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\markboth{Homework \#3, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#3, Phys623, Spring 1998, Prof.~Yakovenko}
\begin{document}
\thispagestyle{empty}
\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#3} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, February 23, 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 Variational Principle (Chapter 11.2)}
\end{center}
Using the variational principle, one finds the energy and the wave
function of the ground state by minimizing the following expression
(see Eq.\ (11.16)):
\begin{equation}
E(\mu)=\frac{\langle\psi_\mu|\hat{H}|\psi_\mu\rangle}
{\langle\psi_\mu|\psi_\mu\rangle}
=\frac{\int dx\,[-\frac{\hbar^2}{2m}\psi_\mu^*(x)\psi_\mu''(x)
+V(x)|\psi_\mu(x)|^2 ]}
{\int dx\,\psi_\mu^*(x)\psi_\mu(x)},
\label{''}
\end{equation}
where the primes denote derivatives in $x$.
Integrating the first term in denominator by parts, Eq.\ (\ref{''})
can be equivalently written as (show this!)
\begin{equation}
E(\mu)=\frac{\int dx\,
[\frac{\hbar^2}{2m}|\psi_\mu'(x)|^2 + V(x)|\psi_\mu(x)|^2 ]}
{\int dx\,\psi_\mu^*(x)\psi_\mu(x)}.
\label{'}
\end{equation}
I strongly recommend you to use form (\ref{'}), rather than form
(\ref{''}). In form (\ref{'}) you need less derivatives to take, and
the kinetic energy term is manifestly positive. I also strongly
recommend to use Mathematica or a similar program to take the
integrals and, perhaps, even to minimize $E(\mu)$.
\bigskip
\begin{enumerate}
\item {\bf [12 points]} Schwabl's Problem 11.7. {\bf There is a typo in
the formulation of this problem. You should consider the
potential from Problem 3.12 (a half of the oscillator
potential), not Problem 3.13 (penetration through a barrier).}
\item {\bf [6 points]} Schwabl's Problem 11.8.
By definition, the Airy function Ai($x$) satisfies the equation
\begin{equation}
[d^2/dx^2 -x]{\rm Ai}(x)=0
\end{equation}
and vanishes when $x\rightarrow+\infty$. It oscillates when
$x<0$ and vanishes at a sequence of points $x_n$ called
the zeros of the Airy function:
\begin{equation}
{\rm Ai}(x_n)=0,\quad x_1=-2.338,\quad x_2=-4.088,\quad x_3=-5.521,\quad\ldots
\end{equation}
In part (a) of this Problem you need to express the energy
levels in terms of the zeros $x_n$ of the Airy function.
In Part (b), you don't need to do all four steps as in Problem
11.7. Just try one variational function for the ground state.
\begin{center}
\section*{Relativistic Corrections (Chapter 12)}
\end{center}
\item A neutrino is a massless spin-1/2 particle. Its wave
function $\xi$ is a spinor with two components and, instead
of the nonrelativistic Schr\"odinger equation, it satisfies
the following {\it Weyl} equation:
\begin{equation}
i\hbar\partial_t \xi= \pm c\,(\bfs\cdot{\bf p})\,\xi,
\label{weyl}
\end{equation}
where $c$ is the speed of light, {\bf p} is the momentum of
neutrino, and $\bfs$ are the Pauli matrices. The two signs in
Eq.\ (\ref{weyl}) correspond to the two possible types of
neutrino. This is a ``Schr\"odinger equation" with Hamiltonian
$\pm c\,(\bfs\cdot{\bf p})$ instead of ${\bf p}^2/2m$. It is
Lorentz covariant, because the time and space derivatives enter
on equal footing at the same first order.
\begin{enumerate}
\item {\bf [3 points]} Find the energy eigenstates of
(\ref{weyl}). Is the energy spectrum bounded from below?
Compare your result with the classical energy dispersion
relation of a massless particle $E=c|{\bf p}|$.
{\em In the quantum field theory the states with positive
energy correspond to particles, whereas the states with
negative energies correspond to antiparticles.}
\item {\bf [3 points]} Shows that the Weyl equation (\ref{weyl})
is not parity invariant. The + ($-$) sign Hamiltonian
describes right (left) handed neutrinos. Show that if the
$+$ sign is chosen in (\ref{weyl}), the spin is parallel
to the momentum for positive energy momentum eigenstates.
\setcounter{continue}{\value{enumii}}
\end{enumerate}
Now consider a right-handed neutrino-like particle characterized
by the two-component spinor $\xi$ and a left-handed
neutrino-like particle characterized by the two-component spinor
$\eta$ that are coupled together by the following equations:
\begin{eqnarray}
i\hbar\partial_t \xi &=&~~ c(\bfs\cdot{\bf p})\xi + mc^2\eta,
\label{1dirac} \\
i\hbar\partial_t \eta &=&- c(\bfs\cdot{\bf p})\eta + mc^2\xi.
\label{2dirac}
\end{eqnarray}
\begin{enumerate}
\setcounter{enumii}{\value{continue}}
\item {\bf [3 points]} \label{E} Show that Eqs.\ (\ref{1dirac}) and
(\ref{2dirac}) has energy eigenvalues $E=\pm\sqrt{{\bf
p}^2+m^2}$, thus $m$ can be interpreted as the mass of the
particle {\bf(see Hints)}. Eqs.\ (\ref{1dirac}) and
(\ref{2dirac}) are the {\em Dirac equation} in the {\em spinor}
representation.
