*Check out complex numbers and potential wells if you haven't already!* # Intro to Quantum Mechanics The two fundamental units of a quantum particle are its **position** and **momentum** respectively! This is a definition thing. Screw you, lazy predecessors that found a solution that worked anyways! This article will go over the basics to quantum mechanics. From the Bohr model to Schrödinger's equation, we've got it all! (Though as of writing, this is STILL unfinished, so take this with a grain of salt haha) ![[schrodingers-cat.png]] *Schrödinger's cat. No opening, lest the cat be frozen in place.* *For the Bohr Model, take a look at this* # Max Planck and his Constant *TBA* # The Heisenberg Uncertainty Principle Quantum postulates state that we can't ever know the position OR momentum of a quantum particle exactly - there's a limit to the precision of the fundamental constants! $\sigma_{x} \sigma_{p} \leq \frac{\hbar}{2}$ We can instead choose to write this as a $\Delta x\Delta p \leq \frac{\hbar}{2}$ %%visualisation?%% The case where $\Delta x\Delta p = \frac{\hbar}{2}$ is known as the quantum harmonic oscillator, where the resolution of both the momentum and position are at their best. ## Kinetic Energy of Confinement We need to set the scene for the Schrödinger Equation first - by finding the **confinement energy** of a particle - or how much energy it will take to remove it from its potential well. First things first, let's lay down some ground rules; remember that $E = \frac{p^2}{2m}$ where $p = \frac{\hbar}{2\Delta x}$. Therefore, we can just do a simple substitution to get: $E_{K} = \frac{\hbar^2}{2m(\Delta x)^2}$ Within a potential well of width $a$, the particle can be found at any point between the two (as it is probabilistic under a gaussian distribution, which I need to elaborate on later!), so the positional uncertainty is just the width of the well: $E_{K} = \frac{\hbar^2}{2ma^2}$ The kinetic energy is also known as the confinement energy; this is as the kinetic energy represents the energy required to extract the particle from the well itself. This also means that the uncertainty of the momentum $\Delta p \approx p$, given that the same is true for the position! >[!Abstract]- Understanding "Confinement" > >To understand 'confinement', let's use a method to visualise this particle - the **Potential Well.** > >![[potential-well.png]] >Here, the 'blue' represents where the electron can actually reside, whilst the "classically forbidden region" is where the electron cannot be 'found' - that is, putting the electron's positional uncertainty $\Delta x$ to 0, it can only be found in a linear space from $-L/2$ to $L/2$. > >Still, this begs the question - why is there a potential well in the first place? To answer this, we've got to look at the **electric potential due to the electron**; when we isolate the electron, the electric field due to it creates a region of potential that is much larger than the surrounding space, meaning that an obscene amount of [[Power, Work Done and Energy (Mechanics, %%marked for rewrite + NEEDS LOADS OF DIAGRAMS, ASK EXTERNAL HELP!%%)|work]] must be done on the particle to remove it from the field. As a result, the potential of the 'field' is much higher than its surroundings, giving rise to the base potential well. > >Still, in some cases (VERY high energy environments) enough work can be done to remove the electron from the well, allowing the electron to move into the 'classically-forbidden' regions of space. When an electron goes from a classically-forbidden region back to a normal region, it emits a photon - which mimics an emission spectrum. The [story of Nebulium](https://www.britannica.com/science/nebulium) is a consequence of this! # Building to the Wavefunction ## Mathematical Basics: Operators The algebra only gets worse and worse from here. From 3-D transformations of functions or fourier transforms, the expressions we're left with are often extremely tedious not just to derive but also to look at. As a result, we've decided to use **operators**, which are to functions what functions are to single inputs. In other words, these operators represent a total transformation of a function! They're denoted by a "hat", as you can see in the example below: >[!Success]- A Basic Operator Example > >A derivative is an operator! It's something that transform a function, so as an operator we can write: >$\hat{f} = \frac{d}{dt}$ >And to use this operator, we can write: >$\hat{f}x^2 = 2x$ This notation's useful when we've got to transpose two things together with different dimensions; it's used widely in QM - so keep an eye out for it! # The Schrödinger Equation hurr durr need derivation ... this is a differential operator that when solved will give us the time-dependant (or time-independent) schrodinger equation $\hat{p} = -i\hbar \frac{\delta}{\delta x}$ --> $-\frac{\hbar^2}{2m} \frac{\delta^2\psi}{\delta x^2}$ where $\psi$ (psi) is the wavefunction