\[x=x_{\mathrm{best}}\pm\Delta x\]

\[\mathrm{fractional}=\frac{\Delta x}{|x_{\mathrm{best}}|}\]

\[a_x=a\cos\theta,\qquad a_y=a\sin\theta\]

\[a=\sqrt{a_x^2+a_y^2}\]

\[\vec a=a_x\hat{\imath}+a_y\hat{\jmath}+a_z\hat{k}\]

\[\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\]

\[|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta\]

\[\bar v_x=\frac{\Delta x}{\Delta t}\]

\[v_x=\frac{dx}{dt}\]

\[a_x=\frac{dv_x}{dt}=\frac{d^2x}{dt^2}\]

\[v_x=v_{0x}+a_xt,\qquad x=x_0+v_{0x}t+\frac12a_xt^2\]

\[v_x^2=v_{0x}^2+2a_x(x-x_0)\]

\[a_y=-g,\qquad y=y_0+v_{0y}t-\frac12gt^2\]

\[\Delta x=\int_{t_i}^{t_f}v_x(t)\,dt,\qquad \Delta v_x=\int_{t_i}^{t_f}a_x(t)\,dt\]

\[\vec r=x\hat{\imath}+y\hat{\jmath}+z\hat{k},\qquad \vec v=\frac{d\vec r}{dt}\]

\[v=|\vec v|=\sqrt{v_x^2+v_y^2+v_z^2}\]

\[x=x_0+v_0\cos\theta\,t,\qquad y=y_0+v_0\sin\theta\,t-\frac12gt^2\]

\[R=\frac{v_0^2\sin 2\theta}{g}\]

\[v=R|\omega|,\qquad a_c=R\omega^2=\frac{v^2}{R}\]

\[\vec r_{PA}=\vec r_{PB}+\vec r_{BA},\qquad \vec v_{PA}=\vec v_{PB}+\vec v_{BA}\]

\[\sum\vec F=m\vec a\]

\[\sum F_x=ma_x,\qquad \sum F_y=ma_y\]

\[\vec W=m\vec g,\qquad W=mg\]

\[\vec F_{AB}=-\vec F_{BA}\]

\[W_{\parallel}=mg\sin\theta,\qquad W_{\perp}=mg\cos\theta\]

\[\sum\vec F=\vec 0\]

\[0\le f_s\le\mu_sN,\qquad f_k=\mu_kN\]

\[\sum F_r=m\frac{v^2}{r}=mr\omega^2\]

\[v_{\max}=\sqrt{\mu_sgr}\]

\[\tan\theta=\frac{v^2}{rg}\]

\[W=\vec F\cdot\Delta\vec r=F\Delta r\cos\theta\]

\[W=\int_{\vec r_i}^{\vec r_f}\vec F\cdot d\vec r\]

\[K=\frac12mv^2\]

\[W_{\mathrm{net}}=\Delta K=K_f-K_i\]

\[P=\frac{dW}{dt}=\vec F\cdot\vec v\]

\[U_g=mgy,\qquad \Delta U_g=mg\Delta y\]

\[F_s=-kx,\qquad U_s=\frac12kx^2\]

\[K_i+U_i+W_{nc}=K_f+U_f\]

\[F_x=-\frac{dU}{dx},\qquad \vec F=-\nabla U\]

\[v=\sqrt{\frac{2(E-U)}{m}}\]

\[\vec p=m\vec v\]

\[\vec J=\int_{t_1}^{t_2}\vec F\,dt=\Delta\vec p\]

\[\vec P_f=\vec P_i\quad(\vec J_{\mathrm{ext}}=\vec 0)\]

\[\vec r_{\mathrm{cm}}=\frac{1}{M}\sum_i m_i\vec r_i\]

\[\sum\vec F_{\mathrm{ext}}=M\vec a_{\mathrm{cm}}\]

