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\markboth{Homework \#2, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#2, Phys623, Spring 1998, Prof.~Yakovenko}
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\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#2} --- Phys623 --- Spring 1998 \\
{\bf Deadline: Friday, February 13, 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!}
\begin{center}
\section*{The WKB Method (Chapter 11.3)}
\end{center}
\begin{enumerate}
\item Study the potential of Schwabl's Problem 11.8 in the WKB
approximation.
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\begin{enumerate}
\item {\bf [3 points]} Show that for the potentials of this kind,
which are bounded by a vertical potential wall on one
side, the Bohr-Sommerfeld quantization condition takes the
form
\begin{equation}
\int_a^b dx\, p(x)=\pi\hbar\left(n+\frac34\right).
\label{3/4}
\end{equation}
{\bf Directions:} Take into account that the wave function
must vanish at the point $a$ and perform manipulations
analogous to Eqs.\ (11.36) and (11.37) of Schwabl.
\item {\bf [3 points]} Using the quantization condition
(\ref{3/4}), find the energy levels in the potential of
Schwabl's Problem 11.8.
\end{enumerate}
\item {\bf [7 points]} Using the Bohr-Sommerfeld quantization
condition, find the energies of the bound states of the
one-dimensional potential $V(x)=V_0|x/a|^\nu$, where $U_0>0$ and
$\nu>0$. The result may contain a numerical coefficient
expressed as a definite integral, which you don't need to
evaluate.
Depending on the parameter $\nu$, how does the energy difference
between the neighboring energy levels change when the quantum
number $n$ increases? Do the energy levels become denser or
sparser in energy?
What is the energy density of states as a function of the
energy? The density of states is defined as $\Delta n/\Delta
E$, where $\Delta n$ is the number of states in the energy
interval $\Delta E$.
Compare results with the exact solution for the harmonic
oscillator when $\nu=2$.
\item {\bf [8 points]} Schwabl's Problem 11.6. {\bf Directions:}
Classically, the particles moves on a circle of a radius $R$
with the cyclotron frequency $\omega_c$ (see Eq.\ (7.36)). Take
the integral $\int dx\,p_x$ for the circular motion and
substitute it into the the Bohr-Sommerfeld quantization
condition.
\item {\bf [7 points]} In the WKB approximation, find discrete energy
levels of the attractive Coulomb potential
$U(r)=-e^2/r$. Consider a general case where the angular
momentum $l$ is not equal to zero. Apply the WKB approximation
to the radial motion described by Eq.\ (6.11), where the
effective potential contains the centrifugal term. Since the
potential is singular at $r=0$, apply the Bohr-Sommerfeld
quantization condition in a generalized form $\int
p(r)\,dr=(N+\gamma)\hbar\pi$ with some phase $\gamma$ that may
depend on $l$. {\bf (see Hints)} Compare your result with the
spectrum of the hydrogen atom in the limit where $N,\:l\gg1$.
\label{Coulomb}
\item {\bf [5 points]} A particle stays in a bound state of the energy
$E$. Using the Bohr-Sommerfeld quantization condition, find the
change of the energy of the state $\delta E$ when the potential
$U(x)$ changes by a small variation $\delta U(x)$. Interpret the
obtained result in terms of classical mechanics.
\item {\bf [3 points]} Using the Bohr-Sommerfeld quantization
condition, express the distance between the adjacent energy
levels, $\delta E=E_{n+1}-E_{n}$ in terms of the frequency of
classical motion in the bound state. $\omega(E)$.
\item Using the quasi-classical approach, approximately calculate the
total number of discrete energy levels in the following
cases. All potentials $V$ are assumed to be negative and
approaching zero at infinity; that is, they have a well shape.
\begin{enumerate}
\item {\bf [5 points]} Generic one-dimensional potential $V(x)$.
\item {\bf [2 points]} One-dimensional potential
$V(x)=(-V_0)/(x^2+a^2)^2$.
