Confused on the premise behind De Broglie's hypothesis

[DanielR1]

TL;DR Summary

How does the premise that lamda=h/mv make sense? The derivation doesn't seem mathematically correct.

I'm confused on how De Broglie's hypothesis works. I've attached my thinking in a pdf file below.
I'm not an expert in the field of particle physics(I'm only in high school), so there may be some error in my logic(I really think so, or else De Broglie was wrong!(hint:he isn't))
I've also read that Method 1 is wrong as a whole, but I decided to include it anyways. I think method 2 is the stronger argument though.
Any particle physics expert know the origin of De Broglie's hypothesis, and know the error I have made? I would really like to know, this has been bugging me for a bit.
Thank you!
*edit: I made a small but significant math error, I just fixed it and it should be okay now(but the problem still persists, just slightly differently)

 

[mfb]

E=mc² is a formula for the rest energy. It only applies to particles at rest.
E=hf is a formula for moving things.
Applying these two at the same time cannot work.

p=mv is only an approximation at low velocities, if you want to derive anything use the relativistic $p=\gamma m v$.

$v_p = f \lambda$ measures the phase velocity of particles, this is different from the group velocity (i.e. how fast the particle goes from A to B). You are mixing the two different velocities.
The phase velocity for massive particles is always faster than the speed of light. This is not in conflict with relativity because nothing moves forward at the phase velocity.

 

[DanielR1]

E=mc² is a formula for the rest energy. It only applies to particles at rest.
E=hf is a formula for light. It cannot be at rest.
Applying these two at the same time cannot work.

p=mv is only an approximation at low velocities, if you want to derive anything use the relativistic $p=\gamma m v$.

$v_p = f \lambda$ measures the phase velocity of particles, this is different from the group velocity (i.e. how fast the particle goes from A to B). You are mixing the two different velocities.
The phase velocity for massive particles is always faster than the speed of light. This is not in conflict with relativity because nothing moves forward at the phase velocity.

I understand. Do you know of any way to derive the equation then? I've looked all over the internet and all the derivations are simple and wrong(most of them use E=hf and E=mc^2 in conjunction).
Also, regarding vp=f*lambda, why wouldn't it be the same as kinetic velocity? If the wavelength, a distance, is multiplied by a frequency, wouldn't that yield a velocity identical to that if you multiplied a length by a frequency(or divided by a corresponding time), since it measures how much times per second a particle/wave travels through a distance of a wavelength.

 

[mfb]

You cannot derive it from relativity because it's a formula of quantum mechanics. The formula was developed to match observations.

Also, regarding vp=f*lambda, why wouldn't it be the same as kinetic velocity? If the wavelength, a distance, is multiplied by a frequency, wouldn't that yield a velocity identical to that if you multiplied a length by a frequency(or divided by a corresponding time), since it measures how much times per second a particle/wave travels through a distance of a wavelength.

That's not how waves of massive particles travel.
Here are some animations of waves where phase velocity and group velocity differ.

 

[DanielR1]

You cannot derive it from relativity because it's a formula of quantum mechanics. The formula was developed to match observations.
That's not how waves of massive particles travel.
Here are some animations of waves where phase velocity and group velocity differ.

Oh, so the formula is based off of observations and experiments, not derived from pre-existing formulas?
What I'm asking is, is there any way to make sense of it mathematically, even if not a derivation?(maybe derive it from formulas found later), or is it simply a model to fit our observations?

 

[mfb]

It was largely based on the Bohr model which introduced the concept of a "length scale" of electrons. De Broglie found a relation that interprets this length scale as wavelength, and the same relation also works for light (where we know the wavelength).

Here is a quick derivation that the phase velocity of massive particles must be larger than the speed of light:
$v_p = f \lambda = f \frac{h}{p} = \frac E h \frac h p = \frac E p = \frac {\gamma m c^2}{\gamma m v} = \frac{c^2}{v} = c \frac c v > c$

(I just saw that my comment about E=hf in post 2 was too restrictive, but it's still meaningless for stationary particles so the result doesn't change)

 

[vanhees71]

The problem with the heuristic argument by de Broglie is what Debye told Schrödinger during a colloquium at the univeristy of Zürich: If there are matter waves, better find a wave function! The result was a series of papers worked out during Schrödinger's summer vacation with his (anonymous) mistress ;-)).

Given the Schrödinger equation for a free particle (for simplicity I consider 1D motion only)
$$\mathrm{i} \hbar \partial_t \psi(t,x)=-\hbar^2/(2m) \partial_x^2 \psi(t,x),$$
you can ask for plane-wave solutions,
$$\psi(t,x)=A \exp(-\mathrm{i} \omega t + \mathrm{i} k x).$$
Plugging this into the Schrödinger equation you get
$$\omega = \frac{\hbar}{2m} k^2.$$
That's the dispersion relation for free Schrödinger waves.

Multiplying by $\hbar$ yields
$$\hbar \omega=E=\frac{\hbar^2}{2m} k^2=\frac{p^2}{2m}.$$
This is as for a Newtonian point particle, and de Broglies heuristics was the opposite way, using the idea that $E=\hbar \omega$ and $p=\hbar k$ from "old quantum mechanics" in the context of Planck's and Einstein's "photon concept" concerning em. waves and the "wave-particle dualism".

Characteristically Schrödinger used a more advanced heuristics, using the analogy between how to derive "geometric optics", which is a more "particle like" description of the propagation of light in the sense of "light rays" from "wave optics" or as was clear since Maxwell's and Hertz's theoretical and experimental work on electromagnetic waves, the electromagnetic theory of optics. The answer is "singular perturbation theory" aka. WKB (Wenzel-Kramers-Brillouin) method, i.e., ray optics follows from wave optics using the eikonal approximation for light of wave lengths small compared to all relevant geometry of the obstacles and openings relevant for the propagation of the light waves.

Taken then the corresponding Hamilton-Jacobi partial differential equation, which is the natural "geometric-optics description" of classical mechanics as the eikonal approximation of a yet to be found "wave-optics description" for non-relativistic particles, lead him to his wave equation for non-relativistic matter waves.