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\author{Formulario}
\title{Física cuántica}
\date{Otoño, 2001}


\begin{document}
\maketitle

\noindent
 Ley Stefan: $$R_T = \sigma T^4 = \frac{c}{4} \rho_T = \int_0^\infty \rho_T(\nu)\dd\nu $$
 Ley Plank:  $$\rho_T(\nu) = \frac{8\pi \nu^2}{c^3} \frac{h\nu}{\expon{\frac{h\nu}{k T}}-1} \ ,\ \ \rho_T(\lambda) =
 \frac{8\pi h c}{\lambda} \frac{1}{\expon{\frac{h c}{k\lambda T}} - 1}$$

\noindent
 Fotoeléctrico: $$T = h\nu - \phi \to \nu_0 = \phi / h \ \textrm{, corte\ }V_0 = \frac{h}{e} \nu - \frac{\phi}{e}$$

\noindent
 Compton: $$ \lambda' = \lambda + \frac{h}{m c}(1-\cos\theta) $$
 $$ K = \frac{(h c / \lambda)^2}{\frac{h c}{\lambda} + \frac{mc^2}{1-\cos\theta}} \ ,\ \
 \cotg\varphi = \left(1+\frac{h}{m \lambda c}\right)\tg\frac{\theta}{2} $$

\sep
 Rutherford: $$ \tg\frac{\beta}{2} = \frac{K q Q}{2 b E} \ ,\ \dd n = N b\, \dd b\, \dd \varphi =
 N\sigma(\beta) \dd\Omega$$
 Secc. eff.: $$\sigma(\beta) = \left(\frac{K q Q}{4E}\right)^2 \sin^{-4}(\beta/2)$$
 $$ L = m b v_0 \ ,\ r_\textrm{mín} = K \frac{q Q}{E} \, \ b = \frac{K q Q}{2E}\sqrt{\frac{m}{\mu}}\cotg\frac{\beta}{2}$$

\noindent
 Bohr: $$ r=\frac{L^2}{\mu Z e^2} = \frac{(n\hbar)^2}{m Z e^2} = \frac{n^2 a_0}{Z} \ ,$$ $$ a_0 = \hbar^2/me^2 = 0.529\AAA$$
 $$ E_n = -\frac{\mu Z^2 e^4}{2 n^2\hbar^2} = Z^2 \frac{E_1}{n^2} \ , \ E_1 = -\frac{m  e^4}{2\hbar^2}  = -13.6$$
 $$\nu_{n,m} = \frac{mZ^2 e^4}{2h\hbar^2} \left( \frac{1}{m^2} - \frac{1}{n^2} \right) ,\
 \frac{1}{\lambda} = Z^2R \left( \frac{1}{m^2} - \frac{1}{n^2} \right)$$
 $$\textrm{Rydberg}\ R = \frac{2\pi^2 m e^4}{c h^3} = 109 677,576 \cm^{-1} $$

\noindent
 Franck-Hertz: Hg  $4,9\eV$, $\lambda = 2536\AA$ \\

\noindent
 Moseley (r. X): $$ \frac{1}{\lambda_{K_\alpha}} = R(Z-1)^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) $$

\noindent
 Sommerfeld: $$ E = \frac{\mu e^4}{2\hbar n^2} \ , \ L_z = m \hbar \ , \ |\vL| = l \hbar $$

\sep
 $$ \lambda = \frac{h}{p} \ , \quad \nu = \frac{E}{h} \ , \qquad \Delta E \Delta t \ge \frac{\hbar}{2} \ , \quad \Delta x
 \Delta p \ge \frac{\hbar}{2} $$

\sep
 Ecuación de Schödinger
 $$ i \hbar \parcial[\psi(x,t)]{t} = - \frac{\hbar^2}{2m} \parcial[^2\psi(x,t)]{x^2} + V(x) \psi(x,t) $$
 $$ -\frac{\hbar^2}{2\mu} \deriv[^2 X(x)]{x^2} + V(x) X(x) = E X(x) \ , \ \ \psi(x,t) = X(x) \expon{-i {E}/{\hbar} t} $$
 $$ p \to - i \hbar \parcial{x} \ ,\ E = \frac{p^2}{2\mu} + V \to - \frac{\hbar^2}{2\mu} \parcial[^2]{x^2} + V $$
 $$ E \to i \hbar \parcial{t} \ , \quad \langle E \rangle = \int \psi^*(x,t) E \psi(x,t) \dx $$
 $$ \langle E^l \rangle = \sum |a_n|^2 E_n^l \ ,\ \Delta E^2 = \langle E^2 \rangle - \langle E \rangle^2 $$

