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\title{física eléctrica}
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%\begin{center}{\huge{\textbf{Física nuclear:}}\hspace{130mm}\normalsize{Leire Zufia}}\\
%\end{center}
\begin{multicols*}{3}

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%--------------------------
\begin{tikzpicture}
\node [TEMA1] (box){%
    \begin{minipage}{0.3\textwidth}
\text{Operador paridad: } $\Pi\psi(\vec{r})=\psi(-\vec{r}) \hspace{3mm} \Pi^2=1$\\
$\Pi Y_l^m(\hat{n})=Y_l^m(-\hat{n})=(-1)^lY_l^m(\hat{n})$\\
$\Pi_\pi=-1\hspace{2mm}\Pi_n=\Pi_p=+1$\\
\text{Espacio de carga:} $p=\begin{pmatrix}
    1\\
    0
\end{pmatrix}\hspace{3mm}n=\begin{pmatrix}
    0\\
    1
\end{pmatrix}$\\
\text{Operador carga:}$Q=\begin{pmatrix}
    1&0\\
    0&0
\end{pmatrix}=\frac{1}{2}(1+\tau_z), \hspace{2mm}$\\
$Q=\begin{pmatrix}
    1&0\\
    0&0
\end{pmatrix}=\frac{1}{2}(1+\tau_z), \hspace{2mm}$\\
$\tau_+=\begin{pmatrix}
	0&1\\
	0&0
\end{pmatrix}=\tau_x+i\tau_y\hspace{3mm}\tau_-=\begin{pmatrix}
0&0\\
1&0
\end{pmatrix}=\tau_x-i\tau_y$\\
\text{Isoespin: } $\vec{t}=\frac{1}{2}\sigma\hspace{3mm}\text{Carga isospin: }Z=\frac{1}{2}+\tau_z$\\
$t_3p=\frac{1}{2}p\hspace{3mm}t_3n=-\frac{1}{2}n$\\$\pi^+\rightarrow t_z=1,\pi^0\rightarrow t_z=0,\pi^-\rightarrow t_z=-1$\\
$B_p=B_n=1\hspace{3mm}B_\pi=0\hspace{3mm}Z=t_z+\frac{B}{2}$\\
$\text{Isospin total: } \vec{T}=\sum_a \vec{t_a}\hspace{3mm}\vec{T}^2=T(1+T)\hspace{3mm}[\vec{T},H_s]=0$\\
$Z=\sum_a\left( \frac{1}{2}+t_{za}\right) =\frac{A}{2}+T_z\hspace{3mm}T_z=\frac{1}{2}(Z-N)$\\
\text{Momento dipolar magnético: }$\vec{\mu}=\frac{e}{2mc}\vec{L}\hspace{3mm}\vec{\mu_L}=\frac{eg_L}{2mc}\vec{L}$\\
$p\rightarrow g_L=1\hspace{2mm}n\rightarrow g_L=0\hspace{5mm}g_s=2\hspace{3mm}\vec{\mu_s}=\frac{eg_s}{2mc}\vec{S}$\\
$\mu_N=\frac{e\hbar}{2mc}\hspace{3mm}\vec{\mu_L}=\mu_N g_L\frac{\vec{L}}{\hbar}\hspace{3mm}\vec{\mu_s}=\mu_N g_s \frac{\vec{S}}{\hbar}\quad \alpha=\frac{e^2}{\hbar c}$\\
Compton: $\lambda'=\lambda+\frac{h}{mc}(1-cos\theta)$
    \end{minipage}
};
%---------------------------------
\node[TEMA1tit, right=10pt] at (box.north west) {TEMA 1.1: Hadrones};
\end{tikzpicture}

%---------------------------
\begin{tikzpicture}
\node [TEMA1] (box){%
    \begin{minipage}{0.3\textwidth}
    	$\nabla^2 \psi+\frac{2M}{\hbar^2}(E-V)\psi=0$\\
    	\text{Func. ond. d: } $\psi=\Psi_{space}\chi_s^{ms}\phi_T^{T_z}\hspace{3mm}\Psi_{space}=\frac{u_l(r)}{r}Y_l^{m}$\\
    	\text{Paridades: }$\Pi_S=(-1)^{S+1}\hspace{3mm}\Pi_T=(-1)^{T+1}\hspace{3mm}\Pi_L=(-1)^L$\\
    \text{Monopolo electrostático }$l=0$\\$\hspace{2mm} M(E0,0)=\frac{1}{\sqrt{4\pi}}\sum_a e_a=\frac{1}{\sqrt{4\pi}}\int d^3rp(\vec{r})=\frac{1}{\sqrt{4\pi}}Z_e$\\
    \text{Momento dipolar: } $l=1$\\
    $M(E1,m)=\sqrt{\frac{3}{4\pi}}\sum_a e_a r_a(n_a)_m=\sqrt{\frac{3}{4\pi}}d_m$\\
    \text{Momento cuadrupolar: }$Q=\bra{\psi}\hat{Q}\ket{\psi}$\\
    $Q=\int\psi^*(r)(3z^2-r^2)\psi(r)d^3r$\\
    \text{Momento magnético: } $\vec{\mu}=\sum_a\left( g^s\vec{s}+g^l\vec{l}\right) $\\
    $\vec{j}=\vec{l}+\vec{s}\hspace{3mm}\vec{j}^2=j(j+1)\hspace{3mm}\vec{l}^2=l(l+1)\hspace{3mm}\vec{s}^2=s(s+1)$\\
    $\vec{j}\cdot\vec{l}=\frac{j(j+1)+l(l+1)-s(s+1)}{2}\hspace{3mm}\vec{j}\cdot\vec{s}=\frac{j(j+1)+s(s+1)-l(l+1)}{2}$\\
    $\vec{s}\cdot \vec{l}=\frac{\vec{j}^2-\vec{s}^2-\vec{l}^2}{2}$\\
    \text{Op. mom. mag.: }$\vec{\mu}=g^s\vec{s}+g^l\vec{l}=g(j,l,s)\vec{j}$\\
    \text{Landé: }$g(j,l,s)=g^l\frac{\langle\vec{j}\cdot\vec{l}\rangle}{j(j+1)}+g^s\frac{\langle\vec{j}\cdot\vec{s}\rangle}{j(j+1)=}$\\
    $=\frac{1}{2j(j+1)}\left\lbrace (g^l+g^s)j(j+1)+(g^l-g^s)[l(l+1)-s(s+1)]\right\rbrace $\\
    $g_p^s=2\mu_p=5.58\quad g_n^s=-3.82\quad g_p^l=1\quad g_n^l=0$\\
    $u_0=Asin(Kr) \quad r<R\quad Be^{-\kappa r}\quad r>R$\\
    K=$\frac{1}{\hbar}\sqrt{2M(V_0-E)}\quad \kappa=\frac{1}{\hbar}\sqrt{2ME_B}$
    
    
\end{minipage}
};
%---------------------------------
\node[TEMA1tit, right=10pt] at (box.north west) {TEMA 1.2: Sistema de dos nucleones};
\end{tikzpicture}

%---------------------------
\begin{tikzpicture}
\node [TEMA1] (box){%
    \begin{minipage}{0.3\textwidth}
\text{Mom. mag. deuterón: } $\mu_d\simeq\mu_p+\mu_n=0.8797\mu_N$\\
\text{Pot. local: }$V(1,2)=V_c+V_s(\vec{\sigma_1}\cdot\vec{\sigma_2})+V_TS_{12}(\vec{r})+V_T'S_{12}(\vec{p})+V_{LS}\vec{L}\cdot\vec{S}+V_Q(\vec{L}\cdot\vec{S})^2$\\
$S_{12}=3\left( \vec{\sigma_1}\cdot\frac{\vec{r}}{r}\right)\cdot\left( \vec{\sigma_2}\cdot\frac{\vec{r}}{r}\right)-\left( \vec{\sigma_1}-\vec{\sigma_2}\right) \hspace{3mm}(\vec{r}\cdot\vec{p})^2=r^2p^2-L^2$\\
$\vec{S_i}^2=\frac{3}{4}\hbar^2\hspace{3mm}\vec{S_1}\cdot\vec{S_2}=\frac{1}{4}\vec{\sigma_1}\cdot\vec{\sigma_2}\hspace{3mm}\vec{S_1}\cdot\vec{S_2}=\frac{1}{2}(\vec{S}^2-\vec{S_1}^2-\vec{S_2}^2)$\\
\text{Pot. Paris: }$S=1$ $V(1,2)=V_{c1}(r)+V_T(r)S_{12}(\vec{r})+V_{so}(r)\vec{L}\cdot(\vec{\sigma_1}+\vec{\sigma_2})$\\
$S=0$   $V(1,2)=V_{co}(r)$\\
\text{Reid soft: } $V=V_c(\mu r)+V_{12}(\mu r)S_{12}+V_{LS}(\mu r)\vec{L}\cdot\vec{S}$\\
\text{Yukawa: } $V_g^2\frac{e^{-\mu r}}{r}$\\
\text{Lennard Jones: } $V_{LJ}(r)=4 E_{PO}\left[\left( \frac{\sigma}{r}\right)^{12}-\left( \frac{\sigma}{r}\right)^6\right]$

