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\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
 
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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