obtained a spin potential V S = - S ∇ 2 S m that depends on S which is attributed to the particle spin. The spin force is found to give rise to dissipative viscous force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass m c 2 with an internal frequency of 2 m c 2 / ℏ , satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.
By expressing the Schrödinger wave function in the form ψ = R e i S / ℏ , where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, viz., V Q = - ℏ 2 2 m ∇ 2 R R , we have obtained a spin potential V S = - S ∇ 2 S m that depends on S which is attributed to the particle spin. The spin force is found to give rise to dissipative viscous force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass m c 2 with an internal frequency of 2 m c 2 / ℏ , satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.
By expressing the Schrödinger wave function in the form ψ=ReiS/ℏ, where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, \emph{viz.}, VQ=−2mℏ2R∇2R\,, we have obtained a phase potential VS=−mS∇2S that depends on the phase S derivative. The phase force is a dissipative force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass mc2 with an internal frequency of 2mc2/ℏ, satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
In the search for a relativistic wave equation for a free particle of rest mass m one must demand that the energy(E)-momentum (p) [dispersion] relation E2 = m2c4 + p2c2 (25.1) is satisfied. By the substitutions E → i~ ∂ ∂t , p→ −i~∇ (25.2) in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, gives us the Klein-Gordon wave equation for a spinless particle. The Klein-Gordon equation is of second order in the space (r) and time (t) coordinates. If one seeks a covariant first-order differential equation, r and t must play symmetric roles. A natural starting point therefore would be a linear dispersion relation of the form E = βmc2 + cα · p, (25.3) where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compatible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given in Eqs. (10.77)-(10.79)]: β2 = 1, (25.4) αiβ + βαi = 0, (25.5) αiαj + αjαi = 2δij , (25.6) where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot be numbers, but one can find matrices which satisfy these equations. Since the Hamiltonian appearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian, the αi and β matrices also have to be Hermitian, i.e., α†i = αi, i = 1− 3, (25.7) β† = β. (25.8) One can show that the matrices must have even rank and that the rank must be at least 4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs. (25.4)-(25.8). Representations of a given rank are not unique.
A formulation is proposed for the behavior of a particle in a box based on two assumptions: (1) particles have an associated phase which changes periodically with time as defined by de Broglie, and (2) the transmission of phase information follows optical principles, including reflection at potential boundaries and relativistic Doppler effects. This approach defines optimal trajectories by matching the phase of a source and its reflections. The momentum of the particle is guided by a de Broglie wave which follows the equation R exp {i[w0 (1-v^2/c^2)^-1/2][(t - vz)/c^2]} cos {[w0(1-v^2/c^2)^-1/2/c](vt-z)} This wave is generated by reflections occurring an even number of times, the phase and amplitude of which correspond to Lorentz transforms relative to the time and position, respectively. Nodes are produced by reflections occurring an odd number of times, which can be conceived to generate a wave traveling in the opposite direction and shifted by a phase constant π. Application to an individual particle in an infinite potential well affords the same results for energies and nodes as derived by the Schrödinger equation.