%% This document created by Scientific Notebook (R) Version 3.0 \documentclass[12pt,thmsa]{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{sw20jart} %TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart} %TCIDATA{} %TCIDATA{Created=Mon Aug 19 14:52:24 1996} %TCIDATA{LastRevised=Wed Feb 28 12:33:19 2001} %TCIDATA{CSTFile=Lab Report.cst} %TCIDATA{PageSetup=0,0,0,0,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\hfill \thepage} %} \input{tcilatex} \begin{document} \subsection{\protect\vspace{1pt}Espectroscopia Molecular} $E=h\widetilde{\nu }=\frac{hc}{\lambda }=hc\upsilon =pc$ $h\widetilde{\nu }=E\ \ h\widetilde{\nu }=E$ \ \ $N_{0}\ h\widetilde{\nu }% =N_{0}E$ \ \ $\widetilde{\nu }=\frac{E}{h}$ \ \ $T=\frac{E}{K}\ $\ \ $% \lambda =\frac{hc}{E}$ \vspace{1pt}$c=2.99792458\times 10^{8}\unit{m}\unit{s}^{-1}$ $e=1.60217733\times 10^{-19}\unit{C}$ $h=6.6260755\times 10^{-34}\unit{J}\unit{s}$ \vspace{1pt}$e=4.8030\times 10^{-10}u.e.e.$ \vspace{1pt}N$_{A}=6.0225.10^{23}mol^{-1}$ R$_{H}=109677.6cm^{-1}$ R$_{\infty }=109737.3cm^{-1}$ \vspace{1pt}$\frac{\ \ 9.143\,\allowbreak 6\ \times 10^{3}\ \ \ \ \ \ \ \ \ }{8.617385\times 10^{-5}}=\allowbreak 1.\,\allowbreak 061\,1\times 10^{8}$ N$_{0}$E \ \ ev/mol/Julios/mol\ \ \ \ \ \ \ \ E/k \ $\ {{}^{o}}K$\ \ \ \ \ \ \ \ \ \ \ \ \ E \ \ $ev$\ $\ \ \ \ \ \ \ \ \ \ $\ \ \ \ \ \ \ \ \ \ \ \ E/h \ $\frac{c}{\unit{s}}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ hc/E\ \ m\ \ 1 atm $\times $ litro\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $7.\,\allowbreak 339\,8\times 10^{24}$\ \ \ \ \ \ \ \ \ \ \ \ \ $\ 6.325\times 10^{20}\ \ \ \ \ \ \ \ \ \ \ \ \ 1.529\times 10^{35}\ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \ \ 1.960\,4\times 10^{-27}$ 1 caloria \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\allowbreak 3.\,\allowbreak 031\,1\times 10^{23}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ $% 2.612\times 10^{19}\ \ \ \ \ \ \ \ \ \ 6.\,\allowbreak 313\,6\times 10^{33}\ \ \ $\ $\ \ \ \ \ \ \ \ \ \ \ \ \ \allowbreak 4.748\,3\times 10^{-26}$ 1 Julio $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.\,\allowbreak 243\,5\times 10^{22}\ \ \ \ \ \ \ \ \ \ \ \ \ 6.242\times 10^{18}$ \ \ \ \ \ \ \ \ $1.\,\allowbreak 509\,2\times 10^{33}$\ \ \ \ \ \ \ $\ \ \ \ \ \ \ \ \ \ 1.\,\allowbreak 986\,5\times 10^{-25}$ X\ \ blandos\ \ \ \ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.\,\allowbreak 061\,1\times 10^{8}\ \ \ \ \ \ \ \ \ \ \ 9.143\,\allowbreak 6\ \times 10^{3}\ \ \ \ \ \ \ \ \ \ \ 2.\,210\,9\times 10^{18}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.356\times 10^{-10}$\ \ \ \ R$_{\infty }$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $1.\,\allowbreak 578\,9\times 10^{5}$ \ \ \ \ \ \ \ \ \ $\ \ 1.3\,\allowbreak 606\times 10^{1}\allowbreak $ $\ \ \ \ \ \ \ \ \ \ 3.\,\allowbreak 289\,9\times 10^{15}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $% 9.\,\allowbreak 112\,7\times 10^{-8}\ $ R$_{H}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .\,\allowbreak 157\,8\times 10^{5}\ \ \ \ \ \ \ \ \ \ \ \ 1.\,\allowbreak 359\,8\times 10^{1}\ $\ \ \ \ $\ \ \ \ \ \ 3.\,\allowbreak 288\,1\times 10^{15}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ 9.\,1176\times 10^{-8}$ \ \ \ \ \ \ \ \ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ $\ \ \ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.\,\allowbreak 160\,4\times 10^{5}\allowbreak $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\times $\ $10^{1}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ 2.\allowbreak 418\times 10\allowbreak ^{15}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.\,\allowbreak 239\,8\times 10^{-7}$ ultravioleta\ \ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.\,175\,8\times 10^{-6}\ \ \ \ \ \ \ \ \ \ \ 4.\,843\,3\times 10^{1}\ \ \ \ \ \ \ \ \ \ \ \ \ 1.1711\times 10^{15}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.560\times 10^{-7}\ \ \ \ $ visible$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \allowbreak 3.\,\allowbreak 301\,5\times 10^{5}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.\,\allowbreak 845\times 10^{1}\ \ \ \ \ \ \ \ \ \ \ 6.\,\allowbreak 879\,1\times 10^{14}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4.358\times 10^{-7}\ $ media amarilla sodio$2.\,\allowbreak 440\,6\times 10^{5}\ \ \ \ \ \ \ \ \ \ \ 2.\,103\,2\ \times 10^{1}\ \ \ \ \ \ \ \ \ \ \ 5.\,085\,5\times 10^{14}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.895\times 10^{-7}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ visible\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\allowbreak 1.