Physics Notation and Symbols

\usepackage{amsmath}     % Essential math
\usepackage{amssymb}     % Extra symbols
\usepackage{physics}     % Physics shortcuts
\usepackage{siunitx}     % SI units
\usepackage{tensor}      % Tensor notation
\usepackage{braket}      % Quantum mechanics

% Different vector styles
\vec{F} = m\vec{a}              % Arrow notation
\mathbf{F} = m\mathbf{a}        % Bold notation
\boldsymbol{\tau} = \vec{r} \times \vec{F}  % Bold Greek

% Vector operations
\vec{A} \cdot \vec{B}           % Dot product
\vec{A} \times \vec{B}          % Cross product
|\vec{v}| \text{ or } \|\vec{v}\|  % Magnitude

% Unit vectors
\hat{i}, \hat{j}, \hat{k}       % Cartesian
\hat{r}, \hat{\theta}, \hat{\phi}  % Spherical

% Tensor indices
T^{\mu\nu}                      % Contravariant
T_{\mu\nu}                      % Covariant
T^{\mu}_{\nu}                   % Mixed

% Einstein notation
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}

% Christoffel symbols
\Gamma^{\lambda}_{\mu\nu}

% Riemann tensor
R^{\rho}_{\sigma\mu\nu}

% Basic bra-ket
\ket{\psi}                      % Ket
\bra{\phi}                      % Bra
\braket{\phi|\psi}              % Inner product
\braket{\phi|H|\psi}            % Matrix element

% Operators
\hat{H}\ket{\psi} = E\ket{\psi} % Eigenvalue equation
\hat{p} = -i\hbar\frac{\partial}{\partial x}

% Commutators
[\hat{x}, \hat{p}] = i\hbar
\{\hat{a}, \hat{a}^{\dagger}\} = 1  % Anticommutator

% Schrödinger equation
i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi

% Plane wave
\psi(x,t) = Ae^{i(kx - \omega t)}

% Spherical harmonics
Y_{\ell}^m(\theta, \phi)

% Probability density
|\psi(x,t)|^2 = \psi^*(x,t)\psi(x,t)

% Lagrangian
L = T - V = \frac{1}{2}m\dot{x}^2 - V(x)

% Euler-Lagrange equation
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0

% Hamiltonian
H = \sum_i p_i\dot{q}_i - L

% Hamilton's equations
\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad
\dot{p}_i = -\frac{\partial H}{\partial q_i}

% Differential form
\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}
\nabla \cdot \vec{B} = 0
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}

% Integral form
\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}
\oint_C \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}

% Electromagnetic tensor
F^{\mu\nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}

% Four-potential
A^{\mu} = (\phi/c, \vec{A})

% Lorentz force
\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

% First law
dU = \delta Q - \delta W = TdS - PdV

% Partial derivatives
\left(\frac{\partial U}{\partial S}\right)_V = T
\left(\frac{\partial U}{\partial V}\right)_S = -P

% Maxwell relations
\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V

% Basic units
\SI{3e8}{m/s}                   % Speed of light
\SI{6.626e-34}{J.s}             % Planck constant
\SI{9.81}{m/s^2}                % Acceleration

% Complex units
\SI{13.6}{eV}                   % Energy
\SI{2.5}{kg.m/s}                % Momentum
\SI{1.23e-4}{N.m}               % Torque

% Uncertainties
\SI{9.81 \pm 0.02}{m/s^2}

% Four-vectors
x^{\mu} = (ct, \vec{x})
p^{\mu} = (E/c, \vec{p})

% Minkowski metric
\eta_{\mu\nu} = \text{diag}(1, -1, -1, -1)

% Lorentz transformation
x'^{\mu} = \Lambda^{\mu}_{\nu} x^{\nu}

% Invariant interval
ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2

% For derivations
\begin{align}
  F &= ma \\
  &= m\frac{dv}{dt} \\
  &= \frac{dp}{dt}
\end{align}

% For definitions
\begin{equation}
  \boxed{E = mc^2}
\end{equation}

% For multiple cases
\begin{cases}
  \psi(x) = Ae^{ikx} + Be^{-ikx} & \text{for } x < 0 \\
  \psi(x) = Ce^{-\kappa x} & \text{for } x > 0
\end{cases}