>[!summary] If we assume two particles are indistinguishable in every way. You get two different wave functions. > Fermions: Have a wave function as $\psi(a,b) = -\psi(b,a)$ Only exist as spin with half integers (1/2, 3/2,...) Cant exist on the same wave-function (Pauli Exclusion Principle ) > Boson: Have a wave-function $\psi(a,b) = \psi(b,a)$ Can exist on the same wave-function > Pauli Exclusion Principle: A principle that forbids two fermions to exist on the same wavefunction # Deriving Fermions & Bosons >[!warning] Assumption We will assume two particles are indistinguishable in every way. We get a function of this $\psi(a,b) = \psi(b,a)$. If we want to find the probability of finding this wave function we get: $\psi^2(a,b) = \psi^2(b,a)$ If we take the square root of the probability function we get two different states a function can be in: $\psi(a,b) = \psi(b,a)$ and: $\psi(a,b) = -\psi(b,a)$ ## Fermion When a particle exist as a fermion its quantum state whose spin are half integer ($1/2, 3/2$). $\psi(a,b) = -\psi(b,a)$ ## Bosons Bosons particles are particles that can exist in the same wavefunction. $\psi(a,b) = \psi(b,a)$ # Pauli Exclusion Principle This principle forbids fermion to exist in the same wavefunciton. This mean that for a fermion to exist on the same quantum state it must have a different quantum property. In quantum mechanics this simplifies to:![[fe_1.png]] having two particles with opposite wave function meaning they can exist in two ways. If we were to swap the positions of the wave function the wave. Function would be exactly opposite of what it is. # Extra Resources I found understanding Pauli Exclusion principle & finding fermions and bosons (deriving) from [This video by Parth G](https://www.youtube.com/watch?v=INYZy6_HaQE&t=286s&ab_channel=ParthG) --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidan-MathMatter) with a star and help others learn more easily. ---