The fundamental concept of this theory is that luminous energy appears as packets called photons or quantum of light, each with the value given by Planck's relationship:
E = h.n ==> n = c/l,
Where, n is the frecuency; h, the Planck's constant; c, the speed of light; and l, the wavelegth.
The analysis done by Planck concludes this concept, and it has been confirmed by experimental tests.
Also, we have obtained that the energy of photon can be expressed by the following relationship:
E = h.n = p.c = p.l.n, where p is the momenutm of photon and c, its speed.
From here h, is equal to:
h = p.l
This indicates that product of photon's momentum times its wavelength, according to Planck, is constant.
These two relationships indicate that particles in motion have two patterns: their wave and matter character. As we have observed, relativistic interpretation does not take in account the wave character, so in this sense it is incomplete.
Nevertheless, given that momentum is continuosly variable, energy has a continuos character, in the sense that it can take all the possible values. Such continuity can be observed also in the quantum of light through the continuous variability of frequency.
What is noticeable from Planck's result is that each photon has a packetized energy, giving as consecuence that total energy conveyed by a stream of photons, at a determined frequency, equals the discrete sum of all these elemental packets, and furthermore, that there do not exist fractional values of these elemental packets of energy, at that frequency. Namely, the packetized character of energy is a natural characteristic, as it is the constancy of speed of light. This character is also viewed in the electric charge as another presentation of the quantization in nature. Namely, the universe has its mysteries that by little and little we are discovering and giving us the way to correctly interpret it.
These facts, may be, lead De Broglie in 1924 to make his proposition of the wave-particle duality of matter, which we will discuss now.
De Broglie's Wave-Particle Duality (1924).
De Broglie in 1924 established a wavelength associated to a mass that should satisfy the same relationship as that for photon. Let's see, for photon:
E = h.n = p.c = p.n.l ==> p = h/l
From these facts experimentally demonstrated, he extrapolated that a particle, with no null mass at rest, should have a wavelength l and a frequency n given by:
l = h/p and n = E/h
where, h is the Planck's constant, E is the particle's kinetic energy and p is the particle's momentum.
Until now, everything seems to be "reasonable deductions", and experimental tests had demonstrated that particles as electrons, protons, neutrons, etc. showed features of waves.
Nevertheless, let's discuss the De Broglie's statement.
The wave associated to photons, with null mass at rest has a constant velocity of 299,790,000 m/s, a characteristic that results tracendent in nature.
Because the energy of a photon at rest is null, the energy of a photon is kinetic pure. So, a correct way to express it, is:
K = h.n
But, when we refer to particles with no null mass at rest, according to our obtained expression in Part 2, kinetic energy is given by:
K = Mo.c² - m.(c² - 2.v²) = 2.p.v. - m. c² + Mo.c²
Which reduces, at low speed of particles, to Newtonian kinetic energy: m.v²/2.
By now, let's take the Newtonian kinetic energy as a starting point for moving particles, in order not to establish differences with Quantum Mechanics.
It is highly known the relationship between the speed with wavelenght and the frequency :
v = l.n
The mathematical analysis of the displacement of a wave in the space (light, sound, fluids, etc.) at a velocity v, leads to this conclusion. Let's establish that, because wave is another characteristic of the mass in motion, it, obviously, should travel at the same velocity of mass. Additionally, De Broglie stated that the product of the wavelength times the momentum, for any particle, is constant, extrapolating what was established indirectly by Plank for photon:
h = p.l
If we introduce the relationship of the wave's speed with its wavelength and frequency, v = l.n, together with h = p.l, in the expression of particle's kinetic energy K = h.n, we must obtain:
K = p.l.n = p.v = m.v²
Which doubles the value of the Newtonian kinetic energy!. Obviously, this is an erroneous result. Then, necessarily the acception of the constant product of the momentum times the wavelength is not correct. We believe that the error had been coming from the erroneous definition of Energy given by Einstein: E = m.c² - Mo.c², which could lead to K = m.c², for any particle. Then, De Broglie actually assumes that K = p.v, because it is apparent that there is not any reason for not to do it. From here, accepting this way of reasoning, it is easy to conclude that h = p.l. After this De Broglie's statement, contradicting the very most simplest acception of kinetic energy, it was invented the concept of group velocity of the matter different of that of the phase velocity of a wave, in order to smooth such contradiction!. It seems that, sometimes, Physics accepts anything, in order to justify any brlliant idea (De Broglie's statement was a smart one, although not completely exact), that differs from basic physical concepts!.
Schrödinger's Equation (1926)
Schrödinger established another equation for the movement of a particle. We believe, that his original purpose was to correct and enlarge the wave-particle interpretation done by De Broglie in 1924. After his work, defenders of quantum theory gave to Schrödinger's equation a probabilistic interpretation with which Shchrödinger, its creator, was not in agreement (See the opposition of Schrödinger to Born about this aspect in: Einstein versus Bohr, The Continuing Controversies in Physics, page 93, Mendel Sachs, 1988).
In a similar way, as De Broglie did the relation of the electromagnetic wave with particle's momentum starting from photon's momentum, we believe that Schrödinger's primary idea was to relate particle's momentum with the wave number appearing in the wave equation of a string, air or electromagnetic waves. In all these cases the equation is the same.
