Atomic Physics

Rydberg

\tilde{\nu} = \frac 1 {\lambda} = R\left(\frac 1{n^2} - \frac 1{m^2}\right)Z^2

E_n = -hcR\frac{Z^2}{n^2}

hcR_\infty = \frac{m_e(e^2/4\pi\epsilon_0)^2}{2\hbar^2} = 13.606 \;\textup{eV}

R = R_\infty\cdot\frac{M_N}{m_e + M_N}

n^* = n - \delta_l

E = -hcR_\infty \frac{Z_0^2}{n^{*2}}

\Delta E_{FS} = -\frac{Z_i^2Z_0^2}{n^{*3}l(l + 1)}\alpha^2hcR_\infty

V(r) = -\frac{Ze^2}{4\pi\epsilon_0r} = -\frac{Ze_0^2}r

a_0 = \frac{\hbar^24\pi\epsilon_0}{m_ee^2}

In Bohrs model of the atom:

r_n = a_0n^2/Z

R_{1,0} = \left(\frac Z{a_0}\right)^{3/2}2e^{-Zr/a_0}

R_{2,0} = \left(\frac Z{2a_0}\right)^{3/2}2\left(1 - \frac{Zr}{2a_0}\right)e^{-Zr/2a_0}

R_{2,0} = \left(\frac Z{2a_0}\right)^{3/2}\frac{2}{\sqrt{3}}\frac{Zr}{2a_0}e^{-Zr/2a_0}

\def\arraystretch{2.5} \begin{array}{ccc} \hline l & m & \Upsilon_l^m(\theta, \varphi) \\ \hline 0 & 0 & \Upsilon_0^0 = \frac 1{\sqrt{4\pi}} \\ 1 & 0 & \Upsilon_1^0 = \sqrt{\frac 3{4\pi}}\cos\theta\\ 1 & \pm1 & \Upsilon_1^{\pm1} = \pm\sqrt{\frac 3{8\pi}}\sin\theta e^{\pm i\varphi}\\ 2 & 0 & \Upsilon_2^0 = \sqrt{\frac 5{16\pi}}\left(3\cos^2\theta - 1\right)\\ 2 & \pm1 & \Upsilon_2^{\pm1} = \pm\sqrt{\frac {15}{8\pi}}\sin\theta\cos\theta e^{\pm i\varphi}\\ 2 & \pm2 & \Upsilon_2^{\pm2} = \sqrt{\frac {15}{32\pi}}\sin^2\theta e^{\pm 2i\varphi}\\ \hline \end{array}

\bm H = \sum_{i=1}^N\left(-\frac{\hbar^2}{2m}\nabla_i^2 - \frac{Ze^2/4\pi\epsilon_0}{r_i} + \sum_{j>i}^N\frac{e^2/4\pi\epsilon_0}{r_{ij}}\right)

\braket{LM_L|l_1|LM_L} = \frac{\braket{l_1\cdot\bm L}}{L(L + 1)}\braket{LM_L|\bm L|LM_L}

\textup{Terms:}\quad \begin{cases} L = |l_1 - l_2|,...,l_1 + l_2 \\ S = |s_1 - s_2|,...,s_1 + s_2 \end{cases}

\textup{Levels:} \quad J = |L - S|,...,L + S

E_{ZE} = \begin{cases} g_J\mu_BBM_J & \textup{(fine structure)} \\ g_F\mu_BBM_F & \textup{(weak field, hfs)} \\ g_J\mu_BBM_J + AM_IM_J & \textup{(strong field},\; \mu_BB > A) \end{cases}