    \end{minipage}
};
%---------------------------------
\node[TEMA1tit, right=10pt] at (box.north west) {TEMA 1.3: Interacción nucleón-nucleón:};
\end{tikzpicture}

%---------------------------
\begin{tikzpicture}
\node [TEMA2] (box){%
    \begin{minipage}{0.3\textwidth}
$A\simeq \frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3}\hspace{3mm}R\simeq r_o\cdot A^{1/3} \hspace{3mm}r_0>r\sim 1fm$\\
\text{Densidad de carga: }$\rho(r)= \frac{\rho_0}{1+e^{\frac{r-R_{1/2}}{a}}}\hspace{3mm}A>20\hspace{3mm}t=4.4a$\\
$R_s=1.128A^{1/3}\quad R_{1/2}=R_s-0.89A^{-1/3}$\\
$\rho_0=0.17 \frac{Z}{A}fm^{-3}\quad a=0.54 fm$\\
\text{Energia enlace: }$B(Z,N)=[Zm_p+Nm_n-m(Z,N)]c^2$\\
\text{Energia separación: }$S_m(Z,N)=[m(Z,N-1)+m_n-m(Z,N)]c^2=B(Z,N)-B(Z,N-1)$\\
\text{E.pairing: }$\delta_n(Z,N)=S_n(Z,N)-S_n(Z,N-1)\hspace{2mm}$  N par\\
\text{Números mágicos: } $2,8,20,50,82,(126,184)^*$ * n\\
$l=\left\{\begin{array}{cc}
n\hbar\hspace{3mm} impar-impar\\
(n+1\frac{1}{2})\hbar\hspace{3mm}nucleos-impares\\
0\hspace{3mm}par-par
\end{array} \right.$\\
$V=\int\frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'\hspace{3mm}V\simeq\frac{q}{r}+\frac{\vec{\rho}\cdot\vec{x}}{r^3}+\frac{Q_{ij}x_ix_j}{r^5}+...\hspace{3mm}\vec{A}(\vec{r})\simeq\frac{\vec{\mu}\times\vec{r}}{r^3}+...$\\
Mom. mag. nuclear: $\mu=g^lj+(g^s-g^l)<s_z>$\\
$\mu=g^l(j-\frac{1}{2})+\frac{1}{2}g^s\hspace{3mm}j=l+\frac{1}{2}$\\
$\mu=g^l\frac{j(j+\frac{3}{2})}{j+1}-\frac{jg^s}{2(j+1)}\hspace{3mm}j=l-\frac{1}{2}$\\
Tensor 4pol: $Q_{ij} = \int \left( 3x'_i x'_j - r'^2 \delta_{ij} \right) \psi^*(\mathbf{r}') \psi(\mathbf{r}') \, d\mathbf{r}'$\\
$V_Q = Ze \frac{z^2}{2r^5} Q - Ze \frac{x^2 + y^2}{4r^5} Q, \quad$\\$ Q = \int (3z'^2 - r'^2) \psi^*(\mathbf{r}') \psi(\mathbf{r}') \, d\mathbf{r}'$\\
Funciones spin angular:\\ $\mathcal{Y}^{j = \ell \pm 1/2,\, m}_\ell = \frac{1}{\sqrt{2\ell + 1}} \left[ 
\pm \sqrt{\ell \pm m + \tfrac{1}{2}} \, Y^{m - 1/2}_\ell (\theta, \phi) \chi_{\text{spin}+} 
+\right.$\\$\left.+\sqrt{\ell \mp m + \tfrac{1}{2}} \, Y^{m + 1/2}_\ell (\theta, \phi) \chi_{\text{spin}-} 
\right]$\\
$\chi_{\text{spin}+} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \quad 
\chi_{\text{spin}-} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$\\
$Q=-\langle r^2 \rangle \frac{2j - 1}{2j + 2}  \quad 
\langle r^2 \rangle = \int_0^{\infty} u_\ell^2(r) \, r^2 \, dr\quad\langle r^2\rangle=R^2$\\
$Q = Q_{\text{sp}} \left( 1 - 2 \frac{n - 1}{2j - 1} \right) \quad 
Q_{\text{sp}} = -\langle r^2 \rangle \frac{2j - 1}{2j + 2}$\\
$\beta^-$ decay: $_Z^A X_N\rightarrow _{Z+1}^AY_{N-1}+e^-+\bar{\nu}$\\
$\beta^+$ decay: $_Z^A X_N\rightarrow _{Z-1}^AY_{N+1}+e^++\nu$
\end{minipage}
};
%------------ campo titulo---------------------
\node[TEMA2tit, right=10pt] at (box.north west) {TEMA 2.1: Propiedades generales de los núcleos};
\end{tikzpicture}
%---------------------------------
%\bigskip
%---------------------------
\begin{tikzpicture}
\node [TEMA2] (box){%
    \begin{minipage}{0.3\textwidth}
Ex de m: $\Delta (Z,A)=M(Z,A)-Am_u=(M(Z,A)(u)-A)u$\\
Masa nuclear: $m(Z,A)=M(Z,A)-Zm_e+\frac{1}{c^2}\sum_{i=1}^ZB_i$\\
Energía enlace: $B / c^2 = Zm_p + N m_n - m(Z, A)$\\$ = Zm_p + N m_n - [M(Z, A) - Z m_e] $\\$= Z m(^{1}\!H) + N m_n - M(Z, A)$

    \end{minipage}
};
%------------ potencial titulo  ---------------------
%\node[TEMA5tit, right=10pt] at (box.north west) {TEMA 5: ESTADISTICAS CUANTICAS + MATE};
\end{tikzpicture}

