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Quantum Mechanics considers the duality wave-particle through the interpretation proposed by de Broglie. The diffraction has been detected for the elementary particles, as electrons, protons, neutrons, molecules. Considering these experiments, we show here that there is a grave incompatibility between this solution of Quantum Mechanics and the Michelson-Morley experiment, if we replace the light by protons, and Michelson’s interferometer is replaced by a crystal.

#### Davison-Germer experiment

When an electron crosses a crystal, it can suffer diffraction according to the Bragg’s relation, which is: n? = 2.d. sen? .... [2.1]

Davisson, Germer and Thomson made experiments with ? = 65o , d = 0,91Å , and electrons with kinetic energy 54eV.

Through the expression 2.1 we get: ?= 1,65 Å .... [2.2]

The wavelength of de Broglie, for the electrons with energy 54eV used at the Davisson-Germer-Thomson experiment, is:

? = h/p = 6,6x10-34j-s/4,0x10-24kg-m/s = 1,65 Å .... [2.3]

Electrons with kinetic energy 54eV have approximately a speed 4.000km/s. As we see, the postulate of de Broglie gets the same result of the Bragg’s relation. According to the authors Robert Eisberg & Robert Resnick( 1 ), the electron suffers diffraction into the crystal because “there is a constructive interference of waves spread by the periodic arrangement of the atoms in the planes of the crystal ?. So, this constructive interference is a consequence of: d=0,91Å within the crystal, and the electron’s speed 4.000km/s.

In the experiments of diffraction electrons are used with speed 4.000km/s. But instead of using electrons we can replace them by protons. As the proton has a mass 2.000 times greater than the electron, then de Broglie’s wavelength of a proton with speed 2km/s will be 1,65Å. Then let us imagine Michelson-Morley experiment, made with a proton with speed 32km/s.

We will consider the Sun as a reference at rest. And in order to simplify the explanation, let's consider that the Earth's translation velocity around the Sun is 30km/s. So the crystal in our laboratory has a speed of 30km/s with regard to the Sun. But the Bragg’s relation does not depend on the speed of the crystal, in order that through his relation we get the value ?=1,65Å.

Now let us submit the protons to the experiment, when they are emitted in two directions. Let us analyze the two different directions of the proton’s motion in the experiment.

#### Michelson-Morley experiment for protons

In a new version of Michelson-Morley experiment, we replace the light by a flux of protons, and the Michelson's interferometer by a Davison-Germer crystal. Let's analyse such new version of the experiment.

1- First let us consider that the flux of protons is emitted with 32km/s in contrary direction of the Earth’s motion. The speed of the protons with regard to the Sun is 32km/s - 30km/s = 2km/s. So, by de Broglie’s relation we get a wavelength ?=1,65Å , and by the Bragg’s relation we also have ?=1,65Å. This means that the proton shall be submitted to the diffraction effect into the crystal, and we can detect the proton’s duality by the experiment.

2- Now consider the flux of protons emitted with 32km/s in the same direction of the Earth’s motion. The speed of the protons with regard to the Sun is 32km/s + 30km/s = 62km/s. Then, the proton has a de Broglie’s wavelength ?=1,65Å/31 = 0,055Å, while from the Bragg’s relation for the crystal ?=1,65Å. Therefore such proton cannot suffer diffraction into the crystal.

This is the result that we have to expect from the concepts of Quantum Mechanics. But suppose that we make this Michelson-Morley experiment for protons, and we get a result showing that the speed 30km/s of the Earth does not have influence on the proton’s diffraction, no matter the direction of the flux of protons with regard to the Earth’s motion. Clearly this experimental result does not fit to the concepts of Quantum Mechanics, as has been shown above. One can say that there is no paradox, because it is necessary to consider the velocity of the crystal with regard to the proton, i.e., actually it would be necessary to consider the relation ?= h/m(V-v), where V is the velocity of the proton, and v is the velocity of the crystal. With such argument, actually we are introducing the Doppler effect between the proton and the crystal. However such argument is valid only for pure waves, it is not valid for the de Broglie’s idea of duality. Let us show why.

Consider a proton with speed 30km/s. Its wavelength h/mv is ?= 0,11Å. And if we use a crystal with distance d= 0,06Å , from the Bragg’s relation we get ?= 0,11Å, and therefore in the laboratory we must detect the proton’s diffraction. This is the prediction according to de Broglie’s interpretation. But now consider that we make such experiment with the proton going in contrary direction of the Earth’s motion around the Sun. Therefore, with regard to the Sun, the velocity of the proton is Vp= 0. In such experiment, the proton is at rest, while the crystal has a velocity Vc=30km/s toward the direction of the proton. Unmistakably the proton is stopped with regard to the Sun, and this means that it does not have wave feature. The proton with Vp= 0 is 100% corpuscular, and therefore it cannot suffer diffraction into the crystal. So, the de Broglie’s interpretation is wrong.

Obviously we have a paradox. The duality, according to the interpretation of de Broglie, is not compatible with the Michelson-Morley experiment for protons. Let us call it Michelson-deBroglie Paradox. It shows that it is not correct the de Broglie's interpretation for the relation ?=h/p.

#### New interpretation for duality wave-particle

Instead of being a property of the matter, it's possible the duality wave-particle may be a property of the helical trajectory of elementary particles as the electrons. The helical trajectory is known as zitterbewegung, which appears in the Dirac's equation of the electron.

From such new interpretation, the duality wave-particle is not a manifestation of the matter. Actually it's a property of the helical trajectory.

The de Broglie's intepretation on the duality wave-particle is related to the EPR paradox. Such relation is analysed in the book Quantum Ring Theory( 2 ), where it's proposed the hypothesis that EPR paradox can be solved from a model of photon composed by a particle and its anti-particle moving with helical trajectory

The de Brogi'es interpretation is related to several other paradoxes in Quantum Mechanics, since the de Broglie's postulate is the fundamental background of the theory. It is related to the inversion of logic in Schrödinger equation, and also to a paradox of the Bohr theory of the hydrogen atom (see Successes of Bohr's model in the Discussion of the article Bohr model : http://en.wikipedia.org/wiki/Talk:Bohr_model )

1- Cold fusion, Don Borghi's Experiment, and hydrogen atom: http://peswiki.com/index.php/PowerPedia:Cold_fusion%2C_Don_Borghi%27s_Experiment%2C_and_hydrogen_atom

2- Successes of the Bohr atom: http://peswiki.com/index.php/PowerPedia:Successes_of_the_Bohr_atom

#### Deleted from Wikipedia

See discussion for deletion:

Wikipedia:Articles for deletion/New de Broglie's paradox: