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\title{PHYS 2380 - Assignment 3}
\author{Due March 1, 2017}
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Note: an electron volt (eV) is equivalent to $1.60 \times 10^{-19}$ J. It may also be convenient to use the combination $hc=1240$ eV$\cdot$nm.

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Recalling that an object smaller than the wavelength illuminating it cannot be ``seen'', what is the minimum kinetic energy of electrons needed in an electron microscope in order to ``see'' an atom whose diameter is $0.1$ nm, about the size of a silicon atom?
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This is question 5-41 in Tipler \& Llewellyn.
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The wavelength of light emitted by a ruby laser is $694.3$ nm. Assuming that the emission of a photon of this wavelength accompanies the transititon of an electrion from the $n=2$ level to the $n=1$ level of an infinite square well, compute $L$ for the well. 
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This is question 6-16 in Tipler \& Llewellyn.
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An electron moving in a one-dimensional infinite square well is trapped in the $n=5$ state. 
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a.) Show that the probability of finding the electron between $x=0.2L$ and $x=0.4L$ is $1/5$. 
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b.) Compute the probability of finding the electron within the ``volume'' $\Delta x = 0.01L$ at $x=L/2$.
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This is question 6-20 in Tipler \& Llewellyn.
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a.) Find the wavefunction $\psi(x)$ by computing the Fourier transform of the function 
\begin{equation}
g(k)=A e^{\frac{-|k|}{a}}
\end{equation}
with $A$ and $a$ constant. You can deal with the absolute value by splitting the domain of integration in half, taking $g(k)=Ae^{\frac{k}{a}}$ for $k<0$, and $g(k)=Ae^{\frac{-k}{a}}$ for $k>0$. 
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b.) Normalize the resulting wavefunction to find the constant $A$.
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In class we calculated the Fourier transform of a Gaussian centered on the origin. Now let us consider a Gaussian centered at an arbitrary wave number $k_0$, such that 
\begin{equation}
g(k)=A\exp\left( -\frac{(k-k_0)^2}{4 \sigma_k^2}\right).
\end{equation}
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a.) Calculate the resulting wavefunction. What is the difference in the wavefunction between the two cases, and is this difference observable? 
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b.) Does the uncertainty relationship we derived in class change if the Gaussian is offset from the origin in momentum space?
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c.) Normalize the wavefunction to determine the constant $A$. 
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d.) For the Gaussian centered at $0$, use the wavefunction we derived in class to find the expectation value of the momentum, $<p>$.
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e.) For the Gaussian centered at $k_0$, derived in part a, calculate the expectation value of the momentum, $<p>$.
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f.) Compare your answers in d and e. What does the expectation value tell us about the momentum distribution $g(k)$?

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