[Diogo_Afonso_Leitao (OP)]

Hello! My name is Diogo, I'm 14 years old and a couple months ago I was studying the mathematics of Quantum Physics to make a video about the subject and I found a new equation that describes the relation between frequency, wavelength and velocity of a particle with mass. This equation, however, gives matter's probability wave an impossible shape, as the frequency calculated seems too high to fit in the wavelength value. Here is the maths:

I was combining Einstein's equation for energy with Planck's equation for energy, which must give equal values because a particle's energy can be calculated with its mass or frequency.

E = √m²c^4 + p²c²
E = hf

√m²c^4 + p²c² = hf

In the particular case in which m = 0,

√p²c² = hf
pc = hf
hc/λ = hf
c/λ = f
fλ = c

I realized that this was the general equation for waves. But then I thought, why could I only find this equation when mass equals zero? In the equation, "c" can't possibly represent velocity, but only the speed of light, after all mass equals zero. Why wouldn't I get fλ = v if I accounted for mass? This intrigued me and kept me going. What if I make a relation between linear momentum in classical physics and in quantum physics? If I did that, I'd obtain:

mv = h/λ
mvλ = h
λ = h/mv

Amazing! I just deducted De Broglie's Equation. This means that the relation between the linear momentum in classical physics and in quantum physics has to be correct. But something always bothered me about this equation. What about light? Particles without mass would not have a definite value for wavelength. Something was wrong. This is when I thought about using "Energy" instead of mass.

E = mc² for rest mass.
m = E/c².

Then:
mv = h/λ
Ev/c² = h/λ
Evλ = hc²
E = hc²/λv

Wohoo! I just deducted a new equation for energy. I couldn't possibly know if this was correct, so I ran the mathematics. I calculated the energy of an electron moving at 3 . 10^5 with Einstein's Equation, with Planck's equation and with my equation. All of them gave the same result. I unfortunately lost the note block in which I noted all the results, but if this hypothesis shows any promise I'd love to run it again and post here.

But if that is the case, then Planck's E = hf is equal to my energy equation. In other mathematical words,

hc²/λv = hf
c²/λv = f
fλv = c²

Apparently, this is the general equation for waves with mass that I was looking for. Notice that if, and only if v = c, we get fλ =c. I figured that if scientists used fλ = v to convert frequency to wavelenght and vice-versa they would be making a mistake in their calculations, so this could be important. But then I noticed something very interesting. fλ = v represents all possible values for frequency and wavelength that allow something to have the shape of a wave. For instance, you could have a wave with frequency of 10Hz, with wavelength 10 centimeters, moving at 1 meter per second. But you cannot move at that same speed and have the same frequency but having a wavelength ten times longer. It'd be simply an impossible shape.

However, when I did the maths, I found out something awesome:
For an electron moving at 3 . 10^5, if we apply the normal equation:

fλ = 3.10^5. Using De Broglie's equation, we can calculate that the wavelength of the electron, which is:

λ = 6.626070040 . 10^-34 j.s/ 9.10938356 × 10-31 kilograms . 3.10^5 m/s.

This gives us a value of 2,4246316982 . 10^-9 meters. Now, if the electron has a possible wave shape, its frequency is:

f.2,4246316982 . 10^-9 = 3.10^5
f = 3.10^5/2,4246316982 x 10^-9
f = 1,2373013197 x 10^14

However, what if we use our equation for energy to calculate the frequency?

f.λ.v=c²
f.2,4246316982 . 10^-9 . 3x10^5 = 8,9875517874 x 10^16
f.7,2738950946.10^-4 = 8,9875517874 x 10^16
f = 8,9875517874 x 10^16/7,2738950946.10^-4
f = 1,2355899653 . 10^20

In other words, if the equation I deducted is indeed correct, particles with mass have an exotic shape that is not exactly a wave, because frequency and wavelength can't possibly fit together to form the typical wave shape. I'm not a fan of extra dimensions, but maybe this shape is possible if there are extra dimensions of space. Therefore, this could be a mathematical evidence of extra dimensions. In general, I'm still thinking about how to solve this problem =)


I also obtained the result above when I calculated the energy of this particular electron with Einstein's equation and then used the equation E = hf to calculate its frequency.

Also, we can "correct" De Broglie's Equation to make it work for light and non-massive particles too.

Evλ = hc²
λ = hc²/Ev.