\[m_1u_1+m_2u_2=m_1v_1+m_2v_2,\qquad K_i=K_f\]

\[\Delta v=v_e\ln\left(\frac{m_0}{m_f}\right)\]

\[\omega=\frac{d\theta}{dt}=2\pi f=\frac{2\pi}{T}\]

\[\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}\]

\[\omega=\omega_0+\alpha t,\qquad \theta=\theta_0+\omega_0t+\frac12\alpha t^2\]

\[\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)\]

\[s=r\theta,\qquad v_t=r\omega,\qquad a_t=r\alpha\]

\[a_c=r\omega^2=\frac{v_t^2}{r}\]

\[I=\sum_i m_ir_{\perp i}^2,\qquad I=\int r_\perp^2\,dm\]

\[I=I_{\mathrm{cm}}+Md^2\]

\[K_{\mathrm{rot}}=\frac12I\omega^2\]

\[K=\frac12Mv_{\mathrm{cm}}^2+\frac12I_{\mathrm{cm}}\omega^2\]

\[\vec\tau=\vec r\times\vec F,\qquad \tau=rF\sin\phi\]

\[\sum\tau_z=I\alpha\]

\[W=\int_{\theta_i}^{\theta_f}\tau\,d\theta,\qquad P=\tau\omega\]

\[\vec L=\vec r\times\vec p,\qquad L=I\omega\]

\[\vec\tau_{\mathrm{ext}}=\frac{d\vec L}{dt}\]

\[\tau_{\mathrm{ext}}=0\Rightarrow L_i=L_f\]

\[\Omega=\frac{\tau}{L}=\frac{Mgd}{I_s\omega_s}\]

\[\sum\vec F=\vec 0,\qquad \sum\vec\tau=\vec 0\]

\[x_{\mathrm{cg}}=\frac{\sum_i m_ix_i}{\sum_i m_i}\]

\[\sigma=\frac{F_\perp}{A},\qquad \epsilon=\frac{\Delta L}{L_0}\]

\[\tau=\frac{F_\parallel}{A},\qquad \gamma=\frac{\Delta x}{h}\]

\[Y=\frac{\sigma}{\epsilon}=\frac{FL_0}{A\Delta L}\]

\[B=-\frac{\Delta p}{\Delta V/V}\]

\[u=\frac12\sigma\epsilon\]

\[\rho=\frac{m}{V}\]

\[p=\frac{F_\perp}{A}\]

\[p=p_0+\rho gh\]

\[F_B=\rho_fgV_{\mathrm{disp}}\]

\[\rho_1A_1v_1=\rho_2A_2v_2,\qquad A_1v_1=A_2v_2\ \text{if incompressible}\]

\[p+\frac12\rho v^2+\rho gy=\mathrm{constant}\]

\[\tau=\eta\frac{dv}{dy}\]

\[\mathrm{Re}=\frac{\rho vL}{\eta}\]

\[F_g=G\frac{m_1m_2}{r^2}\]

\[\vec g=-G\frac{M}{r^2}\hat e_r,\qquad \vec F_g=m\vec g\]

\[v=\sqrt{\frac{GM}{r}},\qquad T=2\pi\sqrt{\frac{r^3}{GM}}\]

\[T^2=\frac{4\pi^2}{GM}a^3\]

\[v_{\mathrm{esc}}=\sqrt{\frac{2GM}{r}}\]

\[r_s=\frac{2GM}{c^2}\]

\[f=\frac1T,\qquad \omega=2\pi f=\frac{2\pi}{T}\]

\[a=-\omega^2x\]

\[\omega=\sqrt{\frac{k}{m}},\qquad T=2\pi\sqrt{\frac{m}{k}}\]

\[x=A\cos(\omega t+\phi)\]

\[E=\frac12mv^2+\frac12kx^2=\frac12kA^2\]

\[\omega=\sqrt{\frac{g}{L}},\qquad T=2\pi\sqrt{\frac{L}{g}}\]

