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Gentoo's Bugzilla – Attachment 8231 Details for
Bug 15629
Fonts messed up in lyx 1.3.0
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ps3.lyx
ps3.lyx (text/plain), 8.90 KB, created by
Grant Goodyear (RETIRED)
on 2003-02-13 09:05:23 UTC
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Description:
ps3.lyx
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Creator:
Grant Goodyear (RETIRED)
Created:
2003-02-13 09:05:23 UTC
Size:
8.90 KB
patch
obsolete
>#LyX 1.2 created this file. For more info see http://www.lyx.org/ >\lyxformat 220 >\textclass article >\begin_preamble >\usepackage{times} >\usepackage[T1]{fontenc} >\usepackage[latin1]{inputenc} >\pagestyle{empty} >\usepackage{babel} > >\makeatletter > >\setlength{\parindent}{0in} >\setlength{\parskip}{.25in} >\setlength{\textheight}{9.3in} >\setlength{\topmargin}{-.6in} >\textwidth=6.5in >\addtolength{\oddsidemargin}{-.85in} >\makeatother >\end_preamble >\options english >\language english >\inputencoding latin1 >\fontscheme times >\graphics default >\paperfontsize 10 >\spacing single >\papersize letterpaper >\paperpackage a4 >\use_geometry 0 >\use_amsmath 0 >\use_natbib 0 >\use_numerical_citations 0 >\paperorientation portrait >\secnumdepth 3 >\tocdepth 3 >\paragraph_separation indent >\defskip medskip >\quotes_language english >\quotes_times 2 >\papercolumns 1 >\papersides 1 >\paperpagestyle default > >\layout Standard > >Problem Set #3 >\hfill >Chem 332 --- Spring 2003 >\newline > >\series bold >Due: 4 March 2003 >\series default > >\hfill >Grant Goodyear >\newline > >\begin_inset ERT >status Collapsed > >\layout Standard > >\backslash >rule{6.5in}{.0625in} >\end_inset > > >\layout Enumerate > >A general property of solutions of the time-independent Schrodinger equation > is that wavefunctions must be >\emph on >orthonormal >\emph default >: >\begin_inset Formula \[ >\int d\vec{r}\: \psi _{n}^{*}(\vec{r})\psi _{m}(\vec{r})=\left\{ \begin{array}{cc} > 1, & n=m\\ > 0, & n\neq m\end{array}\right..\] > >\end_inset > >Let's introduce a bit of notational shorthand: >\begin_inset Formula \[ >\delta _{nm}\equiv \left\{ \begin{array}{cc} > 1, & n=m\\ > 0, & n\neq m\end{array}\right.,\] > >\end_inset > >where >\begin_inset Formula $\delta _{nm}$ >\end_inset > > is called the >\begin_inset Quotes eld >\end_inset > >Kronecker delta >\begin_inset Quotes erd >\end_inset > >. > Explicitly writing out the integrals is also a bit of a pain, so >\begin_inset Quotes eld >\end_inset > >bra >\begin_inset Quotes erd >\end_inset > >-c- >\begin_inset Quotes erd >\end_inset > >ket >\begin_inset Quotes erd >\end_inset > > notation was developed: >\begin_inset Formula \[ ><n|m>\equiv \int d\vec{r}\: \psi _{n}^{*}(\vec{r})\psi _{m}(\vec{r}),\] > >\end_inset > >and >\begin_inset Formula \[ ><n|\hat{o}|m>\equiv \int d\vec{r}\: \psi _{n}^{*}(\vec{r})\hat{o}\psi _{m}(\vec{r}),\] > >\end_inset > >where >\begin_inset Formula $<n|m>$ >\end_inset > > and >\begin_inset Formula $<n|\hat{o}|m>$ >\end_inset > > are both referred to as >\begin_inset Quotes eld >\end_inset > >matrix elements >\begin_inset Quotes erd >\end_inset > >. > Show that the particle-in-a-box wavefunctions satisfy >\begin_inset Formula $<n|m>=\delta _{nm}$ >\end_inset > >. >\layout Enumerate > >If an electron confined in a box were not quantum mechanical, it would have > an equal probability for being anywhere in the box. > Assuming a box of length L, >\begin_deeper >\layout Enumerate > >What would P(x), the probability density, be in this classical case? >\layout Enumerate > >What would >\begin_inset Formula $\sigma _{x}$ >\end_inset > > (the root-mean-square fluctuation in position) be? >\layout Enumerate > >Compute what the same root-mean-square fluctuation in position would be > if the electron really were quantum mechanical but the electron happened > to be in a state with an infinitely large quantum number n. > What is the relationship with the answer you found for part (b)? Why? >\end_deeper >\layout Enumerate > >Retinal, the molecule responsible for human and animal vision, has 6 conjugated > double bonds (the backbone is C=C-C=C-C=C-C=C-C=C-C=C). > Retinal, of course, absorbs visible light. > Using a particle-in-a-box model, how many cojugated double bonds would > be needed to see infrared light ( >\begin_inset Formula $\lambda \geq 800$ >\end_inset > > nm)? >\layout Enumerate > >S&A, probs. > 9.12, 9.14, 9.15 >\layout Enumerate > >The experimental vibrational frequency for an >\begin_inset Formula $\mathrm{H}_{2}$ >\end_inset > > molecule is >\begin_inset Formula $\omega /2\pi c=4395.2\: \mathrm{cm}^{-1}$ >\end_inset > >. > Use this experimental result to predict the vibrational frequency of >\begin_inset Formula $\mathrm{D}_{2}$ >\end_inset > >. > (Hint: when you change isotopes from H to D, what should change and what > should stay the same?) >\layout Enumerate > >The rate at which a molecule absorbs light is determined by the value of > the matrix element >\begin_inset