Основные факты и формулы
в одном месте

\[ i\hbar\,\frac{\partial \Psi}{\partial t}=\hat H\,\Psi \]

\[ \hat H\,|n\rangle = E_n\,|n\rangle \]

\[ \sum_n |n\rangle\langle n| = \mathbf{1} \]

\[ \dot{\hat A}=\frac{\partial \hat A}{\partial t} + \frac{i}{\hbar}[\hat H,\hat A] \]

\[ \hat{\mathbf p}=-i\hbar\nabla,\qquad [\hat p_x,x]=-i\hbar. \]

\[ \hat U(t)=\exp\!\left(-\frac{i}{\hbar}\hat H t\right). \]

\[ \hat H=\frac{\hat p^2}{2m}+\frac{m\omega^2 x^2}{2}. \]

\[ \varepsilon_0=\hbar\omega,\quad x_0=\sqrt{\frac{\hbar}{m\omega}},\quad p_0=\frac{\hbar}{x_0}=\sqrt{\hbar m\omega}. \]

\[ [\hat a,\hat a^\dagger]=1,\qquad \hat H=\hbar\omega\left(\hat a^\dagger\hat a+\frac12\right). \]

\[ E_n=\hbar\omega\left(n+\frac12\right),\quad n=0,1,2,\dots \] \[ |n\rangle=\frac{(\hat a^\dagger)^n}{\sqrt{n!}}\,|0\rangle. \]

\[ E_n^{(1)} = V_{nn}. \]

\[ E_n^{(2)}=\sum_{m\ne n}\frac{|V_{nm}|^2}{E_n^{(0)}-E_m^{(0)}}. \]

\[ dw_{fi}=\frac{2\pi}{\hbar}|V_{fi}|^2\,\delta(E_f-E_i)\,d\nu_f. \]

\[ [\hat \ell_\alpha,\hat \ell_\beta]= i\,\varepsilon_{\alpha\beta\gamma}\hat \ell_\gamma,\qquad [\hat \ell^2,\hat \ell_\alpha]=0. \]

\[ \ell^2 = \ell(\ell+1),\ \ \ell=0,1,2,\dots;\qquad m=\ell,\ell-1,\dots,-\ell. \]

\[ \psi_{nlm}(\mathbf r)=R_{nl}(r)\,Y_{lm}(\theta,\varphi). \]

\[ \hat H=\frac{\hat p^2}{2m}-\frac{e^2}{r},\qquad a_B=\frac{\hbar^2}{me^2}. \] \[ \varepsilon_n=-\frac{1}{2n^2},\quad n=1,2,\dots \]

\[ \{\sigma_\alpha,\sigma_\beta\}=2\delta_{\alpha\beta},\quad [\sigma_\alpha,\sigma_\beta]=2i\varepsilon_{\alpha\beta\gamma}\sigma_\gamma, \] \[ \sigma_\alpha\sigma_\beta=\delta_{\alpha\beta}+i\varepsilon_{\alpha\beta\gamma}\sigma_\gamma. \]

\[ \hat U_{\mathbf n}(\varphi)=e^{\,i(\mathbf n\cdot\boldsymbol{\sigma})\varphi/2} =\cos\frac{\varphi}{2}+i(\mathbf n\cdot\boldsymbol{\sigma})\sin\frac{\varphi}{2}. \]

\[ \mu_B=\frac{|e|\hbar}{2mc}. \]

\[ i\hbar\frac{\partial \phi}{\partial t} =\frac{(\mathbf p-\frac{e}{c}\mathbf A)^2}{2m}\phi-\frac{e\hbar}{2mc}\,\boldsymbol{\sigma}\mathbf H\,\phi+e\Phi\,\phi. \] \[ H_{so}=-\frac{e\hbar}{4m^2c^2}\,(\mathbf E\times \mathbf p)\,\boldsymbol{\sigma}. \]

\[ \hat\rho = |\psi\rangle\langle\psi|,\qquad \rho_{a'a}=\psi_{a'}\psi_a^*. \]

\[ \rho_A=\mathrm{tr}_B\,\rho_{A+B},\qquad (\rho_A)_{aa'}=\sum_b (\rho_{A+B})_{ab;a'b}. \]

\[ \langle Q\rangle=\mathrm{tr}\,(\hat Q\,\hat\rho). \]

\[ \hat\rho^\dagger=\hat\rho,\qquad \mathrm{tr}\,\hat\rho=1,\qquad \hat\rho\ge 0. \]

\[ \psi(\mathbf r)=e^{i\mathbf k\cdot \mathbf r}+\frac{f(\theta)}{r}\,e^{ikr}. \]

\[ d\sigma = |f(\theta)|^2\,d\Omega. \]

\[ \sigma=\int |f(\theta)|^2\,d\Omega. \]

\[ f(\theta)=-\frac{m}{2\pi\hbar^2}\,V_{\mathbf q},\qquad \mathbf q=\mathbf k'-\mathbf k \] \[ f(\theta)=-\frac{m}{2\pi\hbar^2}\int d\mathbf r\,V(\mathbf r)\,e^{-i\mathbf q\cdot \mathbf r}. \]

\[ \operatorname{Im} f(\mathbf n,\mathbf n)=\frac{k}{4\pi}\,\sigma. \]

\[ \hat a_i|\,\dots,n_i,\dots\rangle=\sqrt{n_i}\,|\,\dots,n_i-1,\dots\rangle,\quad \hat a_i^\dagger|\,\dots,n_i,\dots\rangle=\sqrt{n_i+1}\,|\,\dots,n_i+1,\dots\rangle, \] \[ [a_i,a_j]=0,\ [a_i^\dagger,a_j^\dagger]=0,\ [a_i,a_j^\dagger]=\delta_{ij}. \]

\[ \{a_i,a_j\}=0,\ \{a_i^\dagger,a_j^\dagger\}=0,\ \{a_i,a_j^\dagger\}=\delta_{ij}. \]

\[ H=\int dx\,\hat\psi^\dagger(x)\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+U(x)\right)\hat\psi(x) +\frac12\int\!\!\int dx\,dy\,\hat\psi^\dagger(x)\hat\psi^\dagger(y)V(x,y)\hat\psi(y)\hat\psi(x). \]

\[ \frac{\partial \hat A}{\partial t}=\frac{i}{\hbar}(\hat H\hat A-\hat A\hat H), \qquad \frac{d\rho}{dt}=\frac{i}{\hbar}[\rho,H], \] \[ \frac{d}{dt}|\psi_{\text{int}}\rangle =-\frac{i}{\hbar}V_{\text{int}}(t)|\psi_{\text{int}}\rangle,\ \ V_{\text{int}}(t)=e^{iH_0 t/\hbar} V e^{-iH_0 t/\hbar}. \]