\[T=2\pi\sqrt{\frac{I}{mgd}}\]

\[\omega_d=\sqrt{\omega_0^2-\beta^2}\]

\[v_w=f\lambda=\frac{\lambda}{T}\]

\[y(x,t)=A\cos(kx-\omega t+\phi)\]

\[k=\frac{2\pi}{\lambda},\qquad \omega=\frac{2\pi}{T}\]

\[v_w=\frac{\omega}{k}\]

\[v_w=\sqrt{\frac{T}{\mu}}\]

\[\langle P\rangle=\frac12\mu A^2\omega^2v_w\]

\[\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nv_w}{2L}\]

\[y=y_1+y_2\]

\[v=f\lambda=\frac{\omega}{k}\]

\[v=\sqrt{\frac{B}{\rho}},\qquad v=\sqrt{\frac{\gamma RT}{M}}\]

\[I=\frac{P}{A},\qquad I=\frac{P}{4\pi r^2}\]

\[\beta=10\log_{10}\left(\frac{I}{I_0}\right)\]

\[\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nv}{2L}\]

\[\lambda_n=\frac{4L}{2n-1},\qquad f_n=\frac{(2n-1)v}{4L}\]

\[f_{\mathrm{beat}}=|f_1-f_2|\]

\[f_L=f_S\frac{v+v_L}{v-v_S}\]

\[M=\frac{v_S}{v},\qquad \sin\theta=\frac1M\]

\[T=T_C+273.15\]

\[Q=mc\Delta T\]

\[Q=\pm mL\]

\[\sum_iQ_i=0\]

\[\Delta L=\alpha L_0\Delta T,\qquad \Delta V=\beta V_0\Delta T\]

\[H=kA\frac{T_H-T_C}{L}\]

\[H=e\sigma AT^4,\qquad H_{\mathrm{net}}=e\sigma A(T^4-T_s^4)\]

\[pV=nRT=Nk_BT\]

\[N=nN_A,\qquad m=nM\]

\[Q=mc\Delta T=nC_m\Delta T\]

\[pV=\frac13Nm_0v_{\mathrm{rms}}^2\]

\[\langle K_{\mathrm{tr}}\rangle=\frac32k_BT\]

\[v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}\]

\[\frac{dp}{dT}=\frac{L}{T\Delta v}\]

\[pV=nRT\]

\[W=\int_{V_i}^{V_f}p\,dV\]

\[\Delta U=Q-W\]

\[\Delta U=nC_V\Delta T,\qquad U=\frac{f}{2}nRT\]

\[Q_V=nC_V\Delta T,\qquad Q_P=nC_P\Delta T\]

\[C_P-C_V=R,\qquad \gamma=\frac{C_P}{C_V}\]

\[pV^\gamma=\mathrm{constant},\qquad TV^{\gamma-1}=\mathrm{constant}\]

\[W=nC_V(T_i-T_f)\]

\[\Delta S=\int_i^f\frac{\delta Q_{\mathrm{rev}}}{T}\]

\[\Delta S=\frac{Q_{\mathrm{rev}}}{T}\]

\[\Delta S=\frac{mL}{T}\]

\[\Delta S=nC_V\ln\left(\frac{T_f}{T_i}\right)+nR\ln\left(\frac{V_f}{V_i}\right)\]

\[\Delta S_{\mathrm{univ}}=\Delta S_{\mathrm{sys}}+\Delta S_{\mathrm{surr}}\ge0\]

\[q=Ne,\qquad e=1.60\times10^{-19}\,\mathrm C\]

\[F=k\frac{|q_1q_2|}{r^2},\qquad k=\frac{1}{4\pi\epsilon_0}\]

\[\vec E=\frac{\vec F_0}{q_0},\qquad \vec F=q\vec E\]

\[\vec E=k\frac{Q}{R^2}\hat R\]

