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.