You might have expected a simulation here. This is actually a rederrivation of quantum mechanics, because there are already suprisingly good simulations of quantum computing and quantum particles, and since I currently do not know how to use WebGL and WebGPU doesn't work on my chromebook, I'm not making a simulation for now. Maybe someday. Hopefully, you can actually understand some quantum mechanics here. Anyways, onto the stuff.

Our only special starting assumption is that E=hf. E is a particle's energy, h is a really small constant, and f is a frequency. A frequency of what? Well, this equation (the planck relation) is supposed to be for light, so it'd be the frequency of the electric (or magnetic) field, but we'll assume that every particle has some "frequency" of something. If this seems familliar, yes, it is, as this is what de Borglie did.

To find the frequency instead of the energy, we can divide by h to get f=E/h>. Since h is really, really small (in SI), f must be really, really large. In fact, for a single cat (assume the cat is exactly 4.082 kg), using E=mc^2 to get energy, we get...

5.5368×10⁵⁰ Hz.

Our cat's period, then, is less than a single QUECTOSECOND, which is the smallest SI unit of time. It's actually less than a single Planck time, but we haven't rederrived that yet and that's quantum mechanics. Still, my cat may be fast, but not THAT fast. How about we try a single electron? We get...

8.09329978 zs.

8 zeptoseconds is very small. However, we must remember that small things are fast: a protien can fold in literal femtoseconds, so this isn't unreasonable for an electron. In fact, it's actually more than a single Planck time, but again, we haven't derrived that yet. But how should we even describe an electron?! Well, we need something that waves (clearly), and it should be a field (electrons have interference). Since the electrons are wavy, we'll call this a wavefunction, and denote it as ψ, because it's a kinda wavy letter... ok actually I'm choosing these out of convention but oh well. Good thing someone tried to solve a weird cubic a long time ago, and made:

Complex Numbers!

The guy solving a cubic had to take the square root of a negative number: find a number that, when squared, equaled a negative number. After he failed but was able to find a solution by trial-and-error, he took the square root of -1, failed, and then just set i equal to the square root of -1... well actually, a square root, since x² = (-x)². Thus, we got i² = -1. From this, we can derrive that (a+bi)+(c+di)=(a+c)+(b+d)i (addition is componentwise), (a+bi)(c+di)=(ac-bd)+(ad+bc)i (by FOIL), a+bi/c+di=(ac+bd)/(c²+d²)+(bc-ad)/(c²+d²)i, and also add a new operation, where (a+bi)*=(a-bi). Yes, * is now used to do that. Anyways, a useful thing to do is graph (a+bi) as (a,b) (in cartesian coordinates) to make the complex plane: addition is moving, multiplication is rotation and scaling, and * (conjugation) is flipping across x.

For exponents, it's really hard, however we can somehow use mathematical wizardry to expand e^x into an infinite polynomial and find that the imaginary component (in (a+bi), b) of a complex number causes rotation when you use it as an exponent. This leads to e^iπ=-1. However, if we just increase t in e^it we get a counterclockwise rotation of t radians (where e^iτ does nothing and τ=2π). At last, we found it: a way to make a wavy thing with math! And no, I will not make the exact math too necessary whenever it's beyond algebra plus the stuff explained here.

Let's say our wavefunction takes in position and time, and outputs a complex number to repersent our electron: we'd write it as ψ(x,y,z,t). We'll say 0 is no electron, e^it for any t is yes electron, and the probability of finding an electron is the content (in 3d space) of a graph of ψ*ψ (* is always conjugate now, ψ*ψ is the squared distance from 0 to the value of ψ, oh and yes this graph would be 4-dimensional) that is only plotted in the region we want to find the electron. In way simpler and more useful words, the further ψ is from 0, the more electron there is there, and you shouldn't expect an electron to be found right in the near-exact middle of a really really small box even if ψ is high there. Ok, really I shouldn't say "more electron", but "higher chance of an electron being in a region near a given point" is very long to say type.

