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\markboth{Homework \#8, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#8, Phys623, Spring 1998, Prof.~Yakovenko}
\begin{document}
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\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#8} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, April 6, 1998.} \\
Return homework in class, by e-mail, or \\
put in the box on the door of Phys 2314.
\end{minipage}
\hfill
\begin{minipage}[t]{2.9in}
\raggedleft
Victor Yakovenko, Assistant Professor \\
Office: Physics 2314 \\
Phone: (301)--405--6151 \\
E-mail: yakovenk@physics.umd.edu
\end{minipage}
\medskip
\centerline{\bf Do not forget to write your name and the homework
number!}
\centerline{Equation numbers with the period, like (3.25), refer to the
equations of Schwabl.}
\centerline{Equation numbers without period, like (5), refer to
the equations of this homework.}
\begin{center}
\section*{Time-Dependent Phenomena (Chapters 16.1--16.3)}
\end{center}
\bigskip
\begin{enumerate}
\item {\bf [5 points]} Schwabl's Problem 16.1.
\underline{Directions:} Solve this problem using the first-order
perturbation theory (Ch.\ 16.3.2).
\item {\bf [5 points]} Schwabl's Problem 16.2.
\underline{Directions:} The energy eigenfunctions of an
oscillator in an applied electric field are the same as the
those without electric field, but displaced by a certain
distance. Conversely, the original ground-state wave function
$\psi_0(x-l)$ is displaced relative to the new energy basis
$\psi_n(x)$. The coefficients of expansion of $\psi_0(x-l)$ in
terms of $\psi_n(x)$ are discussed in Ch.\ 3.1.4, ``Coherent
States'', without using Hermite polynomials.
\item {\bf [5 points]} Schwabl's Problem 16.3
\underline{Directions:} In this problem, an atom has one
electron, and the nuclear charge suddenly changes from $Z$ to
$Z+1$. Calculate the probability the electron remains in the
ground state 1s and the probability of transition to the 2s
state. Don't calculate the probability for 3s state.
\item A quantum particle is in an eigenstate $\psi_0({\bf r})$ of
Hamiltonian $\hat{H}$ with the energy $E_0$:
\begin{eqnarray}
&&\hat{H}_0=\frac{\hat{\bf p}^2}{2m}+V_0({\bf r}), \\
&&\hat{H}\psi_0({\bf r})=E_0\psi_0({\bf r}).
\end{eqnarray}
For example, you may think of an electron exposed to the
electric potential of a proton in the ground state of a hydrogen
atom.
Now, let us consider a problem, where the potential $V$ moves
with a constant velocity {\bf v}:
\begin{equation}
V({\bf r},t)=V_0({\bf r}-{\bf v}t).
\end{equation}
For example, the proton of the hydrogen atom may move with the
velocity {\bf v}.
\begin{enumerate}
\item {\bf [5 points]} Using the function $\psi_0({\bf r})$,
construct a solution of the time-dependent Schr\"{o}dinger
equation {\bf(see Hints)}: \label{Galilean}
\begin{eqnarray}
&& i\hbar\frac{\partial\psi({\bf r},t)}{\partial t}=
\hat{H}(t)\psi({\bf r},t),
\label{Schrod} \\
&& \hat{H}(t)=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial{\bf r}^2}
+V_0({\bf r}-{\bf v}t).
\label{H}
\end{eqnarray}
\item {\bf [3 points]} Compare your result with the rules of the
Galilean transformation in classical mechanics:
\begin{eqnarray}
{\bf r}&=&{\bf r}'+{\bf v}t, \\
t&=&t', \\
{\bf p}&=&{\bf p}'+m{\bf v}, \\
E&=&E'+{\bf vp}'+\frac{m{\bf v}^2}{2},
\end{eqnarray}
where the primed variables refer to the reference frame
moving with the velocity ${\bf v}$ relative to the
laboratory reference frame.
\end{enumerate}
\item {\em Adapted from Qualifier, Fall 1994, II-2.}
Consider decay of an unstable compound nucleus $\rm {^2He^5}$
via the following reaction:
\begin{equation}
\rm {^2He^5} \rightarrow {^2He^4}+n.
\label{reaction}
\end{equation}
(The nucleus $\rm {^2He^5}$, where 2 denotes the charge of the
nucleus and 5 denotes the total number of protons and neutrons
in the nucleus, is produced in thermonuclear fusion of deuterium
$\rm {^1H^2}$ and tritium $\rm {^1H^3}$.)