%---------------------------
\begin{tikzpicture}
\node [TEMA2] (box){%
    \begin{minipage}{0.3\textwidth}
 \textbf{Modelo de gota liquida: }\\
 Energía de volumen: $B_1=a_vA$\\
 Energía de superficie: $B_2=-a_sA^{2/3}\quad a_s=4\pi R^2$\\
 Energía de Coulomb: $B_3=-a_cZ^2A^{-1/3}\quad a_c=\frac{3}{5}(Ze)^2/R$\\
 Término de asimetría: $B_4=-a_A\frac{(Z_A/2)^2}{A}$\\
 Pairing term: $B_5=\left\{\begin{array}{cc}
 	\delta\hspace{3mm}\text{par par}\\
 	0\hspace{3mm}\text{impar}\\
  	-\delta\hspace{3mm}\text{impar impar}
 \end{array} \right.$ $\delta\simeq A^{-1/2}a_p$\\
 $B(Z, A) = a_v A - a_s A^{2/3} - a_c \frac{Z^2}{A^{1/3}} - a_A \frac{(Z - A/2)^2}{A} +$\\$ +\frac{(-1)^Z + (-1)^N}{2} a_p A^{-1/2}$\\
 $m(Z, A) = Z m_p + (A - Z) m_n - a_v A + a_s A^{2/3} + a_c \frac{Z^2}{A^{1/3}} + a_A \frac{(Z - A/2)^2}{A} - \frac{(-1)^Z + (-1)^N}{2} a_p A^{-1/2}$\\
 $a_V = 15.85\ \mathrm{MeV}/c^2$, \quad
 $a_S = 18.34\ \mathrm{MeV}/c^2$\\
 $a_C = 0.71\ \mathrm{MeV}/c^2$, 
 $a_A = 92.86\ \mathrm{MeV}/c^2$, \\
 \ $a_P = 11.46\ \mathrm{MeV}/c^2$\\ 
 $\left. \frac{\partial m(Z, A)}{\partial Z} \right|_{A = \text{const}} = 0$ $\Rightarrow Z_0 = \frac{A}{2} \left( \frac{m_n - m_p + a_A}{a_c A^{2/3} + a_A} \right) = \frac{A}{1.98 + 0.015 A^{2/3}}$\\
 $E_\alpha = \left[ m(Z, A) - m(Z - 2, A - 4) - m_\alpha \right] c^2 > 0$ $A>150$\\
 $E_f = \left[ m(Z, A) - 2m(Z/2, A/2) \right] c^2 > 0$ $A>90$\\
 \textbf{Modelo del gas de Fermi:}\\
 $dn(k) = \frac{a^3}{(2\pi)^3} 4\pi k^2 \, dk\quad$ $dn(E) = \frac{\sqrt{2} \, m^{3/2} a^3}{2\pi^2 \hbar^3} E^{1/2} \, dE$\\
 $2 \int_0^{E_F^{(p)}} dn(E) = \frac{2\sqrt{2} \, m^{3/2} a^3}{3\pi^2 \hbar^3} \left(E_F^{(p)}\right)^{3/2} = Z,\quad \rho_p=Z/a^3$\\
 $2 \int_0^{E_F^{(n)}} dn(E) = \frac{2\sqrt{2} \, m^{3/2} a^3}{3\pi^2 \hbar^3} \left(E_F^{(n)}\right)^{3/2} = N.\quad \rho_n=N/a^3$\\
 $E_F(p) = \frac{\hbar^2}{2m} \left(3\pi^2 \rho_p \right)^{2/3}, \quad
 E_F(n) = \frac{\hbar^2}{2m} \left(3\pi^2 \rho_n \right)^{2/3}.$\\
 $k_F=\frac{\sqrt{2mE_F}}{\hbar}\quad \langle E_k\rangle=\frac{3}{5}E_F\quad E_T=\langle E_k\rangle A$\\
 \textbf{Modelo de capas:}
\begin{center}
	\resizebox{8cm}{!} {
		\begin{tabular}{|c|c|c||c|c|c|}
			\hline
			\textbf{Orbital} & \textbf{Acumulado} & \textbf{Energía} & \textbf{Orbital} & \textbf{Acumulado} & \textbf{Energía} \\
			\hline
			$1s_{1/2}$   & 2   & $0$           & $2p_{3/2}$   & 32  & $3\hbar\omega$ \\
			$1p_{3/2}$   & 6   & $\hbar\omega$ & $1f_{5/2}$   & 38  & $3\hbar\omega$ \\
			$1p_{1/2}$   & 8   & $\hbar\omega$ & $2p_{1/2}$   & 40  & $3\hbar\omega$ \\
			$1d_{5/2}$   & 14  & $2\hbar\omega$ & $1g_{9/2}$   & 50  & $3\hbar\omega$ \\
			$2s_{1/2}$   & 16  & $2\hbar\omega$ & $1g_{7/2}$   & 58  & $4\hbar\omega$ \\
			$1d_{3/2}$   & 20  & $2\hbar\omega$ & $2d_{5/2}$   & 64  & $4\hbar\omega$ \\
			$1f_{7/2}$   & 28  & $3\hbar\omega$ & $1h_{11/2}$  & 76  & $4\hbar\omega$ \\
			\hline
	\end{tabular}}
\end{center}

 Núcleos impar-impar $\rightarrow$ Número de Nordheim:\\
 $\mathcal{N} = j_p - \ell_p + j_n - \ell_n$ Si $\mathcal{N}=0$ $j=\abs{j_p-j_n}$, Si $\mathcal{N}=\pm1$ $j=j_p+j_n$ o $\abs{j_p-j_n}$
 
    \end{minipage}
};
%---------------------------------
\node[TEMA2tit, right=10pt] at (box.north west) {TEMA 2.2: Modelos nucleares};
\end{tikzpicture}

%\bigskip\\
%---------------------------
\begin{tikzpicture}
\node [TEMA3] (box){%
    \begin{minipage}{0.3\textwidth}
Número de átomos: $N(t)=N_0e^{-\lambda t}$ $dN(t)=-\lambda N dt$\\
Actividad: $A=-\frac{dN}{dt}\quad\lambda=\lambda_1+\lambda_2+...$\\
Semivida: $N(t_{1/2})=N_0/2\quad t_{1/2}=\frac{ln2}{\lambda}$\\
Mean lifetime: $\tau =\frac{1}{\lambda}$\\ 
Actividad modo k: $A_k=-\left( \frac{dN_k}{dt}\right) _k=\lambda_kN=\lambda_kN_0e^{-\lambda t}$\\
$N_k=\frac{\lambda_k}{\lambda}N_0(1-e^{-\lambda t})\quad \lambda_k=\lambda\cdot P_k$\\
Des. en cadena: $\frac{dN_i}{dt}=\lambda_{i-1}N_{i-1}-\lambda_iN_i$\\
Si $N_2(0)=0,N_3(0)=0...$ $\rightarrow$ $N_i(t)=N_1(0)(h_1e^{-\lambda_1t}+h_2e^{-\lambda_2 t}+...+h_i e^{\lambda_i t})$\\
Equilibrio: $\frac{dN_1}{dt}=\frac{d N_2}{dt}=...\simeq 0$ $\rightarrow$ $\lambda_1N_1=\lambda_2N_2=...$
\textbf{Datación geológica:}\\
Número de isótopos padre e hija: $P+D=P_0+D_0$\\
$P=P_0e^{-\lambda(t-t_0)}\quad e^{-\lambda(t-t_0)}=\frac{P}{P_0}=\frac{1+D_0/P_0}{1+D/P}$\\
Isótopo descendiente $D'\quad \Delta t=t-t_0$\\
$\frac{dD'}{dt}=\lambda P-\lambda_{D'} D'\quad \text{si }\lambda\rightarrow0 \quad\lambda_{D'}=0\ \frac{dD'}{dt}=0$\\
$\frac{P+D}{D'}=\frac{P_0+D_0}{D_0'}$ ya que $D'=D_0'$ $\Rightarrow$ $\frac{D}{D'}=\frac{P}{D'}\left[e^{\lambda(t-t_0)}-1\right] +\frac{D_0}{D_0'}$\\
\textbf{Probabilidad de transición:}\\
Energía: $W=E_0-\frac{\hbar \lambda}{2}$