\sep
 Pozo infinito $(0, a):$
 $$ E_n = \frac{h^2 n^2}{8\mu a^2} \ ,\ \varphi_n(x) = \sqrt{2/a} \sin\frac{n\pi x}{a} $$
 $(-l, l):$ $$ E_n = \frac{\hbar^2 n^2 \pi^2}{8\mu l^2} \ ,\ \varphi_n(x) = \sqrt{1/l} \sin\frac{n\pi}{2l}(x+l) $$

\sep
 Pozo $\ V_0 (-a/2,a/2):  k^2 = \frac{2\mu}{\hbar^2}E \ , \qquad \alpha^2 = \frac{2 \mu}{\hbar^2}(V_0-E) $ \\
 Pares $$ \tg\frac{k a}{2} = \frac{\alpha}{k} \ ,\
  A = \frac{\sqrt{2/a}}{\sqrt{ 1 + \frac{\sin ka}{ka} + \frac{2}{\alpha a} \cos^2\frac{k a}{2}}} $$
 $$ X(|x|)  =  \left\{
 \begin{array}{cccc}
  A \cos k|x| & \textrm{si} & |x| \le & {a}/{2} \\
  A \cos\frac{k a}{2}\expon{-\alpha |x-a/2|}   & \textrm{si} & |x| \ge & {a}/{2}
 \end{array} \right. $$
 Impares $$ \tg\frac{k a}{2} = - \frac{k}{\alpha} \ ,\
  C = \frac{\sqrt{2/a}}{\sqrt{1 - \frac{\sin ka}{ka} + \frac{2}{\alpha a} \sin^2\frac{k a}{2}}}$$
 $$ X(x) = \left\{ \begin{array}{cccc}
  C \sin k|x| & \textrm{si} & |x| \le & {a}/{2} \\
  C \sin\frac{k a}{2} \expon{-\alpha (x-a/2)}   & \textrm{si} & |x| \ge & {a}/{2} \\
  - X(-x) & \textrm{si} & x < 0
 \end{array} \right. $$
 $$ \theta = \frac{k a}{2} \ , \ \theta_0 = \sqrt{\frac{\mu V_0 a^2}{2\hbar^2}} \ ,\  V_0 = \frac{2\hbar^2}{\mu a^2}
 \theta_0^2 \ ,\ E = V_0 \left( \frac{\theta}{\theta_0} \right)^2 $$
 $$ \textrm{p)}\ \tg\theta = \sqrt{ \left( \frac{\theta_0} {\theta} \right)^2 - 1} \ , $$
$$ \textrm{imp)}\ \ \cotg\theta =-\sqrt{ \left( \frac{\theta_0} {\theta} \right)^2 - 1} $$

\sep
 Oscilador armónico $ V(x) = \frac{1}{2} k x^2 = \frac{1}{2} \mu \omega^2 x^2 $
 $$ \alpha = \sqrt{\frac{ 2\mu \omega }{ \hbar }} \ , \qquad E_n = \left( n + \frac{1}{2} \right) \hbar \omega $$
 $$ X_n(x) = \left( \frac{\sqrt{2\pi}}{\alpha}2^n n!\right)^{-1/2}\expon{-\frac{1}{4}\xi^2} H_n \left( \frac{\xi}{\sqrt{2}}
 \right) \ , $$ $$ \quad \xi = \alpha x $$
 Hermite: $$ H_n(z) = (-1)^n \expon{z^2} \deriv[^n]{z^n}\expon{-z^2} $$

\sep
 Dispersión en un pozo $$ k'^2 = \frac{2\mu}{\hbar^2} (V_0 - E) \ ,\ k^2 = \frac{2\mu}{\hbar^2} E $$
 $$ T = \left| \frac{F}{A} \right|^2 = \frac{4k^2k'^2}{(k^2-k'^2)\sin^2 k a + 4 k^2 k'^2} $$
 $$ R = \left| \frac{B}{A} \right|^2 = 1 - T = \frac{(k^2-k'^2)\sin^2 k a}{(k^2-k'^2)\sin^2 k a + 4 k^2 k'^2} $$

