For whatever reason, I was thinking about the most profound idea or concept I've ever heard and learned, and one does stick in my mind: the most famous equation ever written down, E = mc^2.
It certainly does not look like much. Most people you ask to name an equation will state this one, and almost none of them will understand what it means or implies. It is just sort of there, thanks to Einstein. But what does it really mean?
I look at it this way - and it has helped me make sense of a good chunk of modern physics, which includes quantum mechanics. Keep in mind this is coming from special relativity, from 1905.
This states in no uncertain terms that Energy = Matter. It is equivalence. This means that whatever matter can do, so can energy; and whatever energy can do, so can matter. The properties of one must be the properties of the other. The one exception, I suppose, is that energy (i.e. photons) move at the speed of light, whereas matter cannot.
To me, this alone helps me understand why there is particle-wave duality and the uncertainty principle.
For instance, energy has wave properties, in particular diffraction and interference. If this is true for energy, then necessarily it must be true for matter. Particles like electrons have wavelengths and can diffract, and we know this because people use electron microscopes all the time. Another example is matter has momentum, which we have always been taught since Newton is p = mv. This must mean energy has momentum...but wait, energy has no mass! Einstein figured out that p = E/c for photons, and this has been confirmed by something called the Compton effect; NASA also wants to make solar sails, and use the momentum of sunlight to help propel a spacecraft. DeBroglie dervied his famous equation from relativity and this principle for matter waves, wavelength = h/p.
E = mc^s changed geopolitics and the course of human history forever, as nuclear reactions, power plants and bombs literally changed the world in just a few short years in the 1930s and 1940s.
This equation effectively outlines a phase transition, where matter can change to pure energy, or energy can change to matter. This reminds me of ice and steam being able to change back and forth into each other through phase transitions due to temperature. Ice and steam certainly don't look like each other, but upon closer inspection they are two forms of the same stuff, H_2O. Matter and energy are two forms of the same stuff.
But this notion of a phase transition goes even deeper. Our universe exists because of it! The Big Bang was a burst of pure energy. Because of this phase transition and equivalence, as the temperature cooled, energy transformed into matter and the forces of Nature. My own background in particle physics at Fermilab relied on mini-Big Bangs, where protons and anti-protons annihilated into pure energy, and then back into matter. We produced top quarks and hundreds of other particles from this phase transition process on a daily basis!
So let's summarize. E = mc^2 holds the equivalence of matter and energy, explains wave-particle duality and the uncertainty principle, changed our world politically forever through an understanding of nuclear processes, and explains how a phase transition produced our universe. Not bad for one little algebraic formula anyone can quote.
For some derivations as to begin to see where this comes from, check out one of my videos. There is another video on where the notion of antimatter comes from, momentum of photons, and matter waves.