 \end{minipage}
};
%---------------------------------
\node[TEMA3tit, right=10pt] at (box.north west) {TEMA 3.1: Radiación};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA3] (box){%
		\begin{minipage}{0.3\textwidth}
		\textbf{Probabilidad de transición:}\\
		Energía: $W=E_0-\frac{\hbar \lambda}{2}$\\
		$\psi(\mathbf{r}, t) = \psi(\mathbf{r}) e^{-iWt/\hbar} \quad
		|\psi(\mathbf{r}, t)|^2 = |\psi(\mathbf{r}, 0)|^2 e^{-\lambda t}$\\
		$e^{-iWt/\hbar} = e^{-\left(iE_0/\hbar + \lambda/2\right)t} = \int_{-\infty}^{\infty} A(E) e^{-iEt/\hbar} \, dE, \quad t \geq 0$\\
		$\psi(t) = \theta(t) e^{-\left(iE_0/\hbar + \lambda/2\right)t}$\\
		$A(E) = \frac{1}{2\pi\hbar} \int_0^{\infty} e^{\left[i(E - E_0)/\hbar - \lambda/2\right]t} \, dt 
		\quad \Rightarrow$\\$\Rightarrow A(E) = \frac{1}{\hbar \lambda/2 - 2\pi i (E - E_0)}\quad$ $A^*(E) A(E) = \frac{1}{\hbar^2 \lambda^2 / 4 + 4\pi^2 (E - E_0)^2}$\\
		$\Gamma = \hbar \lambda = \hbar / \tau$\\
		Hamiltoniano con perturbación: $H=H_0+V$\\
		$\Psi = \sum_n a_n(t) \, \psi_n \, e^{-i E_n t / \hbar}$ usando $i\hbar\frac{d\Psi}{dt}=H\Psi$\\
		$\dot{a}_k = -\frac{i}{\hbar} \sum_n a_n V_{kn} e^{i \frac{E_k - E_n}{\hbar} t}, \quad 
		V_{kn} = \int \psi_k^* V \psi_n \, d\tau
		$\\
		$a_k = \frac{V_{km} \left(1 - e^{i \frac{E_k - E_m}{\hbar} T} \right)}{E_k - E_m}\quad a_k^* a_k = 4 |V_{km}|^2 \frac{\sin^2\left( \frac{(E_k - E_m) T}{2\hbar} \right)}{(E_k - E_m)^2}$\\
		Regla de oro:\\
		$\lambda(m \to k) = \frac{|a_k(T)|^2}{T} = \frac{2\pi}{\hbar} |V_{km}|^2 \delta(E_k - E_m)$\\
		$\lambda=\sum_{k\neq m}\lambda(m\rightarrow k)$
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA3tit, right=10pt] at (box.north west) {TEMA 3.2: Regla de oro de Fermi};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA3] (box){%
		\begin{minipage}{0.3\textwidth}
			\textbf{$\alpha$ decay:}\\
Probabilidad de decay: $P_\alpha=\lambda\simeq \frac{v}{R}T\quad \tau_0\sim\frac{v}{R}$\\
$t_{nuc}=\frac{2R}{v_F}\simeq2.6\cdot10^{-23}\cdot A^{1/3}$\\
Coef. transmisión: $T_a\propto e^{-2G}$\\
Factor Gamow: $G = \frac{1}{\hbar} \sqrt{\frac{2m}{E}} \, Z_1 Z_2 e^2 \, \gamma\left( \frac{R}{R'} \right)$\\
$\tau=\tau_0\cdot e^{2G}\quad \lambda=\tau_0^{-1}\cdot e^{-2G}$\\
$T_\alpha\simeq\frac{p^2c^2}{2 m_\alpha c^2}\quad\tau_0=\frac{R}{c}\sqrt{\frac{m c^2}{2(V_0+E)}}\quad R=r_0\left(A_{F_1}^{1/3}+A_{F_2}^{1/3}\right) $\\
$\gamma(x)=arc cos\sqrt{x}-\sqrt{x(1-x)}\quad\frac{Z_1Z_2e^2}{R'}=E=\frac{1}{2}mv^2$\\
$x=R/R'=E/V_c\quad V_c=\frac{Z_1 Z_2 e^2}{R}\quad \gamma\approx \pi/2-2\sqrt{x}$\\
\textbf{Fisión: }\\
$\Delta B = 2B(A/2, Z/2) - B(A, Z) =$\\$= a_s \left(1 - 2^{1/3} \right) A^{2/3} - a_c \left(2^{-2/3} - 1 \right) \frac{Z^2}{A^{1/3}}
$\\
$\Delta E = a_s A^{2/3} \left( \frac{2}{5} \epsilon^2 + \dots \right) + a_c Z^2 A^{-1/3} \left( -\frac{1}{5} \epsilon^2 \dots \right)
$\\
$\Delta E<0 \text{ si} \quad \frac{Z^2}{A} > \frac{2 a_s}{a_c} \simeq 51
$\\
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA3tit, right=10pt] at (box.north west) {TEMA 3.3: Desintegración alfa y fisión};
\end{tikzpicture}
\begin{tikzpicture}
	\node [TEMA3] (box){%
		\begin{minipage}{0.3\textwidth}
			$Q_{\beta^-} = \left( M_{Z,A} - M_{Z+1,A} \right)c^2 $\\$ Q_{\beta^+} = \left( M_{Z,A} - M_{Z-1,A} - 2m_e \right)c^2$\\
			$\lambda = \frac{2\pi}{\hbar} |\mathcal{M}_{if}|^2 \frac{dN}{dE_T}, \quad \text{con } \mathcal{M}_{if} = \int \psi_f^* \mathcal{V} \psi_i \, d^3r$\\
			$|M_{if}^F|^2 = \sum_{m_f} \left| \int \psi_f^* \left( \sum_k t_\pm^k \right) \psi_i \, d^3r \right|^2$\\
			$dN_e = \frac{p_e^2 \, dp_e}{2\pi^2 \hbar^3} V = \frac{E_e (E_e^2 - m_e^2 c^4)^{1/2} \, dE_e}{2\pi^2 \hbar^3 c^3} V$\\
			$\frac{d\lambda}{dE_e} = \frac{F(Z, E_e) |\mathcal{M}_{if}|^2}{2\pi^3 \hbar^7 c^6} E_e (E_e^2 - m_e^2 c^4)^{1/2} (E_T - E_e)^2$\\
			Plot Kurie: $\left[ \frac{\lambda'(E)}{F(Z, E) \, E \left( E^2 - m_e^2 c^4 \right)^{1/2}} \right]^{1/2}$\\$\lambda = \frac{m^5 g^2 c^4 |\mathcal{M}_{if}|^2}{2\pi^3 \hbar^7} f(Z, E_T)\quad ft_{1/2} = \frac{(2 \ln 2)\pi^3 \hbar^7}{g^2 m^5 c^4} \cdot \frac{1}{|\mathcal{M}_{if}|^2}$\\
			$f= \frac{1}{m^5 c^{10}} \int_{mc^2}^{E_T} F(Z, E) E \left(E^2 - m^2 c^4\right)^{1/2} (E_T - E)^2 dE$\\
			Transiciones: $\vec{l}_i = \vec{l}_f + \vec{\ell} + \vec{S}$\\ $l_i = l_f + \ell \quad \text{(Fermi)}\quad l_i = l_f + \ell + 1 \quad \text{(Gamow–Teller)}$
			\begin{center}
			\resizebox{8cm}{!} {
				\begin{tabular}{|l|c|c||c|c|}
					\hline
					\textbf{Transición} & $\ell$ & $\Delta\pi$ & $\Delta l$ Fermi & $\Delta l$ Gam-Tell \\
					\hline
					super-permitida     & 0 & $+$ & 0     & 0           \\
					permitida           & 0 & $+$ & 0     & 0, 1        \\
					primera prohibida   & 1 & $-$ & 0, 1  & 0, 1, 2     \\
					segunda prohibida   & 2 & $+$ & 1, 2  & 1, 2, 3     \\
					tercera prohibida   & 3 & $-$ & 2, 3  & 2, 3, 4     \\
					\hline
				\end{tabular}}
			\end{center}
	\end{minipage}
	};
	%---------------------------------
	\node[TEMA3tit, right=10pt] at (box.north west) {TEMA 3.4: Desintegración beta};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA3] (box){%
		\begin{minipage}{0.3\textwidth}
		$|l_i - l_f| \leq l \leq l_i + l_f$\\
		$\Delta \pi$=1$\rightarrow$ E par y M impar\\
		$\Delta \pi$=-1$\rightarrow$ E impar y M par
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA3tit, right=10pt] at (box.north west) {TEMA 3.4: Desintegración gamma};
\end{tikzpicture}


\begin{tikzpicture}
	\node [INT] (box){%
		\begin{minipage}{0.3\textwidth}
			1 fermi= 1fm =$10^{-15}$ m=$10^{-13}$ cm\\
			1 barn $\rightarrow$ $10^{-24} cm^2$=$10^2$ $fm^2$\\
			1 u= (1/12) m ($_6^{12}C$)=1.660538921 $10^-27$ kg= 931.494061 MeV/$c^2$\\
			\textbf{Integrales armónicos esféricos:}\\
			$\int |Y_\ell^\ell(\theta, \phi)|^2 (3 \cos^2 \theta - 1) \sin \theta \, d\theta \, d\phi = -\frac{2\ell}{2\ell + 3}$\\
			$Y_\ell^{\ell - 1}(\theta, \phi) = \sqrt{2\ell + 1} \cos \theta \, Y_{\ell - 1}^{\ell - 1}(\theta, \phi)$\\
			$\int |Y_\ell^{\ell - 1}(\theta, \phi)|^2 (3 \cos^2 \theta - 1) \sin \theta \, d\theta \, d\phi = -1 + \frac{9}{2\ell + 3}$\\
			Para $j=l-1/2$ $\left(-1 + \frac{9}{2\ell + 3}\right) \frac{1}{2\ell + 1} + \left(-\frac{2\ell}{2\ell + 3}\right) \frac{2\ell}{2\ell + 1} = \frac{2 - 2\ell}{1 + 2\ell} = -\frac{2j - 1}{2j + 2}$\\
			\textbf{Unidades radiación: }\\
			$\rightarrow$ Actividad:
			\begin{itemize}
				\item Curie (Ci): 1Ci= $3.7\cdot10^{10}$ desintegraciones/s
				\item Becquerel (Bq) : 1Bq= 1 des/s= $0.27\cdot10^{-10}Ci$
				\item Rutherford (Rd): 1Rd=$10^6$ desintegraciones/s
			\end{itemize}
			$\rightarrow$ Cantidad radiación:
			\begin{itemize}
				\item 1Roentgen= 1R= $1esu/1cm^3$ de aire=\\$=2.58\cdot10^{-4}$ C/kg de aire
			\end{itemize}
			$\rightarrow$Dosis absorbida:
			\begin{itemize}
				\item 1rad=$10^{-2}$ J/kg=100 ergs/g de material
				\item 1Gy=1 J/kg de material =100 rad
			\end{itemize}
		$\rightarrow$ Dosis equivalente:
		\begin{itemize}
			\item 1rem=$w_R\times$ (dose in rads)
			\item Sievert (Sv): 1Sv=$w_R\times$ (dose in Gy)=100rem
		\end{itemize}
		\textbf{Propiedades de las deltas:}\\
		$\delta(x) = \lim_{\lambda \to \infty} \sqrt{\frac{\lambda}{\pi}} e^{-\lambda x^2}$
		$= \lim_{\lambda \to \infty} \frac{\sin \lambda x}{\pi x}$\\
		$= \lim_{\lambda \to \infty} \frac{1}{\lambda \pi} \frac{\sin^2 \lambda x}{x^2}$$= \lim_{\varepsilon \to 0^+} \frac{\varepsilon}{\pi(x^2 + \varepsilon^2)}$\\
		$\delta(ax) = \frac{1}{|a|} \delta(x) \quad \text{a es una constante real cualquiera}$\\
		$\delta(F(x)) = \sum_j \frac{1}{|F'(x_j)|} \delta(x - x_j) \quad  x_j \text{0s simples de F(x)}$\\
		$\lim_{\varepsilon \to 0^+} \frac{1}{x \pm i\varepsilon} = \text{VP} \frac{1}{x} \mp i\pi \delta(x), \quad$\\$ \text{donde } \left\langle \text{VP} \frac{1}{x}, \phi \right\rangle = \lim_{\varepsilon \to 0^+} \int_{|x| \geq \varepsilon} dx\, \frac{\phi(x)}{x}$
		
		
		
		
				
		\end{minipage}
	};
	%---------------------------------
	\node[INTtit, right=10pt] at (box.north west) {UNIDADES Y MATES};
\end{tikzpicture}
\bigskip\\

\newpage

\begin{tikzpicture}
	\node [TEMA4] (box){%
		\begin{minipage}{0.3\textwidth}
$\vec{a} \rightarrow M \vec{a}, \quad M = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},$\\
$\vec{a}^{\mathrm{T}} \rightarrow (M \vec{a})^{\mathrm{T}} = \vec{a}^{\mathrm{T}} M^{\mathrm{T}}, \quad M^{\mathrm{T}} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}.$\\
$\vec{b}^{\mathrm{T}} \vec{a} \rightarrow (M \vec{b})^{\mathrm{T}} M \vec{a} = \vec{b}^{\mathrm{T}} M^{\mathrm{T}} M \vec{a} = \vec{b}^{\mathrm{T}} \mathbb{I} \vec{a} = \vec{b}^{\mathrm{T}} \vec{a}.$\\
Invarianza relativista: $E^2=m^2+\abs{\vec{p}}^2$\\
$p=(E,\vec{p})=(E,p_x,p_y,p_z)\quad p\cdot p=p^2=E^2-\abs{\vec{p}}^2$\\
$p \cdot p \equiv p_\mu \eta^{\mu\nu} p_\nu \equiv p_\mu p^\mu = \sum_{\mu=0}^{3} \sum_{\nu=0}^{3} p_\mu \eta^{\mu\nu} p_\nu = $\\$=p^\mu \eta_{\mu\nu} p^\nu$\\
$\eta^{\mu\nu} = \begin{pmatrix}
	1 & 0 & 0 & 0 \\
	0 & -1 & 0 & 0 \\
	0 & 0 & -1 & 0 \\
	0 & 0 & 0 & -1
\end{pmatrix}^{\mu\nu}\quad p_\mu=\eta_{\mu \nu}p^\nu$\\
Transformadas de Lorentz: $p_\mu \rightarrow \Lambda_\mu^{\ \nu} p_\nu$\\
$\Lambda_\mu^{\ \nu} = \begin{pmatrix}
	\gamma & 0 & 0 & \gamma \beta \\
	0 & 1 & 0 & 0 \\
	0 & 0 & 1 & 0 \\
	\gamma \beta & 0 & 0 & \gamma
\end{pmatrix}$\\
$p \cdot q = p_\mu \eta^{\mu\nu} q_\nu \rightarrow p_\rho \Lambda^\rho_{\ \mu} \eta^{\mu\nu} \Lambda^\sigma_{\ \nu} q_\sigma = p_\mu \eta^{\mu\nu} q_\nu$\\
$p_\mu \Lambda^\mu_{\ \rho} \eta^{\rho\sigma} \Lambda^\nu_{\ \sigma} q_\nu = p_\mu \eta^{\mu\nu} q_\nu\quad \Lambda^\mu_{\ \rho} \eta^{\rho\sigma} \Lambda^\nu_{\ \sigma} = \eta^{\mu\nu}$\\
Ec. Klein-Gordon: $\left( \frac{\partial^2}{\partial t^2} - \nabla^2 + m^2 \right) \phi(\vec{x}, t) = 0$\\
$\phi(\vec{x},t)=e^{-ip\cdot x}\quad p\cdot x=Et-\vec{p}\cdot\vec{x}$\\$(E^2-\abs{\vec{p}}^2-m^2)\phi(\vec{x},t)=0$ sols: $E=\pm \sqrt{m^2+\abs{\vec{p}}^2}$\\
Lagrangiano KG: $\frac{\partial^2}{\partial t^2}\phi=\nabla^2 \phi -m^2\phi$\\
E. cinética: $k=\frac{1}{2}\left(\frac{\partial}{\partial t}\phi \right)^2 \quad$ Hamil: H=$\int d\vec{x} (k+u)$\\
E. total: $k+u=\frac{1}{2}\left(\frac{\partial}{\partial t}\phi \right)^2+\frac{1}{2}\vec{\nabla}\phi\cdot\vec{\nabla}\phi+\frac{m^2}{2}\phi^2$\\
Ec. Dirac: $(i\gamma_\mu\partial^\mu-m)\psi=0$\\
$\gamma_\mu = \begin{pmatrix} 0 & \sigma_\mu \\ \bar{\sigma}_\mu & 0 \end{pmatrix}  $Clifford algebra: $\{\gamma_\mu,\gamma_\nu\}=2\eta_{\mu \nu}$\\
$\sigma_\mu = (\mathbb{I}, \sigma_1, \sigma_2, \sigma_3), \quad \bar{\sigma}_\mu = (\mathbb{I}, -\sigma_1, -\sigma_2, -\sigma_3).$\\
$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$\\
$\sigma_i \sigma_j=\delta_{ij}+i\epsilon_{ijk}\sigma_k\quad det(p\cdot \sigma)=p^2$\\
$\vec{\nabla} \cdot \vec{E} = \rho, \quad \vec{\nabla} \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}, \quad \vec{\nabla} \cdot \vec{B} = 0, \quad \frac{\partial \vec{B}}{\partial t} + \vec{\nabla} \times \vec{E} = 0$\\
$J^\mu=(\rho,\vec{J})^\mu\quad F_{\mu\nu}=-F_{\nu \mu}$\\
$F_{\mu\nu} = \begin{pmatrix}
	0 & E_x & E_y & E_z \\
	-E_x & 0 & -B_z & B_y \\
	-E_y & B_z & 0 & -B_x \\
	-E_z & -B_y & B_x & 0
\end{pmatrix}^{\mu\nu}\quad F_{\mu\nu}\rightarrow \Lambda_\mu^\rho \Lambda_\nu^\sigma F_{\rho \sigma}$\\
$\partial_\mu F^{\mu \nu}=J^\nu$\\
$\partial_\mu F^{\mu 0} = \partial_0 F_{00} - \partial_1 F_{10} - \partial_2 F_{20} - \partial_3 F_{30} = \vec{\nabla} \cdot \vec{E} = \rho$\\
$\partial_\mu F^{\mu \nu}-J^\nu=0\quad \Lambda_\nu^\sigma(\partial_\mu F^{\mu \nu}-J^\nu)=0$
	\end{minipage}
	};
	%---------------------------------
	\node[TEMA4tit, right=10pt] at (box.north west) {TEMA 4.1: Relatividad especial};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA4] (box){%
		\begin{minipage}{0.3\textwidth}
			Identidad Bianchi: $\partial_\mu F_{\nu\rho}+\partial_\rho F_{\mu\nu}+\partial_\nu F_{\rho_\mu}=0$\\
			Potencial : $A_\mu=(V,\vec{A})_\mu\quad F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$\\
			$x'^{\mu}=\Lambda_\nu^\mu x^\nu\quad J=det\frac{\partial x'^\mu}{\partial x^\nu}$\\
			$E=\sqrt{p^2+m^2}\simeq m+\frac{p^2}{2m}$\\
			Partícula sin masa: $\abs{\vec{p}}=E\quad \vec{p_1}\cdot\vec{p_2}=\abs{\vec{p}_1}\abs{\vec{p}_2}cos\theta$\\
		\end{minipage}
	};
	%---------------------------------
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA4] (box){%
		\begin{minipage}{0.3\textwidth}
			1. Elemento identidad: $a\cdot 1=1\cdot a=a$\\
			2. Inversa: $a\cdot a^{-1}=a^{-1}\cdot a=1$\\
			3. Grupo cerrado: $a\cdot b=c$ en el grupo\\
			4. Operaciones asoaciativas: $(a\cdot b)\cdot c=a\cdot(b\cdot c)$\\
			\textbf{Algebra de Lie:} $\mathbb{M}=e^{i\alpha^a T^a}$\\
			Conmutador: $[T^a, T^b]=if^{abc}T^c$\\
			$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\quad [\hat{L}_x,\hat{L}_y]=i\hat{L}_z\quad [\frac{\sigma_i}{2},\frac{\sigma_j}{2}]=i\epsilon_{ijk}\frac{\sigma_k}{2}$\\
			\textbf{Isoespín: }$\braket{p}{p}=\braket{n}{n}=1\quad\braket{p}{n}=\braket{n}{p}=0$\\
			Singlete: $\frac{1}{\sqrt{2}}(\ket{pn}-\ket{np})$\\
			Triplete: $\ket{pp}\quad\frac{1}{\sqrt{2}}(\ket{pn}+\ket{np})\quad \ket{nn}$\\
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA4tit, right=10pt] at (box.north west) {TEMA 4.2: Teoría de grupos};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA4] (box){%
		\begin{minipage}{0.3\textwidth}
			Prob scattering of $p_1,p_2...p_n=|_{out}\braket{p_1p_2...p_n}{p_Ap_B}_in|^2$\\
			De Broglie: $\lambda_{dB}=\frac{\hbar}{\abs{\vec{p}}}$ Compton: $\lambda_C=\frac{\hbar}{mc}\quad v=\frac{p^2}{E}$\\
			pp scattering cross section:\\ $\sigma=\frac{1}{2E_A}\frac{1}{2E_B}\frac{1}{\abs{v_A-v_B}}|_{out}\braket{f}{p_Ap_B}_in|^2$\\
			Regla de oro de Fermi:\\ $\sigma = \frac{1}{2E_A} \frac{1}{2E_B} \frac{1}{|\vec{v}_A - \vec{v}_B|} \int \prod_{i=1}^{n} \left[ \frac{d^4 p_i}{(2\pi)^4} 2\pi \delta(p_i^2 - m_i^2) \right] |\mathcal{M}(A + B \rightarrow 1 + 2 + \cdots + n)|^2 (2\pi)^4 \delta^{(4)}\left(p_A + p_B - \sum_{i=1}^{n} p_i \right)$\\
			$\mathcal{M}(A+B\rightarrow 1+2+...+n)=_{out}\braket{p_1p_2...p_n}{p_Ap_B}_in|^2$
			