We'll also need a way to describe how this function's output changes when its inputs change. Well, this is like asking for slope. We'll write δψ/δ[something] for how much ψ changes (δ is change) when [something] is increased a really small amount. Sometimes, this makes literally no sense to even mathematicians (it's undefined), in which case somehing has gone wrong with the wavefunction. Anyways, δψ/δx is really just saying "if I move in +x, how much times more will ψ change", or looking for the slope of a line that approximates ψ at a specific value of x. It should be noted that δψ/δx is also a function of x, y, z, and t, and we assume y, z, and t aren't changing when doing the math: only the change when x changes matters. We can, of course, do things like δψ/δt, and if we want we can do δ(δψ/δt)/δt, which is basically always written as δψ²/δt². If δ[something]/δ[something else] is only dependent on [something else], then we'll write it as d[something]/d[somehing else], because it's simpler. If you've taken calculus classes, you may notice I just explained partial darrivatives. Ah well.

There's another tool, written as ∫. For example, ∫_a^bf(x)dx basically is the area under a graph of f(x) (other variables would be held constant if it's more like f(x,y,z,t)) only taking from a to b, except if it goes down it's negative, and if you have complex numbers as output it's more like the function's value is a velocity. "dx" basically just says which axis is the x axis. Sometimes, you'll need to use more ∫s if you want to repersent volume under the curve, content (4d volume) under the curve, and so on. How is this relavent? Well, for now, all you need to know is that ∫_-∞^∞(∫_-∞^∞(∫_-∞^∞ψ(x,y,z,t)*ψ(x,y,z,t)dx)dy)dz = 1: you'll find the electron if you look hard enough. It may be 729 quintillion lightyears away, but if you can somehow search that far and beyond to infinity, it's there somewhere. Also, ∫ technically doesn't need extra stuff aside from the d[something], but then you get a +C and stuff, and also yes I am teaching you more calculus, trust me this is important and I'm not trying to give you homework (unless I nerd sniped you a while ago)-

Ok, so finally, we can try to model an electron. From relativity, we know that if we have a lot of things that blink on and off "synchronusly" really fast, then if we move really fast (or if they do so) space and time interchange a bit (not fully!) and we see moving stripes. Thus, an electron that is moving should have a "phase gradient" as all of it moves. You know what else moves? A string! A string's equation in 1d is dy²/dt²=c²dy²/dx², where c is the wave speed in the string. In other words, if there's a curve (dy²/dx²), the string accelerates (dy²/dt²), and the more it accelerates the faster waves travel in it (c²). In 2d it's written as dy²/dt²=c²Δu, where Δ[something] means you sum up the curvatures along each axis. We kinda want our wavefunction to spread too, because it makes things more wavy, and because of complex numbers we'll multiply things by i. Also, if the phase increases, we want the electron to move in that direction. Now, we have:

iδψ/δt=-Δψ

We now need to make the electron go around every zeptosecond if it starts at 1, meaning it would change by iħ/τ:

ih/τδψ/δt=-Δψ

We might as well write h/τ as something like ħ, because how wouldn't it appear again, and also an electron's wavelength should be h/(mv), because of some long math that, since de Borglie already derrived it from E=hf, is not going to be put here... also it involves wizardry of the δs and ∫s or something anyways. However, we get this:

iħδψ/δt=-ħ/(2m)Δψ

As a last touch, we should add a potential, V(x,y,z,t) (because V looks like a potential well I guess):

iħδψ/δt=V(x,y,z,t)ψ-ħ/(2m)Δψ

If you play with iħδψ/δt=V(x,y,z,t)ψ-ħ/(2m)Δψ, you can gain good intuition for how this wavfunction evolves. If WegGL just doesn't work at all and blanks out your screen, basically our electron acts like a really sloshy thing that also makes patterns when it sloshes into its own wave.

However, we are missing one important thing: electric charge- *falls through the ground*