\[\vec E_{\mathrm{net}}=\sum_i\vec E_i\]

\[\vec p=q\vec d,\qquad \vec\tau=\vec p\times\vec E\]

\[U=-\vec p\cdot\vec E\]

\[\Phi_E=\int_S\vec E\cdot d\vec A\]

\[\oint_S\vec E\cdot d\vec A=\frac{q_{\mathrm{enc}}}{\epsilon_0}\]

\[\vec E=0\ \text{inside a conductor},\qquad \vec E_{\mathrm{outside}}=\frac{\sigma}{\epsilon_0}\hat n\]

\[E=\frac{q_{\mathrm{enc}}}{4\pi\epsilon_0r^2}\]

\[E=\frac{\lambda}{2\pi\epsilon_0r}\]

\[E=\frac{\sigma}{2\epsilon_0}\]

\[V=\frac{U}{q},\qquad \Delta U=q\Delta V\]

\[W_{\mathrm{elec}}=-q\Delta V\]

\[U=k\frac{q_1q_2}{r}\]

\[V=k\frac{Q}{r}\]

\[V=k\sum_i\frac{Q_i}{R_i}\]

\[E_x=-\frac{dV}{dx},\qquad \vec E=-\nabla V\]

\[C=\frac{Q}{\Delta V}\]

\[C=\epsilon_0\frac{A}{d}\]

\[C_{\mathrm{eq}}=\sum_iC_i\ \text{(parallel)},\qquad \frac{1}{C_{\mathrm{eq}}}=\sum_i\frac{1}{C_i}\ \text{(series)}\]

\[U=\frac{Q^2}{2C}=\frac12CV^2=\frac12QV\]

\[u_E=\frac12\epsilon_0E^2\]

\[C=\kappa C_0,\qquad \epsilon=\kappa\epsilon_0\]

\[p=q\ell,\qquad U=-\vec p\cdot\vec E\]

\[\vec D=\epsilon_0\vec E+\vec P,\qquad \oint\vec D\cdot d\vec A=q_{\mathrm f}\]

\[I=\frac{dQ}{dt},\qquad J=\frac{I}{A}\]

\[R=\frac{V}{I},\qquad V=IR\]

\[R=\rho\frac{L}{A}\]

\[\vec E=\rho\vec J,\qquad \vec J=\sigma\vec E\]

\[\rho(T)=\rho_0[1+\alpha(T-T_0)]\]

\[P=IV=I^2R=\frac{V^2}{R}\]

\[V_{\mathrm{term}}=\mathcal E-Ir,\qquad I=\frac{\mathcal E}{R+r}\]

\[I=n|q|Av_d,\qquad \vec J=nq\vec v_d\]

\[R_{\mathrm{eq}}=R_1+R_2+\cdots,\qquad \frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots\]

\[R_{\mathrm{eq}}=\frac{R_1R_2}{R_1+R_2}\]

\[\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}},\qquad \sum_{\mathrm{loop}}\Delta V=0\]

\[\tau=RC\]

\[V_C(t)=\mathcal E(1-e^{-t/\tau}),\qquad I(t)=\frac{\mathcal E}{R}e^{-t/\tau}\]

\[V_C(t)=V_0e^{-t/\tau}\]

\[P_{\mathrm{loss}}=I^2R_{\mathrm{line}},\qquad \eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\]

\[\vec F_B=q\vec v\times\vec B,\qquad F_B=|q|vB\sin\theta\]

\[\Phi_B=BA\cos\theta=\int\vec B\cdot d\vec A\]

\[r=\frac{mv_\perp}{|q|B},\qquad \omega=\frac{|q|B}{m}\]

\[\vec F=I\vec L\times\vec B,\qquad F=ILB\sin\theta\]

\[\mu=NIA,\qquad \vec\tau=\vec\mu\times\vec B,\qquad \tau=NIAB\sin\theta\]

\[U=-\vec\mu\cdot\vec B\]

