Correct formulation of wave-particle duality
w: wave speed, f: frequency, λ: wavelength, h: Planck's constant, p: light's momentum, m: inertial mass, C: speed of light, E: energy, M: Gravitational mass.
What is de Broglie's matter wave?
In 1924, de Broglie, in his dissertation "Studies in Quantum Theory", presented a hypothesis of matter waves based on relativistic considerations. He postulated that a mass of energy with a rest mass m₀ has internal oscillations of the frequency ν₀ determined by the formula hν₀ = m₀c², and a moving body of momentum p has a hypothetical incidental wave determined by the formula h/ν = p. The wave rays is in the direction of the orbit of the moving body, and its group velocity is equal to the velocity of the moving body. From this matter-wave hypothesis, the quantum conditions for the Bohr's model of the hydrogen atom could be easily derived.
If we treat the quantum hypothesis as a matter wave rather than an equivalent principle of light momentum, the following paradox arises.
But then it would be natural for students to think that the relationship E = hν between the energy E of a photon and the frequency ν is also applicable to matter waves. They also tend to think that there is a relationship between the frequency νB and the wavelength λB of the matter wave of a particle of velocity v. v = νBλB (or, confused with the photon, c = νBλB). These fall into the following paradox.
This de Broglie wave is also different from the momentum of light (h f／C) of Einstein's light quantum hypothesis.
From the fact that it is Einstein who develops the theory of duality of particles and waves of light and derives that the momentum of a photon is hν/c, E=hν represents the wave nature of the motion of an object. It should be called "Einstein-de Broglie's equation" together with de Broglie's equation λ = h/p. However, I wonder if Einstein's name someday went down and was called "de Broglie's formula", so that it could not be explained at the same time as E=hν?
In the quantum relativity principle
Quantum energy is equivalent to total energy (rest energy + kinetic energy), so in quantum relativity principle, E = M C² = h f.
A kilogram is the mass equivalent to the energy of a photon whose frequency is {(299 792 458)²/6.626 069 57} × 10 ³ ⁴ Hertz [31].
When this is calculated from the equivalent principle of light's momentum (p＝h f／C＝M C＝m w), it becomes the wave speed (w＝f λ＝h f／[m C]). But the de Broglie matter wave wavelength formula (λ＝h／[m v]) is neither the particle velocity (v) nor the light velocity (C). In other words, gravitational mass (M), which is an increase or decrease in energy, and inertial mass (m), which is a change in scale, cannot be distinguished.
If we remove the time component（f） from this wave velocity, we get an uncertainty relationship between the inertial mass and the wavelength（h／C＝δm δλ）. This means that the quantum size spreads indefinitely toward infinitesimal to infinity until time is fixed.
The hesitation of Max Planck.
Incorporating this Planck's constant (h) into the principle of relativity is a top priority for shifting the paradigm from classical theory, and originally Max Planck himself should have incorporated it.
In other words, Planck's quantum of action is 'When one is transferred from the present coordinate system to the coordinate system in motion according to the principle of relativity, in this case, all quantities of space, time, energy, etc., change, but they remain invariant.', which is what attracted him.
On the other hand, Planck had a hesitation that he wanted the Planck constant to be wrong, such as having to reduce to classical theory in the limit of（h→0）.
However, this was a forbidden means that directly violated the consensus that energy was continuous. Planck knew it so much that he later said, "It was a hopeless act," regarding the introduction of this quantum hypothesis.
Planck, who was also an educator, pushed forward Einstein's theory of special relativity so as not to deviate from classical theory, and stopped Einstein not to put his hand into gravity theory.
Those who praised Planck and other relativism early on were those who wrote systematic textbooks such as the "physics course."
Formulation of wave-particle duality.
If the principle of relativity incorporates Planck's constant, it does not violate continuity. Max Planck's worries were completely useless.
Prior to educators, scientists Planck had to promote a paradigm shift by formulating the duality of wave particles before staying in the classics.
Professor Miyazawa says,
The word duality should not be used in science unless its meaning is mathematically defined.
I think that this quantum theory of relativity finally made it possible to formulate the wave-particle duality.
Professor Miyazawa is critical of field quantum theory and string theory, and says, "Currently, it is in the early quantum theory stage on the eve of the establishment of quantum mechanics."
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