물리 상수
플랑크 상수 h = 6.62607015 \times 10^{-34} \textrm{ J s}
볼츠만 상수 k_\textrm{B} = 1.380649 \times 10^{-23} \textrm{ J K}^{-1}
아보가드로 수 N_\textrm{A} = 6.02214076 \times 10^{23}
보어 반지름 a_0 = 5.291772109 \times 10^{-11} \textrm{ m}
진공 중의 광속 c_0 = 2.99792458 \times 10^8 \textrm{ m s}^{-1}
기본 전하량 e = 1.602176634 \times 10^{-19} \textrm{ C}
보어 마그네톤 \mu_\textrm{B} = 5.7883818012 \times 10^{-5} \textrm{ eV T}^{-1}
단위 환산표
\begin{array}{ c | c | c | c | c | c } & \textrm{Hartree} & \textrm{eV} & \textrm{kJ mol}^{-1} & \textrm{kcal mol}^{-1} & \textrm{cm}^{-1} \\ \hline 1\textrm{ Hartree} & 1 & 2.72114 \times 10^1 & 2.62550 \times 10^3 & 6.27509 \times 10^2 & 2.19475 \times 10^5 \\ 1\textrm{ eV} & 3.67493 \times 10^{-2} & 1 & 9.64853 \times 10^1 & 2.30605 \times 10^1 & 8.06554 \times 10^3 \\ 1\textrm{ kJ mol}^{-1} & 3.80880 \times 10^{-4} & 1.03642 \times 10^{-2} & 1 & 2.39006 \times 10^{-1} & 8.35935 \times 10^1 \\ 1\textrm{ kcal mol}^{-1} & 1.59360 \times 10^{-3} & 4.33641 \times 10^{-2} & 4.18400 \times 10^0 & 1 & 3.49755 \times 10^2 \\ 1 \textrm{ cm}^{-1} & 4.55634 \times 10^{-6} & 1.23984 \times 10^{-4} & 1.19627 \times 10^{-2} & 2.85914 \times 10^{-3} & 1 \end{array}
양자역학
운동 방정식
\displaystyle i\hbar \frac{\partial}{\partial t} \Psi(t) = \hat{H} \Psi(t), \quad i\hbar \frac{\partial}{\partial t} \hat{\rho}(t) = [\hat{H}, \hat{\rho}(t)]
\displaystyle \frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{i\hbar} \langle [\hat{A}, \hat{H}] \rangle + \bigg \langle \frac{\partial \hat{A}}{\partial t} \bigg \rangle
양자 조화 진동자
\displaystyle \hat{x} = \sqrt{\frac{\hbar}{2 m \omega}} (\hat{a}^\dagger + \hat{a}), \quad \hat{p} = i \sqrt{\frac{\hbar m \omega}{2}} (\hat{a}^\dagger - \hat{a})
\displaystyle \hat{a} = \sqrt{\frac{m \omega}{2 \hbar}} \bigg( \hat{x} + \frac{i}{m \omega} \hat{p} \bigg), \quad \hat{a}^\dagger = \sqrt{\frac{m \omega}{2 \hbar}} \bigg( \hat{x} - \frac{i}{m \omega} \hat{p} \bigg)
\hat{a} \ket{n} = \sqrt{n} \ket{n-1}, \quad \hat{a}^\dagger \ket{n} = \sqrt{n + 1} \ket{n + 1}
파울리 스핀 연산자
\hat{\sigma}_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \hat{\sigma}_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \hat{\sigma}_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
This page was motivated by the cheat sheet of Subotnik group at University of Pennsylvania.