Done! =)
I wrote down in my board for months trying to disprove the Equation and finding an inconsistency but I just cannot find any. This is why I'm posting my hypothesis in here. Your feedback would be of immense value! Thank you very much for the attention. I wish you all a very nice day.

Sincerely,
Diogo.

[Alex Dullius Siqueira]

 Silence is a good sign...

 One question, have you ever try to consider, that the "impossible shape/configuration" you achieved on you calculations, as not being a "Real final product"?
   What I mean is, perhaps you could try this:
  Something that is impossible to achieve, is impossible, and impossible should not exist...
 Now consider this, something that "lures" at the very edge of the "impossibility frame", and that is constant...

 I mean that your impossible shape for light could be in fact precisely that a Real" impossible shape that is able to exist, because the source (space/C) is constantly "recycling" the attempt of the impossibility....

  I'm suggesting that you could try to consider that photon is infarct impossible, and but since is a constant attempt happening at C, Space, is able to "keep the impossible shape happening" but never "complete", so the impossible shape may be like 99,9% of a particle, and also 99,9% of what is not a wave...

 I can't produce math, nor help on your progress, there are other member that does and will, reasons why I suggested that the silence here is a good sign...
   I do understand, that you are adding something to represent "probability wave", and it seems to you as an error/impossibility. I suggest that n a similar way you could try to add something else to represent the the both wave/particle configuration, as being precisely that, impossible...
 
 Why do you assume, or not, that photon is "complete particle"?
I ask because I relate the mass less state and the propagation with C, as being directly related with the "incomplete particle"...  And something that by any reasons is , on space, a "almost particle", should be something that is half of both, 50% particle and also 50% space property... And also consider that being hybrid, photons would theoretical be able to "temporarily" have more than one point of reference at the same time...
 The wave behavior would be the 50% point to point of space(A to B/Linear C)
 The particle behavior would be using electrons as "anchor"(powering up them), sub consequently electron cannot left being an electron so it releases "the plus energy" the photon would than happen on space and from it around the energy, at C with no delay...

  Try to implement only to see what it takes: The impossible shape becomes "possible", as long, space is "constantly" trying to achieve impossible, keeping the photon wave/particle configuration, at 50/50(probability)... It's a wave "and" a particle. It is a "massive particle", but it is also an "almost particle", the possibility of photon being existing on the "almost" allows it to be impossible and real...
  This is only viable if one accept that "particles" are but configurations of quanta, determinate by C...
  And inside such scenario, particles are all "probabilities" the describe photon on the "almost something" also is the reason of why the electron releases a photon... A photon cannot become an electron, but it can surrender its quanta to it, also an electron cannot become something more than an electron, and by doing so it releases the energy, and with that energy space will make another photon...
 The electron did not released a photon, ti released the quanta of energy, the photon is produced on space at C on that moment of the liberation...
   Photon is not a particle, but a "almost particle" that due quanta cannot fulfill C requirements, so it is energy, and being so, space will maintain the "almost particle" constantly at C, this will enable the photon to exist "still" being mass less, and for this reason it can also experience linear C(Penrose diagram), it does not have a self reference for it had not developed mass, but is constantly "almost doing" so it behaves like a particle, and borrows the electron on mater as its own reference. And on the perfect vacuum it is borrowing space A to B at C as reference...

 W(wave) goes for space(C).
 Q(quanta) goes for the amounght of energy "for" the photon attempt
 Light would be a self propagating "almost particle" that due the "almost" is constantly (achieving and loosing) the proportional mass, and is at C...
  Now if photon was to become a particle it would not cause the "visual effect/light) that is does...
  I sugesting that that effect (light) is the photonic mass that is (being lefting behind at C), light is a visual effect of the visual failed attempt...
 As long photon is still not "enought" to become a particle(as the electron), it will be constantly developing mass at C as it is loosing mass at C, if that speculation is correct, space is not at C but at twice that speed...  I wondering here that if light is visual evidence of a failed atempt into produce a stable photon, that is gaining and loosing mass at C, and this cause the photon to propagate itself along with C/spacetime(Penrose diagram), is to accept that space is not at C, but precisily twice that state...
  If the photon developed mass, it would take time, to loose it, and if it does, the speed of light is half of the speed of space time...
   Perhaps rises the question if Dark matter(visible effects) are not on the effects themselves (mass left behind C) of a hudge amounth of light...

 The mass of the "almost particle" photon, occurs outside the photon, last as long as C on the frame, stays behind of it, allowing it not to travel with C, but instead allows it to stood still from A to B "along with space at C...