g_S = 2

g_J = \frac 32 + \frac{S(S + 1) - L(L + 1)}{2J(J + 1)}

g_F = g_J\frac{F(F + 1) + J(J + 1) - I(I + 1)}{2F(F + 1)}

\bm\mu_I = g_I\mu_N\bm I

\frac{\Delta\omega_D}{\omega_0} = 2\sqrt{\ln 2}\frac u c \approx 1.7\frac u c

u = 2230\sqrt{\frac T{300M}}\quad \textup{m/s}

\delta = kv = \frac{\omega v}c

\Delta\omega_N = \Gamma = A_{21} = 1\frac 1{\tau}

\Delta f_N = \frac{\Delta\omega_N}{2\pi}

H = -\bm\mu_I\cdot\bm B_e = A\bm I\cdot\bm J

A = \frac 23\mu_0g_S\mu_Bg_I\mu_N\frac{Z^3}{\pi a_0^3n^3}

\frac{N_2}{N_1} = \frac{g_2}{g_1}e^{-\Delta E/(kT)}

\int_0^\infty x^ne^{-\alpha x}\: dx = \frac{n!}{\alpha^{n + 1}}

\int_0\infty x^{2n + 1}e^{-\alpha x^2}\: dx = \frac{n!}{2\alpha^{n + 1}}

\int_0^\infty e^{-\alpha x^2}\: dx = \frac 12\sqrt{\frac{\pi}\alpha}

\int_0^\infty x^{2n}e^{-\alpha x^2}\: dx = \frac{(2n - 1)!!}{2(2\alpha)^n}\sqrt{\frac{\pi}{\alpha}}

\bm p = -i\hbar\nabla

\bm L = -i\hbar\bm r\times \nabla

\bm H = -\frac{\hbar^2}{2m}\nabla^2 + V \qquad \textup{(standard)}

\nabla^2 = \frac 1r\frac{\partial^2}{\partial r^2}r + \frac 1{r^2}\left(\frac{\partial^2}{\partial\theta^2} + \frac 1{\tan\theta}\frac{\partial}{\partial\theta} + \frac 1{\sin^2\theta}\frac{\partial^2}{\partial \varphi^2}\right)

<\bm H> = <\psi|\bm H|\psi> = \int_{\mathbb{R}} \psi^* \bm H \psi\: dv

\begin{gathered} [A, B] = AB - BA \\ [A, B] = -[B, A] \\ [A, B + C] = [A, B] + [A, C] \\ [AB, C] = A[B, C] + [A, C]B \end{gathered}

\bm H\psi = E\psi \qquad \textup{(time independent)}

\bm H\psi = i\hbar\frac{\partial}{\partial t}\psi \qquad \textup{(time dependent)}

\def\arraystretch{1.5} \begin{array}{cc} \hline \textup{Configuration} & \prod n_il_i^{\omega_i}\\ \textup{Terms} & L \;\textup{and}\; S(^{2S + 1}L)\\ \textup{Levels} & J\\ \textup{States (ZE-sublevels)} & M_J\\ \textup{Hyperfinivåer} & F\\ \hline \end{array}

\def\arraystretch{1.5} \begin{array}{cccc} \hline 1 & \Delta J = 0,\pm 1 & (J = 0 \nleftrightarrow J' = 0) &\textup{level}\\ 2 & \Delta M_J = 0,\pm 1 & (M_J = 0 \nleftrightarrow M_{j'} = 0 \;\textup{if} \; \Delta J = 0) &\textup{state}\\ 3 & \textup{break parity} & & \textup{configuration}\\ 4 & \Delta l = \pm 1 & & \\ 5 & \Delta L = 0,\pm 1& (L = 0 \nleftrightarrow L' = 0) &\textup{term}\\ 6 & \Delta S = 0& &\textup{term}\\ \hline \end{array}

1, 2

are replaced for similar formulas for

F

and

M_F

F

is a good quantum number.

5, 6

only hold if

L

and

S

are good quantum numbers.

\def\arraystretch{1.5} \begin{array}{ccc} \hline & \textup{Fine Structure - LS} & \textup{Hyper Fine Structure - IJ} \\ \hline \textup{interaction} & \beta\bm L\cdot \bm S & A\bm I \cdot\bm J\\ \textup{moment} & \bm J = \bm L + \bm S & \bm F = \bm I + \bm J \\ \textup{eigen-states} & \ket{LSJM_J} & \ket{IJFM_F} \\ \textup{energy} & \beta/2(J(J + 1) - L(L + 1) - S(S + 1)) & A/2(F(F + 1) - I(I + 1) - J(J + 1))\\ \textup{interval} & E_J - E_{J - 1} = \beta J & E_F - E_{F - 1} = AF\\ & (\textup{if}\; E_{S-O} \ll E_{re}) & (\textup{if}\; A \gg \Delta E_{quadrupole}) \\ \hline \end{array}