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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#8, Rev. A, due Monday, 2015-11-16 }
\end{center}

\vspace{.3 in}

{\bf 1.} We will apply the formalism of quantum mechanics to one the simplest
nontrivial system: the {\bf ammonia molecule}, ignoring rotations and
vibrations.  It has two quantum states $|1\rangle$ and $|2\rangle$,
corresponding the nitrogen atom above ($+$) or below ($-$) the triangle of
hydrogen atoms.  These would be degenerate ground states of energy $E$ were it
not for the small interaction energy $-A$ which couples the two states
(tunnelling amplitude between states).  The Hamiltonian of this system in the
$|1,2\rangle$ basis is $\mathcal{H} = \pmatrix{E_0 & -A \cr -A & E_0}$.

{\bf \quad a)} Calculate the eigenstates $|I,II\rangle$ and energy spectrum
$E_{I,II}$ of $\mathcal{H}$.

{\bf \quad b)} Given the initial state $|\psi_0\rangle = \pmatrix{a\cr b}$ in
the $|1,2\rangle$ basis, change components to the $|I,II\rangle$ basis.
Calculate the time-dependence $|\psi(t)\rangle$ in both the $|I,II\rangle$ and
$|1,2\rangle$ bases.

{\bf \quad c)} What is the probability of measuring $E_I$ and $E_{II}$ at time
$t$?  Calculate $\langle E(t) \rangle$.

{\bf \quad d)} Let the `position' $X$ of the nitrogen atom be $+a$ in state
$|1\rangle$ and $-a$ in the state $|2\rangle$.  Write the matrix of the
operator $\hat X$ in both the $|1,2\rangle$ and $|I,II\rangle$ bases.
Calculate the probability of measuring $X(t)=\pm a$, and $\langle X(t)
\rangle$.  What is the state after measuring $X$?  Describe the possible
results and probabilities of a subsequent measurement of $E$.

{\bf \quad e)} Calculate the emission frequency of transitions in this system
and compare to $\langle X(t) \rangle$.

{\bf 2.  Minimum uncertainty wave packet.}

{\bf \quad a)} Following the Griffiths requirement for equality in the
Heisenberg Uncertainty Principle, that $|g\rangle=ia|f\rangle$, where
$|f\rangle=(\hat x-x_0)|\psi\rangle$ and $|g\rangle=(\hat p-
p_0)|\psi\rangle$, show that the general minimum uncertainty wave packet is
$\psi(x)=A e^{-\frac12(\frac{x-x_0}{\sigma_x})^2}e^{ip_0 x/\hbar}$.  Plot this
wavefunction in the complex plane as a function of $x$.  Show that $\langle x
\rangle=x_0$ and $\langle x^2\rangle-x_0^2=\sigma_x^2$.  What is the relation
between $a$ and $\sigma_x$?  Normalize the wavefunction to obtain the
amplitude $A$ up to an arbitrary phase $e^{i\theta}$.

{\bf \quad b)} Perform the {\it Fourier transform} %
$\phi(p)=\frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}dx\, e^{-ipx/\hbar}
\psi(x)$ to show that the momentum amplitudes $\phi(p)=B
e^{-\frac12(\frac{p-p_0}{\sigma_p})^2}e^{-ix_0 p/\hbar}$ have the same
Gaussian distribution.  Compare the roles of $x$ and $p$ in these two
representations, and explain the new $-$ sign in front of $x_0$.  Plot the
complex function $\phi(p)$.  Calculate $\langle p\rangle$ and $\langle p^2
\rangle$ (you already did the math above!).  Verify that $\sigma_x
\sigma_p=\hbar/2$, and that $\phi(p)$ is still normalized (the Fourier
transform is a {\it unitary} operator).

{\bf \quad c)} Perform the {\it Inverse Fourier transform}
$\psi(x)=\frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}dp\,e^{ipx/\hbar}
\phi(p)$ to recover the original wave function in $x$-space. (you already did
the math above!)

{\bf \quad d)} What is the effect on the Fourier transform of multiplying
$\psi(x)$ by i) a global phase $e^{i\theta}$, ii) a linear phase $e^{ikx}$;
iii) translating $\psi(x) \to \psi(x-dx)$ (operating by $i\hat p dx/\hbar$);
iv) scaling $\psi(x) \to \psi(x/b)$ (changing the uncertainty of $x$)?  Repeat
for the analogous transforms on $\phi(p)$.

{\bf \quad e) [bonus]} Transform this wavefunction into the basis of energy
eigenfunctions $|n\rangle$ of the harmonic oscillator to obtain the
coefficients $c_n(x_0,p_0,\sigma_x)$ and compare with Griffiths 2ed problem
3.35.  Note the significance of $|g\rangle=ia|f\rangle$; you may want to use
the operator algebra method.
                       
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