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA4tit, right=10pt] at (box.north west) {TEMA 4.3: Regla oro Fermi y diagramas Feynmann};
\end{tikzpicture}
\begin{tikzpicture}
	\node [TEMA5] (box){%
		\begin{minipage}{0.3\textwidth}
			Massless Dirac Ec. : $i\gamma\cdot \partial\psi=0$\\
			$\psi=\begin{pmatrix}
				\psi_L\\
				\psi_R
			\end{pmatrix}\quad$ Weyl eqs: $ i\bar{\sigma}\cdot\partial\psi_L=0\quad i\sigma\cdot\partial\psi_R=0$\\
			$\psi_R=u_Re^{-ip\cdot x}\quad p=E(1,sin\theta cos\phi,\sin\theta sun\phi,cos\theta)$\\
			$u_R(p) = v_L(p) = \sqrt{2E} \begin{pmatrix} e^{-i\phi/2} \cos \frac{\theta}{2} \\ e^{i\phi/2} \sin \frac{\theta}{2} \end{pmatrix}$\\$u_L(p) = v_R(p) = \sqrt{2E} \begin{pmatrix} e^{-i\phi/2} \sin \frac{\theta}{2} \\ -e^{i\phi/2} \cos \frac{\theta}{2} \end{pmatrix}$\\
			$u_R^\dagger u_R = u_L^\dagger u_L = v_R^\dagger v_R = v_L^\dagger v_L = 2E\quad u_R^\dagger u_L=v_L^\dagger v_R=0$\\
			$e_L^+ e_R^- \rightarrow \mu_L^+ \mu_R^-, \quad e_L^+ e_R^- \rightarrow \mu_R^+ \mu_L^-,$\\$ \quad e_R^+ e_L^- \rightarrow \mu_L^+ \mu_R^-, \quad e_R^+ e_L^- \rightarrow \mu_R^+ \mu_L^-$\\
			$\sigma = \frac{\pi}{2 E_{\text{cm}}^2} \left( \frac{e^2}{4\pi} \right)^2 \int d\cos\theta \, (1 + \cos^2\theta)$\\
			Cte de estructura fina: $\alpha=\frac{e^2}{4\pi}$
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA5tit, right=10pt] at (box.north west) {TEMA 5.1: Quantum electrodinamics};
\end{tikzpicture}
\begin{tikzpicture}
	\node [TEMA5] (box){%
		\begin{minipage}{0.3\textwidth}
		Differential cross section: \\
		$\frac{d\sigma(e^+ e^- \rightarrow \mu^+ \mu^-)}{d\cos\theta} = \frac{\pi \alpha^2}{2 E_{\text{cm}}^2} (1 + \cos^2\theta)$\\
		\begin{figure}[H]
			\centering
			\includegraphics[width=0.8\linewidth]{1}
			\label{fig:1}
		\end{figure}
		\begin{figure}[H]
			\centering
			\includegraphics[width=0.7\linewidth]{2}
			\label{fig:1}
		\end{figure}
\begin{figure}[H]
	\centering
	\includegraphics[width=1\linewidth]{"Captura de pantalla 2025-06-01 201416"}
	\label{fig:captura-de-pantalla-2025-06-01-201416}
\end{figure}

		$\mathcal{M}(e^+_L e^-_R \rightarrow \mu^+_L \mu^-_R)^* = \mathcal{M}(\mu^+_R \mu^-_L \rightarrow e^+_R e^-_L) = \mathcal{M}(e^+_R e^-_L \rightarrow \mu^+_R \mu^-_L).$\\
$|\mathcal{M}(e^+_R e^-_L \rightarrow \mu^+_R \mu^-_L)|^2 = |\mathcal{M}(e^+_L e^-_R \rightarrow \mu^+_L \mu^-_R)|^2 = e^4 (1 + \cos \theta)^2,$\\
$|\mathcal{M}(e^+_R e^-_L \rightarrow \mu^+_L \mu^-_R)|^2 = |\mathcal{M}(e^+_L e^-_R \rightarrow \mu^+_R \mu^-_L)|^2 = e^4 (1 - \cos \theta)^2.$\end{minipage}
	};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA5] (box){%
		\begin{minipage}{0.3\textwidth}
			$p_1+p_2=p_3+p_4$\\
			Variables de Mandelstam:\\
			$s \equiv (p_1 + p_2)^2 = (p_3 + p_4)^2, $\\$ t \equiv (p_1 - p_4)^2 = (p_2 - p_3)^2, $\\$ u \equiv (p_1 - p_3)^2 = (p_2 - p_4)^2.$\\
			$s+t+u=0$\\
			$\frac{d\sigma(e^+ e^- \rightarrow q \bar{q})}{d\cos\theta} = \frac{3 \pi \alpha^2}{2 E_{\text{cm}}^2} Q_q^2 (1 + \cos^2\theta).$\\
			\textbf{Drell-Yan process:}
			pp$\rightarrow e^+e^-+X\quad q\bar{q}=\rightarrow ee^-$\\
			$\hat{s} = (p + p')^2 = (k + k')^2,\quad$
			$\hat{t} = (p - k)^2 = (p' - k')^2,$\\
			$\hat{u} = (p - k')^2 = (p' - k)^2.\quad \hat{s}=E_{cm}^2$\\
			$\sigma(e^+ e^- \rightarrow q \bar{q}) = 4 \frac{\pi \alpha^2}{\hat{s}} \sum_{\substack{q\ \text{quarks}}} Q_q^2.$\\$\sigma(q\bar{q} \rightarrow e^+ e^-) = \frac{4}{9} \frac{\pi \alpha^2}{\hat{s}} \sum_{\text{q quarks}} Q_q^2$\\