\[V_H=\frac{IB}{nqt}\]

\[v=\frac{E}{B},\qquad p=|q|Br\]

\[\vec B=\frac{\mu_0}{4\pi}\frac{q\vec v\times\hat r}{r^2}\]

\[d\vec B=\frac{\mu_0}{4\pi}\frac{I\,d\vec l\times\hat r}{r^2}\]

\[B=\frac{\mu_0I}{2\pi r}\]

\[B(x)=\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}},\qquad B_{\mathrm{center}}=\frac{\mu_0I}{2R}\]

\[\frac{F}{L}=\frac{\mu_0I_1I_2}{2\pi d}\]

\[\oint\vec B\cdot d\vec l=\mu_0I_{\mathrm{enc}}\]

\[B=\mu_0nI\ \text{(solenoid)},\qquad B=\frac{\mu_0NI}{2\pi r}\ \text{(toroid)}\]

\[\Phi_B=\int\vec B\cdot d\vec A\]

\[\mathcal E=-N\frac{d\Phi_B}{dt}\]

\[|\mathcal E|=B\ell v\]

\[\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\]

\[I_d=\epsilon_0\frac{d\Phi_E}{dt}\]

\[\oint\vec E\cdot d\vec A=\frac{q_{\mathrm{enc}}}{\epsilon_0},\quad \oint\vec B\cdot d\vec A=0,\quad \oint\vec B\cdot d\vec\ell=\mu_0I_{\mathrm{enc}}+\mu_0\epsilon_0\frac{d\Phi_E}{dt}\]

\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]

\[\Phi_0=\frac{h}{2e},\qquad \Phi=n\Phi_0\]

\[L=\frac{N\Phi_B}{I},\qquad \mathcal E_L=-L\frac{dI}{dt}\]

\[L=\mu\frac{N^2A}{l}\]

\[U_B=\frac12LI^2,\qquad u_B=\frac{B^2}{2\mu}\]

\[\tau=\frac{L}{R}\]

\[I(t)=\frac{\mathcal E}{R}(1-e^{-tR/L}),\qquad I(t)=I_0e^{-tR/L}\]

\[\omega_0=\frac{1}{\sqrt{LC}},\qquad T=2\pi\sqrt{LC}\]

\[\frac{q^2}{2C}+\frac12LI^2=\mathrm{constant}\]

\[M=\frac{N_2\Phi_{21}}{I_1},\qquad \mathcal E_2=-M\frac{dI_1}{dt}\]

\[\omega=2\pi f,\qquad V_{\mathrm{rms}}=\frac{V_0}{\sqrt2},\qquad I_{\mathrm{rms}}=\frac{I_0}{\sqrt2}\]

\[X_C=\frac{1}{\omega C},\qquad X_L=\omega L\]

\[Z=\sqrt{R^2+(X_L-X_C)^2}\]

\[I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{Z},\qquad \tan\phi=\frac{X_L-X_C}{R}\]

\[P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi=I_{\mathrm{rms}}^2R\]

\[X_L=X_C,\qquad \omega_0=\frac{1}{\sqrt{LC}},\qquad f_0=\frac{1}{2\pi\sqrt{LC}}\]

\[Q=\frac{\omega_0L}{R}\]

\[\frac{V_s}{V_p}=\frac{N_s}{N_p},\qquad V_pI_p=V_sI_s,\qquad \frac{I_s}{I_p}=\frac{N_p}{N_s}\]