\,\allowbreak 390\,1\times 10^{5}\ \ \ \ \ \ \ \ \ \ \ 1.\,197\,9\ \ \times 10^{1}\ \ \ \ \ \ \ \ \ \ \ 2.89\,65\times 10^{14}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.0350\times 10^{-6}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1ev\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $% \ \ \ 1.1604\ \times 10^{5}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\times 10^{0}\ \ \ \ \ \ \ \ \ \ \ 2.418\,1\times 10^{14}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.239\,8\times 10^{-6}\ \ \ \ \ \ \ \ $ infrarojo $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.1510\ \times 10^{3}\ \ \ \ \ \ \ \ \ \ \ \ \ 9.\,918\,6\times 10^{-2}\ \ \ \ \ \ \ \ \ 2.398\,3\times 10^{13}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.250\times 10^{-5}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ microondas\ \ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \allowbreak 2.\,555\,5\times 10\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.\,202\,2\times 10^{-4}\ \ \ \ \ \ \ \ \ 5.324\,9\times 10^{10}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.563\times 10^{-2}$ \vspace{1pt}separaci\'{o}n de doblete amarillo del sodio$\ \ \ \ \ \allowbreak $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.\allowbreak 527\,9\times 10^{-3}\ \ \ \ \ \ \ \ 3.\,\allowbreak 040\,1\times 10^{-7}\ \ \ \ \ \ \ \ \ \ \ \ \ 1.\,\allowbreak 735\,1\times 10^{9}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.17\,\allowbreak 278\times 10^{0}$ 1cm$^{-1}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ \ \ \ \ \ \ \ \ \ \ \allowbreak 1.\,\allowbreak 438\,7\times 10^{-2}\ \ \ \ \ \ 1.\,\allowbreak 239\,8\times 10^{-6}$ \ \ \ \ \ \ \ $\ \ \ \ \ \ \ 2.\,\allowbreak 997\,9\times 10^{8}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ 1\times 10^{0}$ \vspace{1pt} Formula de Rydberg-Ritz o de Balmer $\nu =R_{H}(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}})=R_{H}(\frac{% n_{2}^{2}-n_{1}^{2}}{n_{1}^{2}\times n_{2}^{2}})$ $\widetilde{\nu }=\frac{1}{R_{H}}(\frac{n_{1}^{2}\times n_{2}^{2}}{% n_{2}^{2}-n_{1}^{2}})=$\ \ \ $\ 911.76\times 10^{-10}$ m\ $\times (\frac{% n_{1}^{2}\times n_{2}^{2}}{n_{2}^{2}-n_{1}^{2}})=\ 911.76\grave{A}$\ $\times (\frac{n_{1}^{2}\times n_{2}^{2}}{n_{2}^{2}-n_{1}^{2}})=$\ \ \ $\ 911.76\times 10^{-10}\times 5.\,\allowbreak 075\,1\times 10^{-5}=\allowbreak $ \vspace{1pt}$E=eV=\frac{1}{2}mv^{2}=\frac{p^{2}}{2m}$ $p=\frac{h}{\lambda }=\sqrt{2mE}=\sqrt{2meV}$ $\lambda =\frac{h}{\sqrt{2meV}}=\frac{6.6260755\times 10^{-34}\unit{J}\unit{s% }}{\sqrt{2\times 9.1093897\times 10^{-31}\unit{kg}1.60217733\times 10^{-19}% \unit{C}\times 100}}=\allowbreak 1.\,\allowbreak 226\,4\times 10^{-10}\left( \unit{kg}\right) \frac{\unit{m}^{2}}{\left( \unit{s}\right) \sqrt{\left( \unit{kg}\unit{A}\unit{s}\right) }}$ $\lambda =\frac{h}{\sqrt{2meV}}=\frac{6.6260755\times 10^{-34}\unit{J}\unit{s% }}{\sqrt{2\times 1.6726231\times 10^{-27}\unit{kg}1.60217733\times 10^{-19}% \unit{C}\times 100}}=\allowbreak 2.\,\allowbreak 862\,1\times 10^{-12}\left( \unit{kg}\right) \frac{\unit{m}^{2}}{\left( \unit{s}\right) \sqrt{\left( \unit{kg}\unit{A}\unit{s}\right) }}$ $E=h\widetilde{\nu }=\frac{hc}{\lambda }=hc\upsilon =pc$ $h\widetilde{\nu }=E\ \ h\widetilde{\nu }=E$ \ \ $N_{0}\ h\widetilde{\nu }% =N_{0}E$ \ \ $\widetilde{\nu }=\frac{E}{h}$ \ \ $T=\frac{E}{K}\ $\ \ $% \lambda =\frac{hc}{E}$ $E=h\widetilde{\nu }=\frac{hc}{\lambda }=hc\upsilon =pc$ $\Delta E\Delta t\leq h$ $\vspace{1pt}\frac{\Delta \lambda c\Delta t}{\lambda ^{2}}\thickapprox 1$ $\frac{hc}{\lambda ^{2}}\Delta \lambda \Delta t\thickapprox h$ $\Delta E\Delta t=h\Delta \widetilde{\nu }\Delta t=\frac{hc}{\lambda ^{2}}% \Delta \lambda \Delta t$ $\Delta \lambda \thickapprox \frac{\lambda ^{2}}{c\Delta t}$ Masa de la particula alfa Masa del nucleon Masa del mes\'{o}n-K Energ\'{i}a liberada en la fisi\'{o}n Masa del pi\'{o}n Masa del mu\'{o}n Energ\'{i}a de enlace de una particula alfa Energ\'{i}a de enlace aproximada por nucle\'{o}n en los n\'{u}cleos Energia aproximada de una particula alfa de Ra2.. Energia del enlace del Deuter\'{o}n Energia liberada en la desintegraci\'{o}n beta del Br... 1Mev Masa del electr\'{o}n (positr\'{o}n) Energ\'{i}a del cuanto gamma emitido por el is\'{o}mero del In114 Raya Ka (rayo X) del tungsteno Energ\'{i}a liberada en la desintegracion beta del tritio 1 \'{a}ngstrom Raya Ka (rayo x) del cobre Temperatura aproximada en el centro del Sol Raya Ka (rayo X) del carbono Energ\'{i}a necesaria para la ionizaci\'{o}n completa del \'{a}tomo de helio Potencial de ionizaci\'{o}n del helio Rinf (potencial de ionizaci\'{o}n del hidr\'{o}geno) Raya ultravioleta en el mercurio Energia de Ferm\'{i} en la plata Energia de enlace de la molecula H2 Raya azul emitida por el mercurio Raya D (amarilla) del sodio Raya roja emitida por el cadmio Calor de vaporizacon del alcohol et\'{i}lico Frecuencia de vibraci\'{o}n de la mol\'{e}cula CO Temperatura Debye en el diamante Calor de fusi\'{o}n del cobre Temperatura de Curie del hierro ''Temperatura normal'' 20${{}^o}$C Frecuencia de vibracion de la mol\'{e}cula Temperatura de Debye en el plomo Punto de ebullici\'{o}n del hidr\'{o}geno Calor de vaporizaci\'{o}n del helio Temperatura a la cual el plomo pasa a superconductor Punto de ebullici\'{o}n del helio He4 Punto lambda del helio Energ\'{i}a de interacci\'{o}n de dos magnetones de Bohr separados por lA Temperatura de transici\'{o}n del cadmio al estado superconductor Transici\'{o}n hiperfina en el cesio Frecuencia de rotaci\'{o}n observada en la mol\'{e}cuja Icl Raya del hidr\'{o}geno de 21 cm (transici\'{o}n hiperfina) Magnet\'{o}n de Bohr en 100 gauss Energ\'{i}a de interacci\'{o}n de un magnet\'{o}n de Bohr y un magnet\'{o}n nuclear separados por 1 A Frecuencia de precesi\'{o}n del prot\'{o}n en 1000 gauss Magnet\'{o}n nuclear en 1000 gauss \vspace{1pt}Problema 6/IV $\int_{0}^{a}\frac{2}{a}x\sin ^{2}\frac{2\pi }{a}xdx=\allowbreak \frac{1}{2}% a $ Problema 7/IV $\int_{0}^{a}\sqrt{\frac{2}{a}}\sin n\pi \frac{x}{a}(-\frac{n^{2}\pi ^{2}% \sqrt{2}}{a^{\frac{3}{2}}}\sin n\pi \frac{x}{a})dx$ ----------------------------------------------------------------- N$_{t}$=N$_{0}e^{-\lambda t}$ \ \ \ \ \ \ \ \ \ \ \ $\tau =\frac{\ln 2}{% \lambda }$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\lambda $\ $=\frac{\ln 2}{\tau }$ $\log 2=\allowbreak .\,\allowbreak 693\,15$ 1000 bolas =$\ 10000\times 4/3\times \pi \times 1^{3}=\allowbreak 41888mm^{3}=41.888cm^{3}$ Rollos de 100 gr de hilo de cobre marca Crovisa/ marca Avisor \ $% 39m.2,4\Omega /m$ $\phi =0.6mm$ $\vspace{1pt}L=5.\,\allowbreak 389\,1\times 10^{-6}Henrios$ $C=4.7\times 10^{-9}Faradios$ $L=$ $\vspace{1pt}C=12\times 10^{-12}Faradios$ $Z=L2\pi f-\frac{1}{C2\pi f}=2\pi L\upsilon -\frac{1}{2\pi C\nu }=0$ $2\pi L\upsilon =\frac{1}{2\pi C\nu }$ $\nu =\frac{1}{2\pi \sqrt{LC}}$ $\sqrt{LC}=\frac{1}{2\pi \nu }=\frac{1}{2\pi 10^{6}}=\allowbreak 1.\,\allowbreak 591\,5\times 10^{-7}$ $L=\frac{2.\,\allowbreak 532\,9\times 10^{-14}}{C}=\frac{2.\,\allowbreak 532\,9\times 10^{-14}}{4.7\times 10^{-9}}=\allowbreak 5.\,\allowbreak 389\,1\times 10^{-6}Henrios$ $Z=2\pi \allowbreak 5.\,\allowbreak 389\,1\times 10^{-6}\upsilon -\frac{1}{% 2\pi 4.7\times 10^{-9}\nu }=$ $\allowbreak y=3.\,\allowbreak 386\,1\times 10^{-5}$Z=$x-\frac{% 3.\,\allowbreak 386\,3\times 10^{7}}{x}$\FRAME{itbpF}{3in}{2in}{0in}{}{}{}{% \special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2in;depth 0in;display "USEDEF";function \TEXUX{$3.\,\allowbreak 386\,1\times 10^{-5}x-\frac{3.\,\allowbreak 386\,3\times 10^{7}}{x}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "-1.56935";xmax "-0.21167";xviewmin "-1.569350";xviewmax "-0.211670";yviewmin "1.8809674570537E7";yviewmax "1.62803567457228E8";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}} $\vspace{1pt}R=39m2.4\Omega /m=\allowbreak 93.\,\allowbreak 6\Omega $ $Z=\sqrt{\left( 93.\,\allowbreak 6\right) ^{2}+(Lw-\frac{1}{Cw})^{2}}=\sqrt{% \left( 93.\,\allowbreak 6\right) ^{2}+(\allowbreak 5.\,\allowbreak 389\,1\times 10^{-6}w-\frac{1}{4.7\times 10^{-9}w})^{2}}$ \FRAME{itbpF}{3in}{2.0003in}{0.0311in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0.0311in;display "USEDEF";function \TEXUX{$\sqrt{\left( 93.\,\allowbreak 6\right) ^{2}+(\allowbreak 5.\,\allowbreak 389\,1\times 10^{-6}w-\frac{1}{4.7\times 10^{-9}w})^{2}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "-0.09491";xmax "9.90509";xviewmin "-0.094910";xviewmax "9.905090";yviewmin "-2.2417654324312664000000E16";yviewmax "1.143748724743846000000000E18";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{w};var1name \TEXUX{$x$};}} $L=\frac{2.\,\allowbreak 532\,9\times 10^{-14}}{C}=\frac{2.\,\allowbreak 532\,9\times 10^{-14}}{4.7\times 10^{-9}}=\allowbreak 5.