		\end{minipage}
	};
	%---------------------------------
	\node[TEMA5tit, right=10pt] at (box.north west) {TEMA 5.2: Quarks y gluones};
\end{tikzpicture}
\begin{tikzpicture}
	\node [TEMA5] (box){%
		\begin{minipage}{0.3\textwidth}
$s \equiv (P_1 + P_2)^2 = 2 P_1 \cdot P_2\quad$
$\hat{s} = x_1 x_2 s = 2 (x_1 P_1) \cdot (x_2 P_2)$\\
$\frac{d^2\sigma(q\bar{q} \rightarrow e^+ e^-)}{dx_1\, dx_2} = \frac{4}{9} \frac{\pi \alpha^2}{\hat{s}} \sum_{\text{q quarks}} Q_q^2 f_q(x_1) f_{\bar{q}}(x_2),$\\
$\frac{d^2\sigma(pp \rightarrow e^+ e^-)}{dx_1\, dx_2} = \frac{4}{9} \frac{\pi \alpha^2}{s} \sum_{\text{q quarks}} Q_q^2 \frac{f_q(x_1)}{x_1} \frac{f_{\bar{q}}(x_2)}{x_2}\quad Q^2=\hat{s}=x_1x_2$\\
Rapidity: $y=\frac{1}{2}log\frac{E+p_z}{E-p_z}=\frac{1}{2}log\frac{x_1}{x_2}$\\
$E = x_1 \frac{E_{\text{cm}}}{2} + x_2 \frac{E_{\text{cm}}}{2} = \frac{x_1 + x_2}{2} E_{\text{cm}},\quad x_1=\sqrt{\frac{Q^2}{s}}e^y$\\
$p_z = x_1 \frac{E_{\text{cm}}}{2} - x_2 \frac{E_{\text{cm}}}{2} = \frac{x_1 - x_2}{2} E_{\text{cm}}.\quad x_2=\sqrt{\frac{Q^2}{s}}e^{-y}$\\
$J = \left| \begin{array}{cc}
	\frac{\partial x_1}{\partial Q^2} & \frac{\partial x_1}{\partial y} \\
	\frac{\partial x_2}{\partial Q^2} & \frac{\partial x_2}{\partial y}
\end{array} \right|
= \left| \begin{array}{cc}
	\frac{e^y}{2\sqrt{s} Q^2} & \sqrt{\frac{Q^2}{s}} e^y \\
	\frac{e^{-y}}{2\sqrt{s} Q^2} & -\sqrt{\frac{Q^2}{s}} e^{-y}
\end{array} \right|
= \frac{1}{s}.$

$\frac{d^2\sigma(pp \rightarrow e^+ e^-)}{dQ^2\, dy} = \frac{d^2\sigma(pp \rightarrow e^+ e^-)}{dx_1\, dx_2} \cdot J$

$= \frac{4\pi \alpha^2}{9\, Q^2 s} \sum_{\text{q quarks}} Q_q^2 f_q\left( \sqrt{\frac{Q^2}{s}} e^y \right) f_{\bar{q}}\left( \sqrt{\frac{Q^2}{s}} e^{-y} \right).$


		\end{minipage}
	};
\end{tikzpicture}


\begin{tikzpicture}
	\node [TEMA5] (box){%
		\begin{minipage}{0.3\textwidth}
			$\psi \rightarrow \mathbb{U} \psi\quad \mathbb{U}=e^{i\alpha^a T^a}\quad [T^a, T^b]=if^{abc}T^c\quad \psi=\mathbb{U}\psi'$\\
			$\mathcal{L}=\bar{\psi}(i\gamma\cdot\partial+g\gamma\cdot A^aT^a)\psi\quad A_\mu^a\rightarrow A_\mu^a+\Delta A_\mu^a$\\
			Covariant dervivative: $D_\mu=\partial_\mu-igA_\mu^aT_\mu^a$\\
			$D_\mu\rightarrow\mathbb{U}D_\mu\mathbb{U}^\dagger\quad \mathcal{L}=\bar{\psi} i\gamma\cdot D\psi$\\
			$D_\mu D_\nu - D_\nu D_\mu = [D_\mu, D_\nu]$
			
			$= \partial_\mu \partial_\nu - i g \partial_\mu A_\nu^a T^a - i g A_\mu^a T^a \partial_\nu - g^2 A_\mu^a A_\nu^b T^a T^b - \partial_\nu \partial_\mu$
			
			$\quad + i g \partial_\nu A_\mu^a T^a + i g A_\nu^a T^a \partial_\mu + g^2 A_\mu^a A_\nu^b T^b T^a$
			
			$= -i g \left( \partial_\mu A_\nu^a T^a - \partial_\nu A_\mu^a T^a - i g A_\mu^a A_\nu^b [T^a, T^b] \right)$
			
			$\equiv -i g F_{\mu \nu}^a T^a.\quad$$F_{\mu \nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c,$\\
			$tr[T^aT^b]=\frac{1}{2}\delta^{ab}$\\ Killing form: $\mathcal{L}_{QCD}=-\frac{1}{4}Fâ_{mu\nu}Fâ_{\mu\nu}F^{\mu\nu a}+\bar{\psi}i\gamma\cdot D\psi$\\
			Gluon: $F_{\mu\nu}^a|_{g=0}=\partial_\mu A_\nu^a-\partial_\nu A_\mu ^a$\\
			$\beta(\alpha)=Q\frac{d \alpha}{dQ}\quad$$\beta_{\text{SM}}(\alpha) = \frac{2}{3} \frac{\alpha^2}{\pi} \sum_{\text{fermions } i} Q_i^2,$\\
			$\alpha(Q) = \frac{\alpha(Q_0)}{1 - \frac{2}{3\pi} \alpha(Q_0) \log\left(\frac{Q}{Q_0}\right)}.$\\
									Strong coupling: $\alpha_s=\frac{g^2}{4\pi}\quad$\\$\alpha_s(Q) = \frac{\alpha_s(Q_0)}{1 + \frac{\alpha_s(Q_0)}{2\pi} \left(11 - \frac{2}{3} n_f \right) \log\left( \frac{Q}{Q_0} \right)},$\\
			Bianchi: $(D_\mu F_{\nu \rho})^a + (D_\rho F_{\mu \nu})^a + (D_\nu F_{\rho \mu})^a = 0.$\\
			$f^{abd} f^{dce} + f^{bcd} f^{dae} + f^{cad} f^{dbe} = 0.$\\
			$[T^a, [T^b, T^c]] + [T^b, [T^c, T^a]] + [T^c, [T^a, T^b]] = 0.$
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA5tit, right=10pt] at (box.north west) {TEMA 5.3: Cromodinámica cuántica};
\end{tikzpicture}
\begin{tikzpicture}
	\node [TEMA6] (box){%
		\begin{minipage}{0.3\textwidth}
	\textbf{Paridad:} $\mathbb{P}\vec{x}=-\vec{x}\quad\mathbb{P}\vec{v}=-\vec{v}\quad\mathbb{P}\vec{L}=\vec{L}$\\
Pseudoscalars: $\mathbb{P}\pi^0=-\pi^0$\\
$\mathbb{P} \vec{E} = -\vec{E}, \quad \mathbb{P} \vec{\nabla} = -\vec{\nabla}, \quad \mathbb{P} \rho = \rho, \quad \mathbb{P} \vec{B} = \vec{B}$\\ $ \mathbb{P} \vec{J} = -\vec{J}, \quad \mathbb{P} \frac{\partial}{\partial t} = \frac{\partial}{\partial t}\quad$$\mathbb{P} \theta = \pi - \theta, \quad \mathbb{P} \phi = \phi + \pi.$\\
$\mathbb{P} \begin{pmatrix} u_R(p) \\ u_L(p) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} u_R(p) \\ u_L(p) \end{pmatrix} = \begin{pmatrix} u_L(p) \\ u_R(p) \end{pmatrix}.$\\
\textbf{Time reversal:} $\mathbb{T}t=-t\quad\mathbb{T}\vec{p}=-\vec{p} $\\
$\mathbb{T} \vec{E} = \vec{E}, \quad \mathbb{T} \vec{\nabla} = \vec{\nabla}, \quad \mathbb{T} \rho = \rho, \quad \mathbb{T} \vec{B} = -\vec{B},$\\$\quad \mathbb{T} \vec{J} = -\vec{J}, \quad \mathbb{T} \frac{\partial}{\partial t} = -\frac{\partial}{\partial t}\quad \mathbb{T} \hat{H}=\hat{H}\quad \mathbb{T}i=-i$\\
$\mathbb{T} u_R = \epsilon u_R = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} u_R.$\\
\textbf{Conjugador de carga:} $\mathbb{C}q=-q$\\
$\mathbb{C} \vec{E} = -\vec{E}, \quad \mathbb{C} \vec{\nabla} = \vec{\nabla}, \quad \mathbb{C} \rho = -\rho, \quad \mathbb{C} \vec{B} = -\vec{B}, $\\$ \mathbb{C} \vec{J} = -\vec{J}, \quad \mathbb{C} \frac{\partial}{\partial t} = \frac{\partial}{\partial t}\quad \mathbb{C}u=\nu\quad \mathbb{C}u_R=\nu:L$\\
$\mathbb{C}\mathbb{P}\mathbb{T}J=H^\dagger$