\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]

\[\frac{E}{B}=c\]

\[c=f\lambda=\frac{\omega}{k},\qquad n=\frac{c}{v}\]

\[\vec S=\frac{1}{\mu_0}\vec E\times\vec B\]

\[I=\frac12c\epsilon_0E_0^2\]

\[p_{\mathrm{rad}}=\frac{I}{c}\ \text{or}\ \frac{2I}{c}\]

\[\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nc}{2L}\]

\[v=f\lambda,\qquad c=3.00\times10^8\,\mathrm{m\,s^{-1}}\]

\[n=\frac{c}{v},\qquad \lambda=\frac{\lambda_0}{n}\]

\[\theta_r=\theta_i,\qquad n_1\sin\theta_1=n_2\sin\theta_2\]

\[\theta_c=\sin^{-1}\left(\frac{n_2}{n_1}\right)\quad(n_1>n_2)\]

\[I=I_0\cos^2\theta,\qquad \tan\theta_B=\frac{n_2}{n_1}\]

\[I_{\mathrm{scattered}}\propto\lambda^{-4}\]

\[\theta_r=\theta_i,\qquad s'=-s,\qquad m=+1\]

\[f=\frac{R}{2},\qquad \frac{1}{s}+\frac{1}{s'}=\frac{1}{f},\qquad m=-\frac{s'}{s}\]

\[\frac{n_1}{s}+\frac{n_2}{s'}=\frac{n_2-n_1}{R}\]

\[\frac{1}{s}+\frac{1}{s'}=\frac{1}{f},\qquad m=-\frac{s'}{s}\]

\[P=\frac1f,\qquad \frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\]

\[M=\frac{N}{f}\ \text{(relaxed)},\qquad M=1+\frac{N}{f}\ \text{(near point)}\]

\[M\approx-\frac{LN}{f_of_e}\]

\[M=-\frac{f_o}{f_e},\qquad L=f_o+f_e\]

\[\Delta\phi=\frac{2\pi\Delta r}{\lambda}\]

\[\Delta r=m\lambda\ \text{(constructive)},\qquad \Delta r=\left(m+\frac12\right)\lambda\ \text{(destructive)}\]

\[d\sin\theta=m\lambda,\qquad y_m\approx\frac{m\lambda L}{d},\qquad \Delta y\approx\frac{\lambda L}{d}\]

\[I=4I_0\cos^2\left(\frac{\delta}{2}\right)\]

\[2nt=\left(m+\frac12\right)\lambda_0\ \text{(constructive)},\qquad 2nt=m\lambda_0\ \text{(destructive)}\]

\[N=\frac{2\Delta L}{\lambda},\qquad \Delta L=\frac{N\lambda}{2}\]

\[N_F=\frac{a^2}{\lambda L}\]

\[a\sin\theta_m=m\lambda,\qquad y_m\approx\frac{m\lambda L}{a}\]

\[I=I_0\left(\frac{\sin\beta}{\beta}\right)^2,\qquad \beta=\frac{\pi a\sin\theta}{\lambda}\]

\[d\sin\theta_m=m\lambda,\qquad d=\frac1{n_L}\]

\[R=mN\]

\[2d\sin\theta=m\lambda\]

\[\theta_R\approx1.22\frac{\lambda}{D},\qquad s_{\min}\approx1.22\frac{\lambda L}{D}\]

\[\beta=\frac{v}{c},\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}\]

\[\Delta t=\gamma\Delta\tau\]

\[L=\frac{L_0}{\gamma}\]

\[x'=\gamma(x-vt),\qquad t'=\gamma\left(t-\frac{vx}{c^2}\right)\]

\[u_x'=\frac{u_x-v}{1-u_xv/c^2}\]

\[f_o=f_s\sqrt{\frac{1+\beta}{1-\beta}}\ \text{(approaching)},\qquad f_o=f_s\sqrt{\frac{1-\beta}{1+\beta}}\ \text{(receding)}\]

\[\vec p=\gamma m\vec v\]

\[E=\gamma mc^2,\qquad E_0=mc^2,\qquad K=(\gamma-1)mc^2\]

\[E^2=(pc)^2+(mc^2)^2\]

\[E=hf=\frac{hc}{\lambda},\qquad p=\frac{h}{\lambda}\]

\[K_{\max}=hf-\phi,\qquad eV_s=K_{\max}\]

