:eigenote: Definitions🪶

:ember: User discretion is advised.

The following definitions/conventions are specific to Scriber Labs projects and are not used in standard literature. Moreover, they are subject to change as I work through projects and update them for the purposes of clarity and consistency among all projects.

:eigenote: Physical Structure🪶

Overview

Unless otherwise specified, the use of the term physical structure will refer to to any mathematical object or constraint that restricts the admissible states or evolutions of a system.

Formal Definition

Suppose $\mathcal{X}$ denotes the state space of all mathematically possible states $\mathcal{x}\in\mathcal{X}$ of a system. Then the physical structure $\mathcal{S}$ determines an admissible subset of states and trajectories that satisfy governing laws, symmetries, invariants, or geometric contraints of the system.

In particular, $\mathcal{S}$ is a collection of constraints, symmetries, or invariants that induces

\[ \mathcal{M}_{\mathcal{S}} \subseteq \mathcal{X} \, .\]

Summary of variables

$\mathcal{X}$: state space of all mathematically possible states $\mathcal{x}\in\mathcal{X}$ of a system
$\mathcal{S}$: physical structure that restricts the admissible states or evolutions of a system
$\mathcal{M}_{\mathcal{S}}$: admissible subset of states and trajectories that satisfy governing laws, symmetries, invariants, or geometric contraints of the system

Examples

$\mathcal{X}$: State Space

:eigenote: Simple Harmonic Oscillator $$ \mathcal{X} = \mathbb{R}^2 \, , \quad \mathcal{x} = (q,p) \quad \text{(phase space)} $$

:eigenote: Quantum Mechanics

Generally, $$ \mathcal{X} = \mathcal{H} \quad \text{(Hilbert space)} $$
For the time-independent Schrödinger equation,

\[ \mathcal{x} \in \Bigg\{ \psi : \bigg[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial q^2} + V(q) \bigg] \psi = E \psi \Bigg\} \]

:eigenote: Probability Distributions

\[ \mathcal{X} = \big\{ \rho : \rho \geq 0 \, , \int{\rho} = 1 \big\} \]