\documentclass{article}
\usepackage{exam2}
\usepackage{amssymb}
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\usepackage{amsmath}
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\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 15}
\time{Start of class on Thurs. Apr 28}
\semester{Spring}
\year{2011}                     % <- default is the current year
%\campus{City}
%\note{ Be sure to show your work and explain what you are doing. (Correct answers, for which we cannot follow the work, are worth no credit). } 
\begin{document}

\begin{questions}

\question  Complete the online  survey at   

{\url{ http://www.colorado.edu/sei/surveys/Sp11/Clicker_Phys2210_sp11-post.html}}  

Note: You get full credit just for inputting your name and ID, but we really appreciate your thoughtful feedback! 



\question
\begin{enumerate}
\item (2 pts)  Show that if a function $f(x)$ is even,  then its Fourier transform  $g(\alpha)$ (as defined in Boas Eq 12.2) is also an even function.
 \item (2 pts) Show that in this case f(x) and g($\alpha$) can be written as Fourier cosine transforms and obtain Boas Eq 12.15 (Assuming  that f(x) is identical to Boas' $f_c(x)$, how is $g_c(\alpha)$ from Boas' Eq 12.15 related to g($\alpha$) of Boas' Eq 12.2? 
 
 {\it Please note the typo in the top equation of Boas 12.15, the argument of $g_c$ should be $\alpha$, not $x$.} 
 \end{enumerate}
 
 \question  For the function shown in Fig.1 
\begin{enumerate}
\item (1 pt) Qualitatively, how do you expect  $g(\alpha)$,  the Fourier transform of $f(x)$, to change if you make $a$ bigger or smaller?
\item(4 pts)  Find the Fourier transform $g(\alpha)$.  
\item (2 pts) Sketch  $g(\alpha)$ (or plot it in Mathematica if you prefer).  Was your prediction in part (a) correct?  Explain.
%\item  (4 pts) Substitute back the function you found for  $g(\alpha)$ into the definition of $f(x)$ and compute $\int_0^\infty \frac{1-\cos\alpha x}{\alpha^2} d\alpha$
    \end{enumerate}

 \begin{figure}[htc]
\centering
{\includegraphics[width=2in]{Hom15Fig1.pdf}}\caption{}
\end{figure}

\question %YOUR CHOICE: Do either part a or part b, you do NOT need to do both! 
%\begin{enumerate}
%\item
 (4 pts) The function  $e^{-c|x|}$ (with c a positive, known constant)  is associated with bound states in quantum mechanics, and (pretty much as always in quantum mechanics) its Fourier transform is important to know. Find the Fourier transform of $e^{-c|x|}$, sketch it, and comment on any key features.     
 
 As a mathematical aside, use your result to (easily!) compute  $\int_0^\infty \frac{\cos(\alpha x)}{\alpha^2+1} d\alpha$.
%\item  Find the Fourier transform of  $ \frac{1}{x^2+1} $. Hint: Use  b).


%\item (CHOOSE THIS, OR THE PREVIOUS PART, you do not have to do both)




\question  The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, 
$$ U(r) = A\left[ \left(e^{(R-r)/S}-1\right)^{2} -1\right] $$
where $r$ is the distance between the two atoms and $A,R,$ and $S$ are positive constants with $S \ll R$.  
\begin{enumerate}
\item (4 pts) Sketch the function for $0<r< \infty$ (or plot in Mathematica if you prefer).  
\item (2 pts) Show that  the equilibrium separation $r_{0}$ at which $U(r)$ is a minimum is given by $r_0=R$. 
\item (2 pts) Based just on your figure/sketch, what behavior do you expect for the system if it starts with no kinetic energy, but at some $r$ close to $r_{0}$?   Describe what this would mean physically for the two atoms in the molecule (i.e. tell the story of what is happening to them).
\item (4 pts)  {\it If our diatomic molecule has one very heavy atom, that atom can be viewed as effectively fixed at the origin, and this U(r) simply describes the potential energy of the second, lighter atom (mass m, a distance r(t)  from the origin.)} 

Approximate $U(r)$ for $r$ close to $r_{0}$. Use this approximation of $U(r)$ to get a differential equation for $r(t)$.  Can you tell from this differential equation if your prediction in part (c) was correct?  Explain.
\end{enumerate}

\ \\

\hrule
\ \\

 {\it Extra Credit} (4 pts) Find the Fourier Transform of a Gaussian function (with standard deviation $\sigma$),  $f(x)=e^{-x^2/(2\sigma^2)}$, and show that it is {\it also} a Gaussian function. (You will need to consult an integral table for this one . See the online lecture notes for more hints) 
 
 Sketch $f(x)$ and $g(\alpha)$. What is the standard deviation of $g(\alpha)$?  Does this agree with your expectations?

\end{questions}
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