Formula $<f|\hat{\mu }|i>$ >\end_inset > >, where >\begin_inset Formula $i$ >\end_inset > > is the initial quantum state of the molecule, >\begin_inset Formula $f$ >\end_inset > > is the final quantum state of the molecule (after the light is absorbed), > and >\begin_inset Formula $\hat{\mu }$ >\end_inset > > is the dipole operator. > For practical purposes, the dipole operator is a constant times the position > operator: >\begin_inset Formula $\hat{\mu }=a\hat{x}$ >\end_inset > >. >\begin_deeper >\layout Enumerate > >Before going further, we need a couple of useful mathematical relationships. > Define >\emph on >even >\emph default > and >\emph on >odd >\emph default > functions according to >\begin_inset Formula $f_{\mathrm{even}}(-x)=f_{\mathrm{even}}(x)$ >\end_inset > > and >\begin_inset Formula $f_{\mathrm{odd}}(-x)=-f_{\mathrm{odd}}(x)$ >\end_inset > > . > Prove the >\begin_inset Quotes eld >\end_inset > >symmetry results >\begin_inset Quotes erd >\end_inset > > >\begin_inset Formula $\int _{-a}^{a}f_{\mathrm{odd}}(x)=0$ >\end_inset > > and >\begin_inset Formula $\int _{-a}^{a}f_{\mathrm{even}}(x)=2\int _{0}^{a}f_{\mathrm{even}}$ >\end_inset > >. > Also prove that >\begin_inset Formula $f_{\mathrm{even}}(x)f_{\mathrm{odd}}(x)$ >\end_inset > > (the product) is an odd function, while >\begin_inset Formula $f_{\mathrm{even}}(x)f_{\mathrm{even}}(x)$ >\end_inset > > and >\begin_inset Formula $f_{\mathrm{odd}}(x)f_{\mathrm{odd}}(x)$ >\end_inset > > are even functions. >\layout Enumerate > >Assume that molecular vibration can be modeled with a simple harmonic oscillator. > In that case symmetry considerations would tell us that light could only > take a molecule starting in the ground state (n=0) to states n=1, 3, 5, > >\begin_inset Formula $\cdots $ >\end_inset > >. > Why? If symmetry were the only consideration, what frequencies >\begin_inset Formula $\upsilon $ >\end_inset > > could be absorbed? >\layout Enumerate > >In fact, when you do IR spectroscopy of vibrating molecules you don't see > all of those frequencies. > We can actually understand this observation using the harmonic oscillator > model because one can use the special properties of harmonic oscillator > wave functions to show that all of the matrix elements of x which connect > to the ground state vanish, except for the one that connects to the first > excited state: >\begin_inset Formula $<f|\hat{x}|0>=0$ >\end_inset > > unless f=1. > Within the harmonic approximation what frequencies of light should be absorbed? >\layout Enumerate > >Let's prove this special property of simple-harmonic-oscillator matrix elements. > Show that with the simple-harmonic-oscillator wave functions >\begin_inset Formula $x\psi _{0}(x)=C\psi _{1}(x)$ >\end_inset > > for some constant >\begin_inset Formula $C$ >\end_inset > >. > Determine >\begin_inset Formula $C$ >\end_inset > >. >\layout Enumerate > >Write out >\begin_inset Formula $<f|\hat{x}|0>$ >\end_inset > > as an integral and substitute in your result from part (c). > What you have written immediately tells you that only the >\begin_inset Formula $f=1$ >\end_inset > > matrix element has a non-zero value. > Why? What's the value of this matrix element? >\end_deeper >\layout Enumerate > >Compute >\begin_inset Formula $\sigma _{x}\sigma _{p}$ >\end_inset > > for the harmonic oscillator ground state. > The harmonic oscillator ground state is often called a >\begin_inset Quotes eld >\end_inset > >minimum uncertainty >\begin_inset Quotes erd >\end_inset > > wavepacket. > Why >\begin_inset Quotes eld >\end_inset > >minimum uncertainty >\begin_inset Quotes erd >\end_inset > >? >\layout Enumerate > >Of the following molecules: >\begin_inset Formula $\mathrm{H}_{2}$ >\end_inset > >, HCl, >\begin_inset Formula $\mathrm{CH}_{4}$ >\end_inset > >, >\begin_inset Formula $\mathrm{CH}_{3}\mathrm{Cl}$ >\end_inset > >, >\begin_inset Formula $\mathrm{H}_{2}\mathrm{O}$ >\end_inset > >, and >\begin_inset Formula $\mathrm{CO}_{2}$ >\end_inset > >, which have rotations that will show up in microwave spectroscopy? Which > have IR active vibrations? Raman active vibrations? >\layout Enumerate > >S&A, probs. > 9.29, 9.36. >\layout Enumerate > >A space probe was designed to look for signs of CO in the atmosphere of > Saturn. > It was decided to employ a microwave technique from an orbiting satellite. > Given the bond length of CO is 112.82 pm, at what wavenumbers do the first > four transitions of >\begin_inset Formula $^{12}\mathrm{C}^{16}\mathrm{O}$ >\end_inset > > lie? What resolution is needed if it was desired to distinguish the 0 >\begin_inset Formula $\rightarrow $ >\end_inset > >1 line in the >\begin_inset Formula $^{12}\mathrm{C}^{16}\mathrm{O}$ >\end_inset > > spectrum from that of >\begin_inset Formula $^{13}\mathrm{C}^{16}\mathrm{O}$ >\end_inset > > in order to examine the relative abundances of the two carbon isotopes? >\the_end
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bug 15629
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