Symmetry and de Broglie’s “wish”

Now, hear me out. I’m sure we’ve all heard about the de Broglie’s relation (or probably wave-particle duality) which related wavelength, classically associated with waves, with the linear momentum, associated with a particle, for all quantum particles.

This expression is more beautiful and that h is the Planck constant, and it shows up pretty much everywhere in quantum mechanics. Now the interesting thing I just found out, is not new or revolutionary in any way, but it really shows a beautiful connection in physics, that echoes in many other occasions.

Now the first things we need to establish are these:

This is a fundamental equation in special relativity that describes the total energy of a body, by relating its rest energy, our classic , with the kinetic energy, represented by , p being the momentum of the particle. For a photon, since it has zero mass, you can even simplify it to be

Another important thing we need to know is the Planck-Einstein relation, that says

This expression represents one of the fundamental principles of quantum physics. Energy is related to frequency by a constant, referring to the quantum nature of light.

Essentially, what you need to know is that light is in discrete packets of energy and obeys special relativity. Cool, both of these expressions are very easy to write down and are very short. Kinda cute, actually. Just for fun, let’s make one equal to the other, after all, they are both equal to energy, right?

Uh-uh, cool, but that is not quite what we had before, right? Sure, that is true, but here is what I didn’t told you before. A couple of decades before anyone had even thought about quantum mechanics or special relativity, there were smart people working out the speed of electromagnetic waves (this fancy club includes light), which happens to be c, and some smarter people than me where able to figure out that

Or that the frequency and wavelength of a electromagnetic wave were related by this constant speed at which they were moving. So replacing c with that expression in the one above and cutting the frequency, we can then get to

And here we go, back to our original expression. All of this made with relatively simple algebra, and no need for advanced mathematics, and yet with these few steps, we pretty much proved the wave-particle duality for light (and other electromagnetic waves). The catch here is that for typical particles, we can’t just eliminate mass for them, so we may not be able to just reach that conclusion through simple methods. Thankfully de Broglie already did that for us, so we know this relation is valid for electrons and whatnot.

Now you could be asking why de Broglie thought of applying the wave-particle duality to regular matter. The quantum nature of light had already been established by Einstein and Planck, but nothing seemed to suggest the same for matter. Heck, there were even those that opposed the atomic interpretation of matter, let alone them also having wave-like properties. The main reason why de Broglie thought there should an equivalent wave-particle duality in matter, was due to symmetry. Hence the title of this post. His quote “My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons.” seems to hint that intention. And one can notice the beautiful parallels that naturally led us to a conclusion that fundamentally changed the perception of reality. The continuous fight in physics is a mere fight for symmetry.

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