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July 17, 2016 at 11:45 am #687
What is energy?
Within the Atomistic Theory of Matter we could clear up what is mass. Now, we shall clear up what is energy and how the stable bound energies relate to other important physical quantities.
The stable elementary particles, i = e, p, P and E, have the elementary masses mP and me and since the elementary particles are not composed of other particles, one can set the gravitational and inertial masses of e, p, P and E to the same value. All other particles system are composed of these elementary particles.
For composite particle system the conserved gravitational mass, mg, and the non conserved inertial masses, mi, are different. These two masses can be expressed with the number of compounding particle, Ni, for an N = NP + NE + Np + Ne particle system and with the elementary masses mP and me. The gravitational mass is
mg(N) = (NP – NE)∙mP + (Np – Ne)∙me.
In the expression for inertial mass, from the sum of elementary masses the bound energy/c^2 must be subtracted
mi(N) = (NP + NE)∙mP + (Np + Ne)∙me – E(N;bound)/c^2 ≥ 0.
The relation of these two masses can be expressed with the mass defect Delta(matter)
mg(matter)/mi(matter) = 1 + Delta(matter).
Phenomenological is known that the mass defect of matter is between that of hydrogen (0.109%) and 56Fe isotope (+0.784%) is.
This inertial mass, in this sense, is the “rest mass” of a particles system with the bound energy E(N;bound). That means, the center of mass of the manyparticles system (COM) is at rest, or it moves with a velocity vCOM << c: the velocity of COM is very much smaller than the propagation of the interactions between the particles. Naturally, the rest COM is a special Lorentz system in which the COM of N = NP + NE + Np + Ne elementary particles is at rest.
We remember, the action integral,
I = ∫Ω L(x) (dx)^4,
for fields and particles was formulated in a Lorentzinvariant form and the subsidiary conditions of fields and particles must be take into account within a finite range of the Minkowski space, Ω. Furthermore, it was assumed that the nonconservative interactions propagate with a constant velocity c. The particle number conservations, as subsidiary conditions for the particle, implied the appearance of Lagrange multipliers, λj, in the equations of particlemotions.
Thus, the bound energy, E(N;bound) of an N particle system must be considered with regard to COM at rest and the amount of energy, E(N;bound), is radiated by the bound particle system. Of course, we are interested on temporally stationary bound states in which no more radiations occur. In this case, there are no free radiations fields present in a finite volume, V, which contains the N particles, and all the other parts of the Lagrange density, L(x), are temporally stationary. Since the parts of the action integral are consisting of the kinetic part and of the mutual potential part, we can write the temporally stationary bound energy as sum of these two temporally stationary parts
– E(N;bound) = E(N;kinetic) – E(N;potetial) < 0.
Since we have point like elementary particles with conserved charges and the charges have an inverse square law of static force we await according to the Virial theorem that
E(N;kinetic) = – E(N;potential)/2.
That is
E(N;bound) = E(N;kinetic)/2.
The half of the kinetic energy of the bound N bound particles is radiated, or, the doubled from the bound energy is stored in the kinetic energy of the bound N particles. Self evident, the mutual potential part of the interactions is not only consisting of the pure static potentials, but it contains also potential parts coming from temporally stationary momentums of the charges motions.
In order to calculate the bound energies with the probability densities of charges and with the vector potentials, from the action integral a variation principle is available, however, one needs also the Langrage multipliers, λj, for computing. The action integral is none an expression of energy.
For twoparticle bound system the reduced mass is mij’ = mi∙mj/(mi+mj). Thus, we can determine the relative velocity of particles according the formula
(vij/c)/(1 –(vij/c)^2)^1/2 = (2∙E(N=2; bound) /mij’∙c2)^1/2,
for the possible occurring velocities vij/c. For instance, for the hydrogen atom in the ground state, calculated with the bound energy
E((P,e);bound) = 13.606 eV,
the velocity of the electron relative to the proton is
v/c ≈ (2∙13.606 eV/ 0.511 MeV)^1/2 = 7,296∙10^3 = 1/137.006
For the stable neutron, N0 = (P,e), the bound energy is E(N0;bound) = 2.04 MeV, calculated with the Lagrange multiplier h0 =h/387, the relative velocity of electron, v/c, is
(v/c)/(1 –(v/c)^2)^1/2 = (2∙2.04 MeV/ mPe’∙c^2)^1/2 ≈(7.906)^1/2 → v/c ≈0.9422.
The electron moves with 94.22% of the velocity of light relative to the proton in the stable neutron, N0.
Generally, the maximal possible radiated energy amount form an N = NP + NE + Np + Ne particle system is
Emax(N;bound) = ((NP +NE)∙mp + (Np +Ne)∙me)∙c^2.
The doubled from this energy amount is stored as kinetic energy of the bound N particles.
We have derived the relations for the bound energy, the inertial mass and the gravitational mass for an N particles bound system. Furthermore, we connected the bound energy to the stored kinetic energy of the N bound particles.
A further interesting relation is an expression between the bound energies and the Lagrange multipliers, λk. In the case of twoparticle systems the relation is
λk= e^2/2∙c∙(mij’∙c^2/2∙E(boundk))^1/2,
assuming different temporally stationary bound states. With the ground state energy of the hydrogen atom, 13.606 eV, we get the value of the Planck constant h,
h = e^2/2∙c∙(mij’∙c^2/2∙13.606 eV)^1/2 = 6.62607004 × 10^34 m^2 kg / s.
We remark, h does not quantizes the energy of an electronproton system; h is a Lagrange multiplier.
Another value for Lagrange multipliers can be obtained from the electronpositron system, if the bound energy is
E((e,p);bound) = 2∙me∙c^2.
In this case, the value of the Lagrange multiplier is
h0 = e^2/2∙c ∙(1/8)^1/2 = h/387.
One obtain the same value, h0, also for the protonelton system at the bound energy
E((P,E);bound) = 2∙mP∙c^2.
With this value, h0 = h/387, the bound energy of the stable neutron N0 =(P,e) is calculated to 2.04 MeV. With h0 one can also calculate the bound states and energies of all atomic nuclei.
For a twoparticle systems the radii of bound states can be calculated according the formula
r = λk^2/(4∙π^2∙mij’∙e^2).
Appropriate calculations show that the elementary particles, e, p, P and E, cannot approach each other nearer than 1017 cm.
Summary
From the Atomistic Theory of Matter, http://www.atomsz.com, we have learned that mass is mass and energy is energy. Matter has two kind of mass: the conserved gravitational mass and the changeable inertial mass and they are different. But, in any case, energy is not a conserved property of matter. The energies of physical systems are not conserved since only nonclosed physical systems exist. The energy is neither equivalent to mass, nor is energy quantized. Quantized are only the elementary charges, qi ={±e} and gi ={±g∙mi}. The interactions between particles are caused by conserved elementary charges and are continuous functions; the interactions are nonconservative and propagate with the constant velocity c.
Gyula I. Szász

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