Let's see as an example of what we state, the "intuitive" deduction of schródinger's equation, as it appears in FISICA Vol. III Fundamentos Cuánticos y Estadísticos. Alonso&Finn, Pag. 61:
The referred wave equation in one dimension (reduced, because the complete wave equation is d²x/dt² - v².Ñ.x = 0, where v is the wave speed) is:
d²x/dx² + k².x = 0, where k = 2.p/l, is the wave number.
whose solutions are: x = ejkx, and x = e-jkx
In where: p = `h.k, and `h = h/(2.p)
As we can observe, Schrödinger also admits that h = p.l, which, as it was concluded, is an erroneous concept.
In this way wave equation becomes:
d²x/dx² + (p²/`h²).x = 0
In classic physics: p² = 2.m.(E - Ep), where E is the kinetic energy plus the potential energy:
E = m.v²/2 + Ep = p²/(2.m) +Ep
As we know from a relativistic point of view there is error in it, because the used expression of energy is not exact. Substituting this "energy" in the wave equation and the "constant" product between the momentum p and the wavelength, as it is done in Alonso&Finn, it is obtained that:
- [`h²/(2.m)].d²x/dx² + Ep.x = E.x
which is the first Schrödinger's equation for one dimension (not dependent of time).
Let's do the following comment: At that time, the reason for using the classical equation of energy in quantum theory is that Einstein's relativistic expression of kinetic energy does not depend directly and explicitly on the velocity of particle. It makes enormously complicated the presentation of quantum theory based on relativistic relationships. In here starts the first differences with relativistic theory, and quantum theory divorces and presents itself as a different conception. Although genius P. A. M. Dirac developed mathematically the Schrödinger's equation based on the Einstein's expression of energy with its solutions!. But, we have shown that Einstein's energy equations are not correct, as it was concluded in Part 2 of this work, and if this work is correct, then Dirac's effort was useless.
But, as we have observed and have presented in Part 8, there are a lot of erroneus acceptions in all this development.
May be, the erroneous concept of h = p.l, introduced in the Wave Equation for obtaining the Schrödinger's equation, as well as the Newtonian expression of the energy, are some of the reasons for not to arrive at exact results, but to approximate ones; a fact, that probably induced to Heisenberg to establish his uncertainty principle, and also to others to give the probabilistic interpretation to the Schrödinger's equation!.
Modified Schrödinger's Equation
Now, let's try to get the Schrödinger's equation, following the same known procedure starting from the reduced wave equation, but in this opportunity using our equations, according to us, correct ones.
As we know from Part 2, kinetic energy is given by:
K = Mo.c² - m.(c² - 2.v²) = 2.p.v - m. c² + Mo.c²
By forming total energy, including the potential energy and the internal energy: E = Mo.c² + K + EP, we obtain:
E = Mo.c² + K + EP = 2.p²/m - c².(m - 2.Mo) + EP
From where, p² = [E - EP + c².(m - 2.Mo)].m/2
On the other hand, let's try to find a relationship between the wave number k = 2.p/l and the momentum p, through the expression of the wavelength l. We can put the expression of kinetic energy as:
K = h.n = 2.p.v - m. c² + Mo.c² = 2.p.n.l - m. c² + Mo.c²
From where,
l = [h + c².(m - Mo)/n]/(2.p)
So,
k = 2.p/l = {2.p/[h + c².(m - Mo)/n]/(2.p)}
Simplifying, and introducing that `h = h/(2.p) and that w = 2.p.n, we arrive at:
k = 2.p/{`h + (m - Mo).c²/w}
By substituting this expression of the wave number in the reduced wave equation for only one dimension, we obtain:
{[`h + (m - Mo).c²/w]²/4}.d²x + p².x = 0
and putting the previous obtained expression for p², p² = [E - EP + c².(m - 2.Mo)].m/2, we arrive at:
Let's continue to arrive at the general expression of Schrödinger's equation (space-time dependent). Following a similar procedure as that presented in Alonso&Finn we will concerned in looking for a wave function depending on space and time, and also for an equation, such that after taking the derivative of this function respect to time we obtain the original spatial equation. May be this was the way used by Schrödinger to obtain his equation. Such equation, as it can be observed, is:
E is the total energy and it preserves constant. Substituting the expressions of the first and the second derivatives in the general equation, we obtain the original spatial equation:
Thus, the Schrödinger's equation expressed with the operator will be:
Hx = E.x
The classical hamiltonian (total energy expressed in function of momentum and the system coordinates), which now can be expressed as a relativistic hamiltonian, given that kinetic energy depends directly on velocity, is:
HRELATIVISTIC =2.p²/m - c².(m - Mo) + EP(r)
Given that the function Y is the same as that known, the following operators are obtained:
Total Energy operator: 2.p²/m - c².(m - 2.Mo) + EP ==> (2.`h²/m).Ñ - c².(m - 2.Mo) + EP
As we observe, only the energy operators are slightly modified, numerically speaking.
CONCLUSION
The new definition of mass obtained in Part 1 of this work, influences all physical magnitudes depending on it, as energy, momentum and angular momentum.
Matter exhibes a double behavior: as a particle and as a wave, and as we have seen, relativistic theory is a correct view of the particle character of matter, which correctly introduced in the basic wave equation leads to a complete interpretation of matter.
This development unifies Quantum and Relativistic theory.
We could introduce the relativistic concepts in the wave equation in a simple way, thanks the direct dependence on velocity of the kinetic energy present in our obtained equations in Part 2. We believe, according to this work, that we can develope the Quantum-Relativistic view, macro and micro, of matter in an unified, exact and simple manner.
Get your own Free Home Page