\sep
 Barrera de potencial $V_0\ (-a/2, a/2):$
 $$ \beta^2 = -k'^2 = \frac{2\mu}{\hbar} (V_0 - E)\ ,\ k^2 = \frac{2\mu}{\hbar} E $$
 $$ (E \le V_0)\ T = \frac{4 k^2 \beta^2}{(k^2+\beta^2)^2 \sinh^2\beta a + 4 \beta^2k^2} \ ,\ R =1-T$$
 $$ (E \ge V_0)\ T = \frac{4 k^2 k'^2}{(k^2-k'^2)^2 \sin^2k'a + 4 k'^2k^2} \ ,\ R =1-T$$

\sep
 $$ \left[ - \frac{\hbar^2}{2\mu} \nabla^2 + V(\vr) \right] X(\vr) = E X(\vr) \ ,\ \vp \to - i \hbar \vnabla $$
 $$ L_x  =  i \hbar \left( \sin\varphi\parcial{\theta} - \cotg\theta \cos\varphi \parcial{\varphi} \right) $$
 $$ L_y  =  i \hbar \left(-\cos\varphi\parcial{\theta} + \cotg\theta \sin\varphi \parcial{\varphi}\right) $$
 $$ L_z  = -i \hbar \parcial{\varphi} \ ,\qquad \nabla^2 = \frac{1}{r^2} \parcial[^2]{r^2} r - \frac{1}{r^2}
   \frac{L^2}{\hbar^2} $$
 $$ L^2 = - \hbar \left[ \frac{1}{\sin^2\theta} \parcial{\theta} \left( \sin\theta \parcial{\theta} \right) +
   \frac{1}{\sin^2\theta}\parcial[^2]{\theta^2} \right] $$
 $$ L_\pm = L_x \pm i L_y \ ,\ L_\pm Y_{l,m} = \sqrt{l(l+1) - m(m\pm 1)} \hbar Y_{l, m\pm1}   $$
 Armónicos esféricos y Legendre
 $$ Y_{lm}(\theta, \varphi) = \sqrt{\frac{2l +1}{4\pi}\frac{(l-m)!}{(l+m)!}} P^m_l(\cos\theta) \expon{i m\varphi} $$
 $$P^m_l(z) = \frac{(-1)^m}{2^l l!}(1-z^2)^{m/2} \deriv[^{l+m}]{z^{l+m}} (z^2-1)^l $$
 Cuantización $$ L^2 Y = l(l+1)\hbar^2 Y \ , \ L_z Y = m\hbar Y \ ,\ |m| \le l$$
 $$ \left[ -\frac{\hbar^2}{2\mu} \deriv[^2]{r^2} + V(r) + \frac{l(l+1)\hbar^2}{2\mu r^2} \right] R_{nl}(r) = E_n
 R_{nl}(r) $$

\sep
 Átomos con un electrón ($l = 0, 1, \ldots n-1$)
 $$X(x)=\frac{R(r)}{r}Y(\theta,\varphi)\ ,\ a=\frac{\hbar}{\mu Z e^2}\ ,\ E=-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n^2}$$
 $$ \left[ \deriv[^2]{\xi^2} - \frac{l(l+1)}{\xi^2} + \frac{2}{\xi} - \frac{1}{\lambda^2}\right] R(r) = 0  \ ,\ \xi=r/a$$
 $$ R_{nl}(r) = \sqrt{ \frac{(n-l-1)!}{2n a [(n+l)!]^3} \left( \frac{2}{n}\right)^{2l+3} } \xi^{l+1}
 \mathcal{L}^{2l+1}_{n+l}\left( \frac{2\xi}{n} \right) \expon{-\xi/n}$$
 Laguerre $$ \mathcal{L}^q_p(z)=\deriv[^q]{z^q} \mathcal{L}_p(z)\, \qquad \mathcal{L}_p(x)=\expon{z} \deriv[^p]{z^p}
 ( z^p \expon{-z} ) $$

\sep
 $ h = 6,626\E{-34}\J\s = 4.136\E{-15}\eV\s $ \\
 $ \hbar = 1,055\E{-34}\J\s = 6,582\E{-16}\eV\s $ \\
 $ \quad |E_1| = \frac{1}{[4\pi \epsilon_0]}\frac{e^2}{2a} = 13,61 \eV $ \\
 $ a_0 = [4\pi \epsilon_0] \frac{\hbar^2}{m_e e^2} = 0.529\AAA$ \\
 $ \lambda_C = \frac{h}{m_e c} = 0,0243 \AAA $ \\
 $ m_e c^2 = 0.511\MeV \ , \quad m_p = 1836 m_e $

\end{document}
