LaTeX for Physics: Equations, Units, and Scientific Notation

\usepackage{siunitx}

\si{\meter}
\si{\kilogram}
\si{\second}
\si{\ampere}
\si{\kelvin}
\si{\mole}
\si{\candela}

% Physical constants
\si{5.97e24}
\si{6.626e-34}
\si{1.602e-19}
\si{8.854e-12}
\si{1.38e-23}

% Density
\si{\kilogram\per\cubic\meter}

% Force
\si{\newton}

% Torque
\si{\newton\meter}

% Heat capacity
\si{\joule\per\mole\per\kelvin}

% Acceleration
\si{\meter\per\square\second}

% Electric field
\si{\volt\per\meter}

\si[per-mode=symbol]{\joule\per\mole\per\kelvin}
\si[per-mode=fraction]{\joule\per\mole\per\kelvin}

% Number formatting
\si[exponent-base=10]{1.602e-19}
\si[exponent-base=10,exponent-to-prefix]{1.602e-19}

% Arrow notation
\vec{v}
\dot{\vec{r}}
\ddot{\vec{x}}

% Hat notation (unit vectors)
\hat{k}
\hat{i}
\hat{j}

% Bold vectors
\mathbf{v}
\boldsymbol{\omega}

% Bar notation
\bar{v}

% Levi-Civita symbol
\epsilon_{ijk}

% Kronecker delta
\delta_{ij}

% Metric tensor
g_{\mu\nu}

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

% Stress-energy tensor
T^{\mu\nu}

% Electromagnetic tensor
F^{\mu\nu}

% Position vectors
\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}

% Momentum
\vec{p} = m\vec{v}

% Angular momentum
\vec{L} = \vec{r} \times \vec{p}

% Newton's second law
\vec{F} = m\vec{a}

% Kinetic energy
E_k = \frac{1}{2}mv^2

% Gravitational force
F = G\frac{m_1 m_2}{r^2}

% Simple harmonic motion
x(t) = A\cos(\omega t + \phi)

% Harmonic oscillator
\frac{d^2x}{dt^2} + \omega^2 x = 0

% Maxwell's equations (differential form)
\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_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\varepsilon_0\frac{\partial \vec{E}}{\partial t}

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

% Electric potential
\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}

% Ideal gas law
PV = nRT

% First law
\Delta U = Q - W

% Entropy
dS = \frac{\delta Q}{T}

% Heat equation
\frac{\partial T}{\partial t} = \alpha \nabla^2 T

% Time-dependent Schrödinger equation
i\hbar\frac{\partial}{\partial t}\Psi(\vec{r},t) = \hat{H}\Psi(\vec{r},t)

% Heisenberg uncertainty principle
\Delta x \Delta p \geq \frac{\hbar}{2}

% Commutation relation
[\hat{x},\hat{p}] = i\hbar

% Dirac notation
\langle\psi|\phi\rangle
\langle\psi|\hat{A}|\phi\rangle

% Hydrogen atom wavefunction
\Psi_{n\ell m}(\vec{r}) = R_{n\ell}(r)Y_{\ell}^{m}(\theta,\phi)

% Lorentz factor
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

% Time dilation
\Delta t' = \gamma\Delta t

% Length contraction
L' = \frac{L}{\gamma}

% Mass-energy equivalence
E = mc^2

% Metric line element
ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{siunitx}
\usepackage{physics}
\usepackage{braket}

\usepackage{braket}

% Dirac notation
\Ket{\psi}
\Bra{\psi}
\Braket{\psi|\phi}
\Braket{\psi|A|\phi}

% States
\Ket{n,\ell,m}
\Ket{0}; \Ket{1}
\Ket{\uparrow}; \Ket{\downarrow}

\usepackage{physics}

% Automatic微分
\dd{x}
\dd[n]{x}

% Derivatives
\dv{x}
\dv[n]{x}{t}
\pdv{F}{x}
\pdv[n]{F}{x}{y}

% Vector operations
\grad{f}
\div{\vec{A}}
\curl{\vec{A}}
\laplacian{f}

\usepackage[separate-uncertainty=true]{siunitx}

% Value with uncertainty
\SI{9.81 +- 0.01}{\meter\per\square\second}

% Compact notation
\SI{9.81(1)}{\meter\per\square\second}

\begin{table}
\centering
\begin{tabular}{
  S[table-format=2.1]
  S[table-format=3.2]
  S[table-format=1.3e-1]
}
\hline
{$t$ (s)} & {$x$ (m)} & {$v$ (m/s)}\\
\hline
0.0 & 0.00 & 0.000\\
0.5 & 1.25 & 2.450\\
1.0 & 4.90 & 4.905\\
1.5 & 11.0 & 7.350\\
\hline
\end{tabular}
\end{table}