>[!summary] Spin number and spin states are two different things. > Spin states are refers to a quantum vector and describes how a spin in orientated. > This note justifies spin states through the Stern-Gerlach experiment and explains that spin states exist in a Hilbert Space where the wave functionn exist. At measurement the wavefunction collapses into either a spin up or down state. # Difference between Spin Number & Spin States Spin number from [[Properties of Quarks]] and spin states are two different things. Spin number refers to a intrinsic property that tell you information about the particle such that how it obeys Pauli Exclusion principle from [[Fermion, Bosons & Pauli Exclusion Principle]] or [[Bose-Eisten Condensate]] Spin state refers to a specific quantum vector that describes how a spin is orientated. This is what this note will be about. # Justifying Spin States From Experiment >[!bug] Experiment jusifcation To justify spin well summarize the Stern-Gerlach experiment: A beam of silver atoms passes through a non-uniform magnetic field and there is two distinct spreads of atoms in two spots. > Therefore spin must be composed of two spins states ($\frac{1}{2}$ or $-\frac{1}{2}$). >[!warning] Assumptions To justify spin states we need to assume the following: >- Electron obey Pauli Exclusion principle, therefore spin states must as well ([[Fermion, Bosons & Pauli Exclusion Principle]]) >- There are two spin states (Obey experiment) >- Rotation of spin is composed of a real and complex plane (Complex Hilbert Space) From [[Single Slit Experiment]], [[Double Slit Experiment]] and the [[Schrödinger Equation]] equation we justified that wave functions have to exist in a complex plane in order to align with those experiment. So they must also be in a complex plane (Hilbert Space) for spin states because quarks make up those particles and are justified through experiment. >[!info] Assumptions of Hilbert space For the 2D Hilbert space well assume the following is true: >- $\psi = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}$ >- $\left | \psi \right | = \left | \alpha \right | ^2 + \left | \beta \right |^2$ >- Its not a real 2D real plane (Its has 4 real dimension of imagery and real) >- $\left | \alpha \right | ^2 + \left | \beta \right |^2 = 1$ >- The state of a wave function returns to its orginal state after a 720° degree rotation In a Hilbert space our wave function looks something like the figure below, where our wave function **before** measurement can be any superposition of states, meaning it can be anywhere on this graph. ![[quar_1.png | 500]] Through a Hilbert space we justify spin charges with the up and down states. We denote the up axis as spin ($1/2$) because in the Stern-Gerlach experiment those wave function are appeared in the upwards clump of atoms. We denote the down axis as spin (-1/2) because in the Stern-Gerlach experiment those wave functions are appeared in downwards clumps of atoms. >[!info] Spin up and down states in Hilbert Spaces The that -Up is analogous to the Up states. (Same for down states) If a particle started on the at the Up axis then moved to the -Up axis we denote this moving a full 360 degrees of motion. ## Spin State of a Electron Example Now that we justified how quarks get spin states from the experiment. We can justify how electron get certain spin states. We know a electron has an intrinsic spin number of -1/2 but its spin state can differ. For example if we effect the electron before measurement so that its already in a spin-down state, the outcome form measurement will be a spin-down state ![[spin_state_1.png |400]] >[!note] Explanation A electron is configured before measurement to exist in a spin-down state as measurement. However if we leave the electron be and suppose its superposition state is in the middle of up and down. When we measure the electron can have a spin state of either up or down. The superposition will either collapse into either of these. ![[spin_state_2.png|400]] >[!note] Explanation Example of particle superposition at measurement collapsing into either up or down states. # Extra Resources In explaining spin-states I found this video helpful called [What is Spin States](https://www.youtube.com/watch?v=pYeRS5a3HbE&t=450s&ab_channel=ScienceClicEnglish) --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidan-MathMatter) to support the project and help others discover it! ---