Once the neutron suddenly leaves the parent nucleus $\rm
{^2He^5}$, the daughter nucleus $\rm {^2He^4}$ starts to move
(non-relativistically) with a velocity $v$ due to recoil.
Suppose, before the decay the nucleus $\rm {^2He^5}$ had one
electron bound in the 1s ($|100\rangle$) state.
\begin{enumerate}
\item {\bf [5 points]} Calculate the probability that the
electron will remain in the 1s state of the moving nucleus
$\rm {^2He^4}$.
\item {\bf [3 points]} Find the formulas for the probabilities
of the electron excitation to the following states of the
moving nucleus $\rm {^2He^4}$:
\begin{enumerate}
\item $|210\rangle$
\item $|200\rangle$
\item $|211\rangle$
\end{enumerate}
Qualitatively sketch the probabilities as functions of
$v$. How do they behave for small $v$, large $v$? what
are the symmetry restrictions? To what power of $v$ are
the probabilities proportional at small $v$? Do not
calculate the integrals in this part.
\end{enumerate}
\item {\em Adapted from Qualifier, September 1992, II-2.}
At the time $t<0$, a system is in a state $|1\rangle$ which has
the same energy as a state $|2\rangle$. At the time $t=0$, a
perturbation $V$, which mixes the states $|1\rangle$ and
$|2\rangle$, is suddenly turned on and remains constant for
$t>0$.
\begin{enumerate}
\item {\bf [5 points]} Find exactly the probability $W(t)$ to
find the system in the state $|2\rangle$ as a function of
time $t$.
\item {\bf [3 points]} Verify that in the limit of small $t$ the
result agrees with the first order of the perturbation
theory.
\item {\bf [2 points]} For which times $t$ does the probability
$W(t)$ vanish?
\end{enumerate}
\item {\em An opposite limiting case to the sudden perturbation is a
very slow, adiabatic perturbation. If the system is initially in
an energy eigenstate of a discrete spectrum, the system remains
in that state (does not make transitions to other states of the
spectrum); however, the states itself gradually evolves into
something different from what it was originally. This
approximation is valid if the characteristic frequency of the
perturbation, $\omega$, is much smaller than the distance
between the energy levels, $\Delta E$: $\hbar\omega\ll\Delta
E$. Adiabatic Approximation is discussed in detail in Ch.\ 10 of
the book by Griffiths.}
{\bf [5 points]} A free two-dimensional rotator (see Problem 5
of Homework 1), which has the moment of inertia $I$ and dipole
moment $d$, is in its ground state at the time $t<0$. At the
time $t>0$, a homogeneous electric field $\vec{\cal E}(t)={\cal
E}(t)\vec{n}$ is gradually turned on as ${\cal E}(t)={\cal
E}_0[1-\exp(-t/\tau)]$. The following conditions are satisfied:
\begin{equation}
\hbar^2/I\ll d{\cal E}_0\ll \tau \hbar^3/I^2,
\label{conditions}
\end{equation}
which means that the field is strong, but is turned on slowly.
Find the probability distribution of the angular momentum $L$
perpendicular to the rotation plane at
$t\rightarrow+\infty$.
\item {\bf [5 points]} Schwabl's Problem 16.10. Assume that the
Hamiltonian has the form $\hat{H}=p^2/2m + V(x)$.
\end{enumerate}
\vfill
\section*{\centerline{Hints}}
\begin{description}
\item[\ref{Galilean}] Let us go the reference frame that moves with
the velocity {\bf v}. In this reference frame, the potential $V$
does not move, thus $\psi_0$ is a solution of the Schr\"{o}dinger
equation. Going back to the laboratory reference frame, one
concludes that the probability density should move together with
the potential as $|\psi_0({\bf r}-{\bf v}t)|^2$.
Thus, it is tempting to say that $\psi({\bf r},t)=\psi_0({\bf
r}-{\bf v}t)$. That is not quite right. The correct relation is:
\begin{equation}
\psi_0({\bf r},t)=f({\bf r},t)\,\psi_0({\bf r}-{\bf v}t),
\label{f}
\end{equation}
where $f({\bf r},t)$ is a phase factor. To find this factor,
substitute the ansatz (\ref{f}) into
Eqs. (\ref{Schrod})--(\ref{H}) and change variables {\bf r} and
$t$ to ${\bf r}'={\bf r}-{\bf v}t$ and $t$.
\end{description}
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