\,\allowbreak 389\,1\times 10^{-6}Henrios$ Esto es para estrenar este metodo de trasmision de ''datos'' a traves del SN. \textquestiondown Pregunta que pasa aqui?. $e^{-iEt/\hbar -t/2\tau }=\int g(E^{\prime })e^{-iE^{\prime }t/\hbar }dE^{\prime }$ $g(E\prime )=\frac{(\hbar /\tau )}{(E-E^{\prime })+\left( \hbar /2\tau \right) ^{2}}$ \textquestiondown Como se puede expresar esto mas correctamente? $\func{assume}(\tau ,\func{positive})$ $\int_{0}^{t}e^{-\frac{t}{\tau }}dt\allowbreak =\allowbreak -\tau e^{-\frac{t% }{\tau }}+\tau $ si $t\rightarrow \infty $ $\ \ \ \ \int_{0}^{t}e^{-\frac{t}{\tau }% }dt\allowbreak \rightarrow \tau $ $\Psi =5\sin \frac{\pi x}{1}+3\cos \frac{\pi x}{1}$\FRAME{itbpF}{3in}{2in}{% 0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2in;depth 0in;display "USEDEF";function \TEXUX{$5\sin \frac{\pi x}{1}+3\cos \frac{\pi x}{1}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "-5";xmax "5";xviewmin "-5.000000";xviewmax "5.000000";yviewmin "-6.064000";yviewmax "6.068461";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}} $\allowbreak \sin x=\frac{e^{ix}-e^{-ix}}{2i}$ $\cos x=\frac{e^{ix}-e^{-ix}}{2}$ $\Psi =e^{ix}=\allowbreak \cos x+i\sin x$ A ver, k es la constante de recuperaci\'{o}n del sistema, es decir, para un desplazamiento x respecto la posici\'{o}n de equilibrio aparece una fuerza del tipo F = - k x. Segun la segunda ley de Newton tenemos, pues $-kx=ma->ma+kx=0->a+k/mx=0$ Sabiendo que a es la derivada segunda de la posici\'{o}n, respecto el tiempo, eso constituye una equaci\'{o}n diferencial de segundo orden homogenia. La soluci\'{o}n general se puede escribir de varias formas, una de ellas es: $x(t)=A\cos (\sqrt{k/m})t+B)$ El termino sqrt(k/m) m\'{u}ltiplica al tiempo en el argumento del coseno, a una cantidad que multiplica al tiempo en el argumento de una funcion sinosoidal se le conoce como freq\"{u}\`{e}ncia angular w. El hecho de que w aumente con la constante k tiene sentido ya que es de esperar ya que un muelle mas ''duro'' da lugar a oscilaciones m\'{a}s r\`{a}pidas. En cambio, en aumentar la masa, cuasta m\'{a}s ''esfuerzo'' realizar el movimiento y por lo tanto la frecuencia sube. El hecho que aparezca la raiz no tiene mas significado que w tenga dimensiones de inversa de tiempo. Por otra parte, mas que la freq\"{u}encia angular, tiene sentido fisica hablar del per\'{i}odo de la oscilacion (o de la frequencia real). Dado que el per\'{i}odo de una funcion trigonometrica es 2 pi, el periodo de la oscilacion sera $T=2\pi /w=2\pi \sqrt{m/k}$ $\Psi =e^{i\sqrt{k/m}}=\cos (\sqrt{k/m})t+i\sin (\sqrt{k/m})t$ --------------------------------------------------------- \vspace{1pt} DATOS tomados de QC: Polares; $x=r\sin \theta \cos \phi $ $\ y=r\sin \theta \sin \phi $ $\ \ z=r\cos \theta $ $\ d\tau =dxdydz=r^{2}\sin \theta drd\theta d\phi $ $\Phi _{nlm_{l}}\left( r,\theta ,\phi \right) =Nr^{n-1}e^{-\frac{\rho }{n}% }Y_{lm_{l}}\left( \theta ,\phi \right) $ \ \ \ \ \ \ \ $\ \rho =\frac{Zr}{% a_{0}}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ \begin{tabular}{lllll} n & l & m & orbital & funcion de onda completa de los hidrogenoides \\ 1 & 0 & 0 & 1s & $\Phi _{1,0,0}=\frac{1}{\sqrt{\pi }}\left( \frac{Z}{a_{0}}% \right) ^{\frac{3}{2}}e^{-\rho }$ \\ 2 & 0 & 0 & 2s & $\Phi _{2,0,0}=\frac{1}{4\sqrt{2\pi }}\left( \frac{Z}{a_{0}}% \right) ^{\frac{3}{2}}\left( 2-\rho \right) e^{-\frac{\rho }{2}}$ \\ 2 & 1 & 0 & 2p$_{z}$ & $\Phi _{2,1,0}=\frac{1}{4\sqrt{2\pi }}\left( \frac{Z}{% a_{0}}\right) ^{\frac{3}{2}}\rho e^{-\frac{\rho }{2}}\cos \theta $ \\ 2 & 1 & $\pm 1$ & 2p$_{x}$ & $\Phi _{2,1,1}=\frac{1}{4\sqrt{2\pi }}\left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho e^{-\frac{\rho }{2}}\sin \theta \cos \phi $ \\ 2 & 1 & $\pm 1$ & 2p$_{y}$ & $\Phi _{2,1,-1}=\frac{1}{4\sqrt{2\pi }}\left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho e^{-\frac{\rho }{2}}\sin \theta \sin \phi $ \\ 3 & 0 & 0 & 3s & $\Phi _{3,0,0}=\frac{1}{81\sqrt{3\pi }}\left( \frac{Z}{a_{0}% }\right) ^{\frac{3}{2}}\rho e^{-\frac{\rho }{2}}\left( 27-18p+2p^{2}\right) e^{-\frac{\rho }{3}}$ \\ 3 & 1 & 0 & 3p$_{z}$ & $\Phi _{3,1,0}=\frac{\sqrt{2}}{81\sqrt{\pi }}\left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\left( 6\rho -p^{2}\right) e^{-\frac{% \rho }{3}}\cos \theta $ \\ 3 & 1 & $\pm 1$ & 3p$_{x}$ & $\Phi _{3,1,1}=\frac{\sqrt{2}}{81\sqrt{\pi }}% \left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\left( 6\rho -p^{2}\right) e^{-% \frac{\rho }{3}}\sin \theta \cos \phi $ \\ 3 & 1 & $\pm 1$ & 3p$_{y}$ & $\Phi _{3,1,-1}=\frac{\sqrt{2}}{81\sqrt{\pi }}% \left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\left( 6\rho -p^{2}\right) e^{-% \frac{\rho }{3}}\sin \theta \sin \phi $ \\ 3 & 2 & 0 & 3d$_{z^{2}}$ & $\Phi _{3,2,0}=\frac{1}{81\sqrt{6\pi }}\left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{5}}\left( 3\cos ^{2}\theta -1\right) $ \\ 3 & 2 & $\pm 1$ & 3d$_{xz}$ & $\Phi _{3,2,1}=\frac{\sqrt{2}}{81\sqrt{\pi }}% \left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}% }\sin \theta \cos \theta \cos \phi $ \\ 3 & 2 & $\pm 1$ & 3d$_{yz}$ & $\Phi _{3,2,-1}=\frac{\sqrt{2}}{81\sqrt{\pi }}% \left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}% }\sin \theta \cos \theta \cos \phi $ \\ 3 & 2 & $\pm $2 & 3d$_{x^{2-y^{2}}}$ & $\Phi _{3,2,2}=\frac{1}{81\sqrt{\pi }}% \left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}% }\sin ^{2}\theta \cos 2\phi $ \\ 3 & 2 & $\pm 2$ & 3d$_{xy}$ & $\Phi _{3,2,-2}=\frac{1}{81\sqrt{2\pi }}\left( \frac{Z}{a_{0}}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}}\sin ^{2}\theta \sin 2\phi $% \end{tabular} \begin{tabular}{llll} n & l & orbital & R$_{n,l}\left( \rho \right) $ \\ 1 & 0 & 1s & $\psi =2\left( \frac{1}{a_{0}}\right) ^{\frac{3}{2}}e^{-\rho }$ \\ 2 & 0 & 2s & $\psi =\frac{1}{2\sqrt{2}}\left( \frac{1}{a_{0}}\right) ^{\frac{% 3}{2}}\left( 2-\rho \right) e^{-\frac{\rho }{2}}$ \\ 2 & 1 & 2p & $\psi =\frac{1}{2\sqrt{6}}\left( \frac{1}{a_{0}}\right) ^{\frac{% 3}{2}}\rho e^{-\frac{\rho }{2}}$ \\ 3 & 0 & 3s & $\psi =\frac{2}{81\sqrt{3}}\left( \frac{1}{a_{0}}\right) ^{% \frac{3}{2}}(27-18\rho +2\rho ^{2})e^{-\frac{\rho }{3}}$ \\ 3 & 1 & 3p & $\psi =\frac{4}{81\sqrt{6}}\left( \frac{1}{a_{0}}\right) ^{% \frac{3}{2}}(6-\rho )\rho e^{-\frac{\rho }{3}}$ \\ 3 & 2 & 3d & $\psi =\frac{4}{81\sqrt{30}}\left( \frac{1}{a_{0}}\right) ^{% \frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}}$% \end{tabular} $\ $ $\ \overline{r}=\frac{n^{2}a}{Z}\left( \frac{3}{2}-\frac{l\left( l+1\right) }{2n^{2}}\right) $ \ \ \ $E=h\widetilde{\nu }=hc\nu =\mu _{z}\frac{e^{4}}{% 2\hbar ^{2}}\frac{Z^{2}}{n^{2}}$ $\ \ \frac{\left( 9.1093897\times 10^{-31}\right) ^{4}}{2\times \left( 1.05457266\times 10^{-34}\right) ^{2}}% =\allowbreak 3.\,\allowbreak 095\,8\times 10^{-53}=\allowbreak 3.\,\allowbreak 095\,8\times 10^{-53}$\ /$\left( 6.6260755\times 10^{-34}\times 2.99792458\times 10^{8}\right) =\allowbreak 1.\,\allowbreak 986\,4\times 10^{-25}$ $\psi =2\left( \frac{1}{a_{0}}\right) ^{\frac{3}{2}}e^{-\rho }$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \psi ^{2}=\left( 2\left( \frac{1}{% a_{0}}\right) ^{\frac{3}{2}}e^{-\rho }\right) ^{2}=\allowbreak \frac{4}{% a_{0}^{3}}e^{-2\rho }$ $\ \rho =\frac{r}{a_{0}}$ $\int_{0.9}^{1.1a}r^{2}\allowbreak \frac{4}{a_{0}^{3}}e^{-2\rho }dr=a^{2}\int_{0.9a}^{1.1a}\rho ^{2}\allowbreak \frac{4}{a_{0}^{3}}e^{-2\rho }d\rho =\allowbreak a^{2}\int_{.\,\allowbreak 9a}^{1.\,\allowbreak 1a}4.0% \frac{\rho ^{2}}{a_{0}^{3}}e^{-2.0\rho }\,d\rho =\allowbreak -.0\,\allowbreak 2a^{2}\frac{121.0\exp \left( -2.\,\allowbreak 2a\right) a^{2}+110.0\exp \left( -2.\,\allowbreak 2a\right) a+50.0\exp \left( -2.\,\allowbreak 2a\right) -81.0\exp \left( -1.\,\allowbreak 8a\right) a^{2}-90.0\exp \left( -1.\,\allowbreak 8a\right) a-50.0\exp \left( -1.\,\allowbreak 8a\right) }{a_{0}^{3}}\allowbreak $ $\psi =2\left( \frac{1}{0.5}\right) ^{\frac{1}{2}}e^{-\rho }$\FRAME{itbpF}{% 3in}{2.0003in}{0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "USEDEF";function \TEXUX{$2\left( \frac{1}{0.5}\right) ^{\frac{1}{2}}e^{-\rho }$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "5";xviewmin "0.000000";xviewmax "5.000000";yviewmin "-0.037130";yviewmax "2.885738";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v961};var1name \TEXUX{$x$};}}1s $\psi =\frac{1}{2\sqrt{2}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\left( 2-\rho \right) e^{-\frac{\rho }{2}}$\FRAME{itbpF}{3in}{2.0003in}{0in}{}{}{}{% \special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "USEDEF";function \TEXUX{$\frac{1}{2\sqrt{2}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\left( 2-\rho \right) e^{-\frac{\rho }{2}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "5";xviewmin "0.000000";xviewmax "5.000000";yviewmin "-0.316015";yviewmax "2.046320";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v961};var1name \TEXUX{$x$};}}2s $\psi =\frac{1}{2\sqrt{6}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\rho e^{-% \frac{\rho }{2}}\cos \theta $\FRAME{itbpF}{3in}{2.0003in}{0.0208in}{}{}{}{% \special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0.0208in;display "USEDEF";function \TEXUX{$\frac{1}{2\sqrt{6}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\rho e^{-\frac{\rho }{2}}\cos \theta $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "6";ymin "0";ymax "15";xviewmin "-0.120000";xviewmax "6.122400";yviewmin "-0.300000";yviewmax "15.306000";zviewmin "-0.438344";zviewmax "0.441171";rangeset"XY";phi -98;theta -5;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle "wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis \TEXUX{v961};var1name \TEXUX{$\theta $};var2name \TEXUX{$\rho $};}}\FRAME{% itbpF}{3in}{2.0003in}{0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "USEDEF";function \TEXUX{$\frac{1}{2\sqrt{6}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\rho e^{-\frac{\rho }{2}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "15";xviewmin "0.000000";xviewmax "15.000000";yviewmin "-0.008495";yviewmax "0.433423";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v961};var1name \TEXUX{$x$};}}2p $\psi =\frac{2}{81\sqrt{3}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}% }(27-18\rho +2\rho ^{2})e^{-\frac{\rho }{3}}\sin \theta $\FRAME{itbpF}{3in}{% 2.0003in}{0.1669in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0.1669in;display "USEDEF";function \TEXUX{$\frac{2}{81\sqrt{3}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}(27-18\rho +2\rho ^{2})e^{-\frac{\rho }{3}}\sin \theta $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "3";ymin "0";ymax "2";xviewmin "0";xviewmax "3";yviewmin "0";yviewmax "2";zviewmin "-0.042826";zviewmax "1.109661";rangeset"XY";phi 95;theta -95;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle "wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis \TEXUX{v961};var1name \TEXUX{$\theta $};var2name \TEXUX{$\rho $};}}\FRAME{% itbpF}{3in}{2.0003in}{0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "USEDEF";function \TEXUX{$\frac{2}{81\sqrt{3}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}(27-18\rho +2\rho ^{2})e^{-\frac{\rho }{3}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "2";xviewmin "0.000000";xviewmax "2.000000";yviewmin "-0.042889";yviewmax "1.111293";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v961};var1name \TEXUX{$x$};}}3s $\psi =\frac{4}{81\sqrt{6}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}(6-\rho )\rho e^{-\frac{\rho }{3}}\sin \theta $\FRAME{itbpF}{3in}{2.0003in}{-0.0519in% }{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth -0.0519in;display "USEDEF";function \TEXUX{$\frac{4}{81\sqrt{6}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}(6-\rho )\rho e^{-\frac{\rho }{3}}\sin \theta $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "3";ymin "0";ymax "6";xviewmin "-0.060000";xviewmax "3.061200";yviewmin "-0.120000";yviewmax "6.122400";zviewmin "-0.004726";zviewmax "0.241138";rangeset"XY";phi 153;theta 101;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle "wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis \TEXUX{v961};var1name \TEXUX{$\theta $};var2name \TEXUX{$\rho $};}}\FRAME{itbpF}{3in}{2.0003in}{0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "USEDEF";function \TEXUX{$\frac{4}{81\sqrt{6}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}(6-\rho )\rho e^{-\frac{\rho }{3}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "6";xviewmin "0.000000";xviewmax "6.000000";yviewmin "-0.004733";yviewmax "0.241495";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v961};var1name \TEXUX{$x$};}}3d $\psi =\frac{4}{81\sqrt{30}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}}\sin \theta $\FRAME{itbpF}{3in}{2.0003in}{-0.0623in}{% }{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth -0.0623in;display "USEDEF";function \TEXUX{$\frac{4}{81\sqrt{30}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}}\sin \theta $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "3";ymin "0";ymax "25";xviewmin "-0.060000";xviewmax "3.061200";yviewmin "-0.500000";yviewmax "25.510000";zviewmin "-0.002477";zviewmax "0.126378";rangeset"XY";phi 58;theta -95;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle "wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis \TEXUX{v961};var1name \TEXUX{$\theta $};var2name \TEXUX{$\rho $};}}\FRAME{% itbpF}{3in}{2.0003in}{0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0in;display "USEDEF";function \TEXUX{$\frac{4}{81\sqrt{30}}\left( \frac{1}{0.5}\right) ^{\frac{3}{2}}\rho ^{2}e^{-\frac{\rho }{3}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "25";xviewmin "0";xviewmax "25";yviewmin "-0.002485";yviewmax "0.126777";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v961};var1name \TEXUX{$x$};}}3s Funciones de onda del rotor rigido; \textsf{V=1/r}\newline \textsf{ \begin{tabular}{llllll} E$\frac{\hbar ^{2}}{2l}$ & l & m & & & \\ 0 & 0 & 0 & $\psi _{0}=\psi _{0,0}$ & $\left( \frac{1}{4\pi }\right) ^{\frac{% 1}{2}}$ & $\left( \frac{1}{4\pi }\right) ^{\frac{1}{2}}$ \\ 2 & 1 & 0 & $\psi _{z}=\psi _{1,0}$ & $\left( \frac{1}{4\pi }\right) ^{\frac{% 1}{2}}\cos \theta $ & $\left( \frac{3}{4\pi }\right) ^{\frac{1}{2}}z$ \\ 2 & 1 & 1 & $\psi _{x}=\frac{1}{\sqrt{2}}(\psi _{1,1}+\psi _{1,-1})$ & $% \left( \frac{3}{4\pi }\right) ^{\frac{1}{2}}\sin \theta \cos \phi $ & $% \left( \frac{3}{4\pi }\right) ^{\frac{1}{2}}x$ \\ 2 & 1 & -1 & $\psi _{y}=-\frac{i}{\sqrt{2}}(\psi _{1,1}-\psi _{1,-1})$ & $% \left( \frac{3}{4\pi }\right) ^{\frac{1}{2}}\sin \theta \cos \phi $ & $% \left( \frac{3}{4\pi }\right) ^{\frac{1}{2}}y$ \\ 6 & 2 & 0 & $\psi _{z^{2}}=\psi _{2,0}$ & $\left( \frac{5}{16\pi }\right) ^{% \frac{1}{2}}(3\cos ^{2}\theta -1)$ & $\left( \frac{1}{4\pi }\right) ^{\frac{1% }{2}}(3z^{2}-1)$ \\ 6 & 2 & +1 & $\psi _{xy}=\frac{1}{\sqrt{2}}(\psi _{2,1}+\psi _{2,-1})$ & $% \left( \frac{15}{4\pi }\right) ^{\frac{1}{2}}\sin \theta \cos \phi $ & $% \left( \frac{1}{4\pi }\right) ^{\frac{1}{2}}xz$ \\ 6 & 2 & -1 & $\psi _{yz}=-\frac{i}{\sqrt{2}}(\psi _{2,1}-\psi _{2,-1})$ & $% \left( \frac{15}{4\pi }\right) ^{\frac{1}{2}}\sin \theta \cos \theta \cos \phi $ & $\left( \frac{1}{4\pi }\right) ^{\frac{1}{2}}yz$ \\ 6 & 2 & 2 & $\psi _{x^{2}-y^{2}}=\frac{1}{\sqrt{2}}(\psi _{2,2}+\psi _{2,-2}) $ & $\left( \frac{15}{16\pi }\right) ^{\frac{1}{2}}\sin ^{2}\theta \cos 2\phi $ & $\left( \frac{1}{4\pi }\right) ^{\frac{1}{2}}(x^{2}-y^{2})$ \\ 6 & 2 & -2 & $\psi _{xy}=\frac{1}{\sqrt{2}}(\psi _{2,2}+\psi _{2,-2})$ & $% \left( \frac{15}{16\pi }\right) ^{\frac{1}{2}}\sin ^{2}\theta \sin 2\phi $ & $\left( \frac{1}{4\pi }\right) ^{\frac{1}{2}}xy$% \end{tabular} } \ \ \ \ INTEGRALES;\ \ \ \ Orbitales hidrogenoideos: $\int_{0}^{\infty }x^{n}e^{-ax}dx=\frac{n!}{a^{n+1}}$ $\ \ $\ $\ \ \ \ \ \ \ \ \int_{0}^{\infty }e^{-ax}dx=\frac{1-e^{-ax}}{a}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \int_{0}^{\infty }x^{-1}e^{-ax}dx=\infty $ \vspace{1pt}\ \ Oscilador armonico; $\int_{0}^{\infty }e^{-ax^{2}}dx=\frac{1}{2}\sqrt{\frac{\pi }{a}}$ $\int_{0}^{\infty }xe^{-ax^{2}}dx=\frac{1}{2a}$ $\int_{0}^{\infty }x^{2}e^{-ax^{2}}dx=\frac{1}{4}\sqrt{\frac{\pi }{a^{3}}}$ $\int_{0}^{\infty }x^{3}e^{-ax^{2}}dx=\frac{1}{2a^{2}}$ $\int_{0}^{\infty }x^{4}e^{-ax^{2}}dx=\frac{3}{8}\sqrt{\frac{\pi }{a^{5}}}$ $\int_{0}^{\infty }x^{5}e^{-ax^{2}}dx=\frac{1}{a^{3}}$ $\int x^{6}e^{-ax^{2}}dx=\frac{15}{16}\sqrt{\frac{\pi }{a^{7}}}$ $.......................$ $\int x^{n}e^{-ax^{2}}dx=\frac{\Gamma \left( \frac{n+1}{2}\right) }{2\sqrt{% a^{n+1}}}$\ \ \ (siendo n\TEXTsymbol{>}0) Si n es par $\frac{n-1!!}{2^{\left( \frac{n}{2}+1\right) }}\sqrt{\frac{\pi }{% \beta ^{n+1}}}$ Si n es par $\frac{\left[ \frac{\left( n-1\right) }{2}\right] !}{2^{\left( \frac{n+1}{2}\right) }}$ Otras; $\int x^{n}\sin x=-x^{n}\cos x+n\int x^{n-1}\cos xdx$ $\int x^{n}e^{nx}dx=-n\int x^{n-1}e^{nx}dx$ $\int \sin ^{2}xdx=-\frac{\sin x\cos x}{2}+cte$ \vspace{1pt} OSCILADOR ARMONICO; $E_{0}=\frac{1}{2}h\nu ................\Psi _{0}=(\sqrt{\beta }/\sqrt{\pi }% )^{\frac{1}{2}}e^{-\frac{\beta x^{2}}{2}% }......................................\alpha =\frac{2mE}{\hbar ^{2}}$ $E_{1}=\frac{3}{2}h\nu ................\Psi _{1}=(2\beta \sqrt{\beta }/\sqrt{% \pi })^{\frac{1}{2}}xe^{-\frac{\beta x^{2}}{2}% }.................................\beta =\frac{\sqrt{mk}}{\hbar }$ $E_{2}=\frac{5}{2}h\nu ................\Psi _{2}=(\sqrt{\beta }/2\sqrt{\pi }% )^{\frac{1}{2}}(2\beta x^{2}-1)e^{-\frac{\beta x^{2}}{2}% }.....................k=4\pi ^{2}\tilde{\upsilon}^{2}m$ $E_{2}=\frac{7}{2}h\nu ................\Psi _{3}=\frac{(\sqrt{\beta }/\sqrt{% \pi })^{-\frac{1}{2}}}{\sqrt{3\pi }}(3\beta ^{\frac{1}{2}}x-2\beta ^{\frac{3% }{2}}x^{3})e^{-\frac{\beta x^{2}}{2}}$ coeficiente inseguro $\ \Psi _{0}=A_{0}e^{-\frac{u^{2}}{2}}=A_{0}e^{-\frac{\beta x^{2}}{2}}$ $\Psi _{1}=A_{1}ue^{-\frac{u^{2}}{2}}=A_{1}ue^{-\frac{\beta x^{2}}{2}}$ $\Psi _{2}=A_{2}(1-2u^{2})e^{-\frac{u^{2}}{2}}$ $\Psi _{3}=A_{3}(3u-2u^{3})e^{-\frac{u^{2}}{2}}$ $\ u^{2}=\beta x^{2}$ $=\left( \alpha x\right) ^{2}$ \vspace{1pt}----------------------------------- $\mathsf{V=kx}^{2}$ $\alpha =\frac{2mE}{h^{2}/4\pi ^{2}}$ $\vspace{1pt}k=4\pi ^{2}m/\lambda $ $\beta =\frac{\sqrt{mk}}{h/2\pi }=2\pi m/\lambda h/2\pi $ $\Psi _{0}=\left( \frac{\sqrt{\beta }}{\sqrt{\pi }}\right) ^{\frac{1}{2}}e^{-% \frac{\beta x^{2}}{2}}$ $\Psi _{1}=\left( \frac{2\beta \sqrt{\beta }}{\sqrt{\pi }}\right) ^{\frac{1}{% 