\[f_0=\frac{\phi}{h},\qquad \lambda_0=\frac{hc}{\phi}\]

\[K=eV,\qquad \lambda_{\min}=\frac{hc}{eV}\]

\[\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)\]

\[E_{\mathrm{th}}=2m_ec^2=1.022\,\mathrm{MeV}\]

\[\Delta x\,\Delta p\ge\frac{\hbar}{2},\qquad \Delta E\,\Delta t\gtrsim\frac{\hbar}{2}\]

\[\lambda=\frac{h}{p}\]

\[\lambda=\frac{h}{p},\qquad \lambda=\frac{h}{\sqrt{2m_eeV}}\]

\[E_\gamma=hf=\frac{hc}{\lambda}=E_i-E_f\]

\[m_evr_n=n\hbar,\qquad r_n=n^2a_0\]

\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2}\]

\[\lambda_{\max}T=b\]

\[r_{\min}=\frac{2kZe^2}{K}\]

\[K\sim\frac{\hbar^2}{2mL^2}\]

\[\int_{-\infty}^{\infty}|\psi|^2\,dx=1,\qquad P_{[a,b]}=\int_a^b|\psi|^2\,dx\]

\[\langle x\rangle=\int_{-\infty}^{\infty}x|\psi|^2\,dx\]

\[i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)\right]\psi\]

\[-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2}+V(x)\phi=E\phi\]

\[\psi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}\]

\[\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar},\qquad T\sim e^{-2\kappa a}\]

\[V(x)=\frac12m\omega^2x^2,\qquad E_n=\left(n+\frac12\right)\hbar\omega\]

\[P_n=|\langle a_n|\psi\rangle|^2,\qquad \langle A\rangle=\langle\psi|\hat A|\psi\rangle\]

\[i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^2}{2m}\nabla^2+U(\vec r)\right]\psi\]

\[\int|\psi|^2\,dV=1\]

\[E=\frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)\]

\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2},\qquad L=\sqrt{\ell(\ell+1)}\hbar,\qquad L_z=m_\ell\hbar\]

\[s=\frac12,\qquad S=\sqrt{s(s+1)}\hbar,\qquad S_z=m_s\hbar\]

\[N_{\mathrm{orbital}}=2,\qquad N_e=2n^2\]

\[\mu_B=\frac{e\hbar}{2m_e},\qquad \Delta E=m_\ell\mu_BB,\qquad \Delta f=\frac{\Delta E}{h}\]

\[E_\gamma=E_i-E_f,\qquad \lambda=\frac{hc}{E_\gamma}\]

\[A=Z+N,\qquad R=R_0A^{1/3}\]

\[\Delta m=Zm_{\mathrm p}+Nm_{\mathrm n}-M_{\mathrm{nucleus}},\qquad B=\Delta mc^2\]

\[N=N_0e^{-\lambda t},\qquad A=\lambda N,\qquad t_{1/2}=\frac{\ln2}{\lambda}\]

\[t=-\frac1\lambda\ln\left(\frac{N}{N_0}\right)\]

\[D=\frac{E}{m},\qquad H=w_RD\]

\[\sum A_i=\sum A_f,\qquad \sum Z_i=\sum Z_f\]

\[Q=(m_i-m_f)c^2\]

\[U_C\approx\frac{1}{4\pi\epsilon_0}\frac{Z_1Z_2e^2}{r}\]

\[\Delta K=qV,\qquad p=|q|Br\]

\[E_0=mc^2\]

\[\sum Q_i=\sum Q_f,\qquad \sum B_i=\sum B_f\]

\[Q_u=+\frac23,\qquad Q_d=Q_s=-\frac13\]

\[v=H_0d,\qquad 1+z=\frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}}=\frac{a_0}{a_{\mathrm{emit}}}\]

\[T(z)=T_0(1+z),\qquad E\sim k_BT\]