Important Equations

Angular Momentum

$L_x = \frac{1}{2}(L_+ + L_-)$

$L_y = \frac{1}{2i}(L_+ - L_-)$

$L_z|l,m\rangle = \hslash m|l,m\rangle$

$L_{\pm}|l,m \rangle = \hslash\sqrt{l(l+1)-m(m\pm 1)}|l,m\pm 1\rangle$

$L_x^2|l,m\rangle = L_x(L_x|l,m\rangle)$

$L_y^2|l,m\rangle = L_y(L_y|l,m\rangle)$

$L_z^2|l,m\rangle = L_z(L_z|l,m\rangle)$

$L^2 = L_x^2 + L_y^2 + L_z^2$

$L^2|l,m\rangle = \hslash^2l(l+1)|l,m\rangle$

Spins

$S_x = \frac{1}{2}(S_+ + S_-)$

$S_y = \frac{1}{2i}(S_+ - S_-)$

$S_z|s,m\rangle = \hslash m |s,m\rangle$

$S_{\pm}|s,m\rangle = \hslash\sqrt{s(s+1)-m(m\pm 1)}|s,m\pm 1\rangle$

$S^2|s,m\rangle = \hslash^2s(s+1)|s,m\rangle$

$S_x = \frac{\hslash}{2}\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}$

$S_y = \frac{\hslash}{2}\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix}$

$S_z = \frac{\hslash}{2}\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$

$S^2 = \frac{3}{4}\hslash^2\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$

$S_+ = \hslash\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}$

$S_- = \hslash\begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}$

$\uparrow = \begin{pmatrix} 1\\0 \end{pmatrix}$

$\downarrow = \begin{pmatrix} 0\\1 \end{pmatrix}$

$P_{+z} = |\langle\uparrow|\chi\rangle|^2 = |\begin{pmatrix} 1&0 \end{pmatrix} \begin{pmatrix} \chi_{0,0} \\ \chi_{1,0} \\ \end{pmatrix}|^2$

$P_{-z} = |\langle\downarrow|\chi\rangle|^2 = |\begin{pmatrix} 0&1 \end{pmatrix} \begin{pmatrix} \chi_{0,0} \\ \chi_{1,0}\\ \end{pmatrix}|^2$

$P_{+y} = \langle \chi_{+y}|\chi\rangle$

$P_{-y} = \langle \chi_{-y}|\chi\rangle$

$S_{y}\chi_{+y} = \frac{\hbar}{2}\chi_{+y}$

$S_{y}\chi_{-y} = \frac{\hbar}{2}\chi_{-y}$

$\langle S_z \rangle = \langle \chi | S_z | \chi \rangle = \chi^*S_z\chi$

$\langle S_x \rangle = \langle \chi | S_x | \chi \rangle = \chi^*S_s\chi$

$\langle S_y \rangle = \langle \chi | S_y | \chi \rangle = \chi^*S_y\chi$

The hydrogen atom

$R_{n,l}(r) = \frac{1}{r}\rho^{l+1}e^{-\rho}v(\rho)$

$\langle\Psi|H|\Psi\rangle = \sum E \times P$

Energy levels: $E_n = \frac{E_0}{n^2}$

Clebsch-Gordan coefficients

$l$ and $s$ tell which table to look at. $m$ and $s_s$ or $s_z$ tell which column to look at.