\item {\bf [7 points]} Generic three-dimensional
spherically-symmetric potential $V(r)$. {\bf (see Hints)}
\label{spherical}
\item {\bf [2 points]} Three-dimensional spherically-symmetric
potential $V(r)=(-V_0)/(r^2+a^2)^2$.
\item {\bf [7 points]} Generic three-dimensional potential
$V({\bf r})$ without a particular
symmetry. \label{non-spherical} {\bf (see Hints)}
\item {\bf [5 points]} Why the formulas are different in cases
\ref{spherical} and \ref{non-spherical}?
\item {\bf [3 points]} Using the obtained results, prove that a
three-dimensional potential $V(r)$, which decreases at
infinity as $1/r^\nu$ with $0<\nu<2$, has an infinite
number of discrete levels.
\end{enumerate}
\item {\bf [7 points]} A one-dimensional potential $U(x)$ consists of
two symmetric potential wells separated by a barrier. If the
barrier were impenetrable, the energy levels corresponding to
the motion of the particle in one or other well would be the
same for both wells. Because the passage through the barrier is
possible, the energy levels split. Within the quasi-classical
approximation, determine the magnitude of the splitting $\Delta
E$. {\bf (see Hints)} \label{double}
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\end{enumerate}
\newpage
\label{Hints}
\section*{\centerline{Hints}}
\begin{description}
\item[\ref{Coulomb}] The following formula may be useful:
\begin{equation}
\int_a^b\frac{dr}{r}\sqrt{(b-r)(r-a)}=\frac\pi2(a+b-2\sqrt{ab}).
\end{equation}
Derive it by contour integration in complex plane.
\item[\ref{spherical}] Count first the number of energy levels of the
radial motion with a given angular momentum $l$. This is,
essentially, a one-dimensional problem. Then, integrate over the
allowed values of $l$. Then, change the order of the integrals
over $r$ and $l$.
\item[\ref{non-spherical}] The quasi-classical quantization rule,
\[
\oint p\,dx=2\pi\hbar(n+\gamma),
\]
can be interpreted in the following way. There is one quantum
state ($\Delta n=1$) per a ``unit cell'' of the phase space
$\Delta p\,\Delta x$ of the volume $2\pi\hbar$. The number of
quantum states in a given region of the phase space is
proportional to the phase volume of that region (we neglect the
constant $\gamma$ of the order of 1, which is small compared to
the big number $n$):
\begin{equation}
n=\int\frac{dp\,dx}{2\pi\hbar}\;.
\label{dpdx1}
\end{equation}
Formula (\ref{dpdx1}), unlike the other quasi-classical formulas,
is straightforwardly generalized to a higher-dimensional case to
become (in three dimensions):
\begin{equation}
n=\int\frac{dp_xdx}{2\pi\hbar}\frac{dp_ydy}{2\pi\hbar}
\frac{dp_zdz}{2\pi\hbar}\;.
\label{dpdx3}
\end{equation}
In problem \ref{non-spherical}, you need to calculate integral
(\ref{dpdx3}) over the region of the phase space occupied by the
bound states. For an element of the coordinate space $d^3${\bf
r} at a given point {\bf r}, find the allowed range of the values
of $|${\bf p}$|$. ($|${\bf p}$|$ cannot be arbitrarily big in a
bound state.) Then, integrate over $d^3${\bf r}.
\item[\ref{double}] In the WKB approximation, transmission amplitude
through a one-dimensional potential barrier is given by the
following expression (see Eq.\ (3.73)):
\begin{equation}
S=\exp\left(-\frac{1}{\hbar}\int_a^b |p(x)|\,dx\right),
\label{T}
\end{equation}
where $a$ and $b$ are the turning points between which the
momentum $p(x)$ is imaginary because the region is classically
forbidden. The transmission rate through a barrier is equal to
the amplitude $S$ times the attempt rate $\hbar\omega/2\pi$,
where $\omega$ is the frequency of classical motion in the bound
state. You answer for the energy levels splitting should be
expressed in terms of $S$ and $\omega$.
\end{description}
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