\item {\bf [3 points]} Show that for ${\bf p}=0$, the eigenstate of
Eqs.\ (\ref{1dirac}) and (\ref{2dirac}) with $E=mc^2$ has the
property $\eta=\xi$. \label{p=0}
\item {\bf [5 points]} Let us make the superpositions of $\xi$ and
$\eta$:
\begin{equation}
\phi=\frac{1}{\sqrt{2}}(\xi+\eta),\qquad
\chi=\frac{1}{\sqrt{2}}(\xi-\eta),
\label{phichi}
\end{equation}
and combine them into a four-component bispinor:
\begin{equation}
\psi={\phi\choose\chi}.
\end{equation}
Show that in this, the so-called {\em standard} representation,
the Dirac equation (\ref{1dirac}) and (\ref{2dirac}) takes the
form
\begin{equation}
i\hbar\partial_t\psi=[c(\bfa\cdot{\bf p})+\beta mc^2]\psi,
\label{dirac'}
\end{equation}
where
\begin{equation}
\bfa=\left(
\begin{array}{cc}
0 & \bfs \\ \bfs & 0
\end{array}
\right)\qquad{\rm and}\qquad
\beta=\left(
\begin{array}{cc}
1 & 0 \\ 0 & -1
\end{array}
\right)
\label{ab}
\end{equation}
are the 4$\times$4 Dirac matrices.
\end{enumerate}
In the presence of an electromagnetic field characterized by the
vector potential {\bf A} and the scalar potential $\Phi$, Eq.\
(\ref{dirac'}) becomes
\begin{equation}
i\hbar\partial_t\psi=
\left[c\bfa\cdot\left({\bf p}-\frac{e}{c}{\bf A}\right)
+\beta mc^2 +e\Phi\right]\psi,
\label{diracA}
\end{equation}
As follows from Problem \ref{p=0} and Eq.\ (\ref{phichi}),
${\chi\ll\phi}$ in the nonrelativistic limit $p\ll mc$. This
allows to reduce the 4$\times$4 Dirac equation to the 2$\times$2
Pauli equation for the two-component spinor $\phi$ and find the
relativistic correction as explained in the enclosed pages from
\emph{Quantum Electrodynamics} by Berestetskii, Lifshitz and
Pitaevskii (vol.\ 4 of the \emph{Course of Theoretical Physics}
by Landau and Lifshitz).
\item {\bf [8 points]} Schwabl's Problem 12.1.
\item {\bf [8 points]} Schwabl's Problem 12.2.
\item {\em Adapted from Qualifier, September 1988, II-1.}
In most calculations of atomic energy levels the nucleus is
taken as a positive point charge $Ze$. Actually, the nuclear
charge is more accurately represented by a uniform charge
distribution reaching to a radius $R\approx Z^{1/3}$ Fermi. (1
Fermi = 10$^{-13}$ cm = 2$\times10^{-5}$ Bohr radius.)
\begin{enumerate}
\item {\bf [5 points]} Find the electrostatic potential produced
by a uniformly changed sphere of the charge $Ze$ and radius
$R$ inside and outside itself. \label{sphere}
\item {\bf [5 points]} Treating the small deviation of the
potential found in Problem \ref{sphere} from the Coulomb
potential in the first order of perturbation theory,
calculate the correction to the energy of a 1s electron
due to the nuclear size effect. How does this correction
depend on the nuclear charge $Z$?
\item {\bf [5 points]} How does this correction compare
numerically with the fine structure, hyperfine structure,
and Lamb shift corrections for the hydrogen atom?
\end{enumerate}
\item {\bf [8 points]} {\em Adapted from Qualifier, August 1997,
II-2.}
Suppose the Coulomb potential $-e^2/r$ in the hydrogen atom is
modified to a Yukawa potential,
\begin{equation}
U(r)=-\frac{e^2}{r} e^{-r/b},
\label{U}
\end{equation}
where $e$ is the electron charge and $r$ is the distance between
the electron and the proton. Assume that the length $b$ is much
greater than the Bohr radius $a$.
Determine the splitting of the degenerate levels of the hydrogen
atom due to potential (\ref{U}) to the lowest nontrivial order
of perturbation theory in $a/b$. Ignore other sources of
splitting discussed in Chapter 12. Restrict you consideration to
the energy levels that are not very close zero energy.
Useful formula for the Coulomb potential: $\langle
nl|r|nl\rangle =a[3n^2-l(l+1)]/2$ (see Eq.\ (6.45) of Schwabl).
\end{enumerate}
%\newpage
\label{Hints}
\section*{\centerline{Hints}}
\begin{description}
\item[\ref{E}] To find solutions of the eigenvalue equation
\begin{eqnarray}
E \xi &=&~~ c(\bfs\cdot{\bf p})\xi + mc^2\eta,
\label{dirac1}\\
E \eta &=&- c(\bfs\cdot{\bf p})\eta + mc^2\xi,
\label{dirac2}
\end{eqnarray}
find $\eta$ from Eq.\ (\ref{dirac1}), substitute it into Eq.\
(\ref{dirac2}), and use Eq.\ (9.18b) to get rid of the Pauli
matrices.
\end{description}
\end{document}