2}}xe^{-\frac{\beta x^{2}}{2}}$ $\Psi _{2}=\left( \frac{\sqrt{\beta }}{2\sqrt{\pi }}\right) ^{\frac{1}{2}% }(2\beta x^{2}-1)e^{-\frac{\beta x^{2}}{2}}$ ------------------------------------------------------------------------------------------------------- $\cos x=\frac{e^{xi}+e^{-xi}}{2}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i\sin x=% \frac{e^{xi}-e^{-xi}}{2}$ $\int udv=uv-\int vdu$ --------------------------------------------------------------------------------------------------------- $\Psi _{n}^{\left( 0\right) }=c_{1}\varphi _{1}+c_{2}\varphi _{2}\rightarrow \Psi _{n}=\left( c_{1}\varphi _{1}+c_{2}\varphi _{2}\right) A\cos wt$ ----------------------------------------------------------------------------------------------------------- Vemos que paridad de $\Psi _{nf}\left( x\right) =\left( -1\right) ^{n}$ Por tanto; \ \ \ \ \ \ A) Si $\Psi _{nf}$ $\left( x\right) $y $\Psi _{ni}\left( x\right) $\ tienen la misma paridad $\rightarrow I_{nfni}=\int_{-\frac{a}{2}% }^{\frac{a}{2}}\Psi _{nf}x\Psi _{ni}dx=0$ \ \ \ \ \ \ B) Y si tienen diferente como por ejemplo; $\ \ \ \ \ \ \ \ \ \ \ \ \ \ I_{30}=\frac{\alpha }{4\sqrt{3\pi }}\int_{-\infty }^{\infty }x\left[ 8\left( \alpha x\right) ^{3}-12\left( \alpha x\right) \right] e^{-\left( \alpha x\right) ^{2}}dx=\frac{2}{\alpha \sqrt{3\pi }}\int_{0}^{\infty }\left( 2y^{4}-3y^{2}\right) e^{-y^{2}}dy=\allowbreak 0$ $\mu \left( v_{30}\right) =\int_{-\infty }^{\infty }\frac{(\sqrt{\beta }/% \sqrt{\pi })^{-\frac{1}{2}}}{\sqrt{3\pi }}(3\beta ^{\frac{1}{2}}x-2\beta ^{% \frac{3}{2}}x^{3})e^{-\frac{\beta x^{2}}{2}}x(\sqrt{\beta }/\sqrt{\pi })^{% \frac{1}{2}}e^{-\frac{\beta x^{2}}{2}}dx=2\frac{1}{(\sqrt{\beta }/\sqrt{\pi }% )^{\frac{1}{2}}\sqrt{3\pi }}(\sqrt{\beta }/\sqrt{\pi })^{\frac{1}{2}% }=\allowbreak \frac{2\sqrt{\beta }}{\sqrt{\beta }\sqrt{3\pi }}% \int_{0}^{\infty }x(3\beta ^{\frac{1}{2}}x-2\beta ^{\frac{3}{2}}x^{3})e^{-% \frac{\beta x^{2}}{2}}e^{-\beta x^{2}}dx=\frac{2}{\sqrt{\beta }\sqrt{3\pi }}% \int_{0}^{\infty }(3\beta x^{2}-2\beta ^{2}x^{4})e^{-\beta x^{2}}dx$ $=\frac{2}{\sqrt{\beta }\sqrt{3\pi }}\int_{0}^{\infty }(3y^{2}-2y^{4})e^{-y^{2}}dy=\allowbreak 0\allowbreak $ \ \ \ \ \ \ \ \ \ \ \ Sin embargo: $I_{10}=\frac{\alpha }{\sqrt{2\pi }}% \int_{-\infty }^{\infty }x\left( 2\left( \alpha x\right) \right) e^{-\left( \alpha x\right) ^{2}}dx=\frac{4}{\sqrt{2\pi }}\int_{0}^{\infty }y^{2}e^{-y^{2}}dy=\allowbreak \frac{1}{\sqrt{2}}$ \ \ \ \ \ \ \ \ \ \ \ Luego la regla de seleccion es; $\Delta n=\pm 1$ $I_{10}=\frac{\alpha }{\sqrt{2\pi }}\int_{-\infty }^{\infty }x\left( 2\left( \alpha x\right) \right) e^{-\left( \alpha x\right) ^{2}}dx=\frac{4\alpha }{% \sqrt{2\pi }}\int_{0}^{\infty }y^{2}e^{-y^{2}}dy=\allowbreak \frac{1}{2}% \alpha \sqrt{2}$ $\left( \frac{d\mu }{dr}\right) _{r=r_{e}}=\left[ \frac{d\left( \frac{\alpha }{\sqrt{2\pi }}\int_{r_{e}}^{r}r\left( 2\left( \alpha r\right) \right) e^{-\left( \alpha r\right) ^{2}}d\right) x}{dr}\right] _{r=r_{e}}=\frac{% \alpha }{\sqrt{2\pi }}r_{e}\left( 2\left( \alpha r_{e}\right) \right) e^{-\left( \alpha r_{e}\right) ^{2}}=\frac{\sqrt{2}}{\sqrt{\pi }}\allowbreak \alpha ^{2}r_{e}^{2}e^{-\alpha ^{2}r_{e}^{2}}$ $\mu =\mu _{r=r_{e}}+\left( \frac{d\mu }{dr}\right) _{r=r_{e}}\left( r-r_{e}\right) =\mu _{r=r_{e}}+\frac{\sqrt{2}}{\sqrt{\pi }}\alpha ^{2}r_{e}^{2}e^{-\alpha ^{2}r_{e}^{2}}\left( r-r_{e}\right) $ --------------------------------------- $I_{30}=\frac{\alpha }{4\sqrt{3\pi }}\int_{-\infty }^{\infty }x\left[ 8\left( \alpha x\right) ^{3}-12\left( \alpha x\right) \right] e^{-\left( \alpha x\right) ^{2}}dx=\frac{2}{\alpha \sqrt{3\pi }}\int_{0}^{\infty }\left( 2y^{4}-3y^{2}\right) e^{-y^{2}}dy=0$ $I_{10}=\frac{\alpha }{\sqrt{2\pi }}\int_{-\infty }^{\infty }x\left( 2\left( \alpha x\right) \right) e^{-\left( \alpha x\right) ^{2}}dx=\frac{4\alpha }{% \sqrt{2\pi }}\int_{0}^{\infty }y^{2}e^{-y^{2}}dy=\allowbreak \frac{1}{2}% \alpha \sqrt{2}$ $\left( \frac{d\mu }{dr}\right) _{r=r_{e}}=\left[ \frac{d\left( \frac{\alpha }{\sqrt{2\pi }}\int_{r_{e}}^{r}r\left( 2\left( \alpha r\right) \right) e^{-\left( \alpha r\right) ^{2}}d\right) x}{dr}\right] _{r=r_{e}}=\frac{% \alpha }{\sqrt{2\pi }}r_{e}\left( 2\left( \alpha r_{e}\right) \right) e^{-\left( \alpha r_{e}\right) ^{2}}=\allowbreak \alpha ^{2}\frac{\sqrt{2}}{% \sqrt{\pi }}r_{e}^{2}e^{-\alpha ^{2}r_{e}^{2}}$ $\mu =\mu _{r=r_{e}}+\left( \frac{d\mu }{dr}\right) _{r=r_{e}}\left( r-r_{e}\right) =\mu _{r=r_{e}}+\alpha ^{2}\frac{\sqrt{2}}{\sqrt{\pi }}% r_{e}^{2}e^{-\alpha ^{2}r_{e}^{2}}\left( r-r_{e}\right) $ \end{document}