		\end{minipage}
	};
	%---------------------------------
	\node[titTEMA6, right=10pt] at (box.north west) {TEMA 6.1: Violación de la paridad};
\end{tikzpicture}

\begin{tikzpicture}
	\node [INT] (box){%
		\begin{minipage}{0.3\textwidth}
			$e^{iX}=\mathbb{I}+iX-\frac{X^2}{2}-i\frac{X^3}{6}+...$\\
			$\int dx \delta(y-f(x))=\int dx \frac{1}{\abs{\frac{df}{dx}}}\delta (x-f^{-1}(y))$\\
			$f(x)=\tau\quad x=f^{-1}(\tau)\quad d\tau=\frac{df}{dx}dx\quad dx=\frac{1}{\frac{df}{dx}}d\tau$\\
			$log_{10}x=\frac{lnx}{ln 10}$
		\end{minipage}
	};
	%---------------------------------
	\node[INTtit, right=10pt] at (box.north west) {UNIDADES Y MATES};
\end{tikzpicture}
\begin{tikzpicture}
	\node [TEMA7] (box){%
		\begin{minipage}{0.3\textwidth}
			\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}}
			
			
			\centering
			\renewcommand{\arraystretch}{1.3}
			\setlength{\tabcolsep}{12pt}
			
			% QUARKS
			\rowcolors{2}{violet!10}{white}
			\resizebox{8cm}{!} {\begin{tabular}{|C{1cm}|C{2cm}|C{1cm}|C{1cm}|}
					\hline
					\multicolumn{4}{|c|}{\cellcolor{violet!30}\textbf{Quarks}} \\
					\hline
					\textbf{Símbolo} & \textbf{Masa (MeV/$c^2$)} & \textbf{Carga} & \textbf{Espín} \\
					\hline
					$u$ & $\approx 2.2$ & $+\tfrac{2}{3}$ & $\tfrac{1}{2}$ \\
					$d$ & $\approx 4.7$ & $-\tfrac{1}{3}$ & $\tfrac{1}{2}$ \\
					$c$ & $\approx 1280$ & $+\tfrac{2}{3}$ & $\tfrac{1}{2}$ \\
					$s$ & $\approx 96$ & $-\tfrac{1}{3}$ & $\tfrac{1}{2}$ \\
					$t$ & $\approx 173100$ & $+\tfrac{2}{3}$ & $\tfrac{1}{2}$ \\
					$b$ & $\approx 4180$ & $-\tfrac{1}{3}$ & $\tfrac{1}{2}$ \\
					\hline
			\end{tabular}}
		\end{minipage}
	};
	%---------------------------------
	\node[TEMA7tit, right=10pt] at (box.north west) {MASAS Y CARGAS};
\end{tikzpicture}

\begin{tikzpicture}
	\node [TEMA7] (box){%
		\begin{minipage}{0.3\textwidth}
\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}}


\centering
\renewcommand{\arraystretch}{1.3}
\setlength{\tabcolsep}{12pt}



% LEPTONES
\rowcolors{2}{lime!10}{white}
\resizebox{8cm}{!} {\begin{tabular}{|C{1cm}|C{2cm}|C{1cm}|C{1cm}|}
	\hline
	\multicolumn{4}{|c|}{\cellcolor{lime!30}\textbf{Leptones}} \\
	\hline
	\textbf{Símbolo} & \textbf{Masa (MeV/$c^2$)} & \textbf{Carga} & \textbf{Espín} \\
	\hline
	$e$ & $0.511$ & $-1$ & $\tfrac{1}{2}$ \\
	$\mu$ & $105.66$ & $-1$ & $\tfrac{1}{2}$ \\
	$\tau$ & $1776.86$ & $-1$ & $\tfrac{1}{2}$ \\
	$\nu_e$ & $<1$ & $0$ & $\tfrac{1}{2}$ \\
	$\nu_\mu$ & $<0.17$ & $0$ & $\tfrac{1}{2}$ \\
	$\nu_\tau$ & $<18.2$ & $0$ & $\tfrac{1}{2}$ \\
	\hline
\end{tabular}}

\vspace{0.5cm}

% BOSONES DE GAUGE
\rowcolors{2}{red!10}{white}
\resizebox{8cm}{!} {\begin{tabular}{|C{1cm}|C{2cm}|C{1cm}|C{1cm}|}
	\hline
	\multicolumn{4}{|c|}{\cellcolor{red!25}\textbf{Bosones de Gauge}} \\
	\hline
	\textbf{Símbolo} & \textbf{Masa (MeV/$c^2$)} & \textbf{Carga} & \textbf{Espín} \\
	\hline
	$\gamma$ & $0$ & $0$ & $1$ \\
	$g$ & $0$ & $0$ & $1$ \\
	$Z^0$ & $91190$ & $0$ & $1$ \\
	$W^\pm$ & $80390$ & $\pm1$ & $1$ \\
	\hline
\end{tabular}}

\vspace{0.5cm}

% BOSÓN DE HIGGS
\rowcolors{2}{yellow!10}{white}
\resizebox{8cm}{!} {\begin{tabular}{|C{1cm}|C{2cm}|C{1cm}|C{1cm}|}
	\hline
	\multicolumn{4}{|c|}{\cellcolor{yellow!30}\textbf{Bosón Escalar}} \\
	\hline
	\textbf{Símbolo} & \textbf{Masa (MeV/$c^2$)} & \textbf{Carga} & \textbf{Espín} \\
	\hline
	$H$ & $124970$ & $0$ & $0$ \\
	\hline
\end{tabular}}

\vspace{0.5cm}

% HADRONES COMPUESTOS
\rowcolors{2}{cyan!10}{white}
\resizebox{8cm}{!} {\begin{tabular}{|C{1cm}|C{2cm}|C{1cm}|C{1cm}|}
	\hline
	\multicolumn{4}{|c|}{\cellcolor{cyan!25}\textbf{Hadrones Compuestos}} \\
	\hline
	\textbf{Símbolo} & \textbf{Masa (MeV/$c^2$)} & \textbf{Carga} & \textbf{Espín} \\
	\hline
	$p$ & $938.272$ & $+1$ & $\tfrac{1}{2}$ \\
	$n$ & $939.565$ & $0$ & $\tfrac{1}{2}$ \\
	$\pi^+$ & $139.570$ & $+1$ & $0$ \\
	$\pi^0$ & $134.977$ & $0$ & $0$ \\
	$\pi^-$ & $139.570$ & $-1$ & $0$ \\
	\hline
\end{tabular}}
\end{minipage}
	};
	%---------------------------------
\end{tikzpicture}

\end{multicols*}


\end{document}
