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\section*{D12-2 - Quantum Mechanics  - due January 20, 2000}
%\section*{D12-2 - Quantum Mechanics  - January 18, 1997}

\noindent{\large\bf I. Path Integrals
for Quantum Amplitudes}\bigskip

\begin{enumerate}

\item For a classical particle moving under conservative forces show
that the momentum of a particle at the endpoint $x_b$ is given by
$p=\partial S_{class}/\partial x_b$ where
$S=\int_{t_a}^{t_b}\,dt\,L(x,\dot{x};t)$ is the action functional.
Also show that the energy of the particle is given by $E=\partial
S_{class}/\partial t_b$. Recall that the energy of a classical particle
is given by $E=L-p\dot{x}$.

\item The path integral for the amplitude a particle to be at $x_b$ at
time $t_b$ given the particle was at $x_a$ at $t_a$ is
\begin{equation}
\phi(x_a,t_a;x_b,t_b)=\int_a^b\,{\cal D}[x(t)]\,
e^{\frac{i}{\hbar}\int_{t_a}^{t_b}\,L(x,\dot{x};t)\,dt}\,.
\end{equation}
\item[i.] Evaluate the Gaussian integral
$\int_{-\infty}^{\infty}\,e^{iax^2}\,dx$ for $Re\, a>0$.  Hint: Change
variables to $t=-iax^2$ and consider the closed contour
in the complex $t$-plane:  $\{t| 0\rightarrow R,R\exp{-i\varphi}\,
[0<\varphi<\pi/2], -iR\rightarrow 0\}$ and $R\rightarrow \infty$.
Consider amplitude for a particle whose motion is derived from a
Lagrangian of the form $L=a(t)\dot{x}^2+b(t)x^2+c(t)x+d(t)$.
\begin{enumerate}
\item[ii.] Show that $\phi(x_a,t_a;x_b,t_b)= {\cal
A}(t_a,t_b)\,e^{iS_{class}[a,b]/\hbar}$ for the general case. Hint:
re-write the path integral in terms of the deviations from the
classical path, $\bar{x}(t)$.
\item[iii.] For a free particle of mass $m$ show that ${\cal
A}(t_a,t_b)={\cal A}(t_b-t_a)=
\sqrt{\frac{m}{2\pi i\hbar (t_b-t_a)}}$.
\end{enumerate}

\item The amplitude for a free particle to be at $x$ at time $t$ given
the particle was at $x=0$ at $t=0$ is
$\phi(x,t;0,0)=\sqrt{\frac{m}{2\pi i\hbar t}}e^{imx^2/2\hbar t}$.

\begin{enumerate}
\item[i.] Consider the spatial variation of $\phi(x,t;0,0)$ at fixed
$t$ and large $x$. Show that the amplitude has a wavelength $\lambda$
given by the de Broglie relation, $p=h/\lambda$, for a particle of
momentum $p$.
\item[ii.] Similarly, consider the temporal variation of $\phi$ at
fixed $x$ and show that the amplitude has a frequency $\omega$ given by
the Einstein relation $E=\hbar\omega$ for a particle with energy $E$.
\end{enumerate}

\item Calculate the amplitude for a particle of mass $m$ moving
in a harmonic potential with natural frequency $\omega$. Express the
result in terms of the initial and final positions, $x_a$ and $x_b$,
and the time interval $\tau=t_b-t_a$.

\end{enumerate}

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