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\pagestyle{myheadings} \markboth{Homework \#2, Phys~623, Spring 2012,
Prof.~Galitski} {Homework \#2, Phys623, Spring 2012, Prof.~Galitski}
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{\bf Homework \#2} --- PHYS 623 --- Spring 2012 \\ {\color{red} Due {\bf
Friday, FEB-10-2012 at 10:00~a.m. in class}}
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Professor Victor Galitski \\
Office: 2330 Physics
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\href{http://terpconnect.umd.edu/~galitski/PHYS623/}{\color{blue} Class web-page:~http://terpconnect.umd.edu/$\sim$galitski/PHYS623/}.
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{\bf Write clearly your name and the homework number and staple the pages together.}
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\begin{itemize}
\item{} Teaching Assistant:~Bhupal Dev\\
Office:~PHS4208, Tel:~301.405.6016 , \href{mailto:bhupal@umd.edu}{e-mail:~bhupal@umd.edu}
\end{itemize}
\noindent Textbooks:
\noindent J. J. Sakurai and J. J. Napolitano, ``Modern Quantum Mechanics'' \\
L. D. Landau and L. M. Lifshitz, Vol.~III:~``Quantum Mechanics Non-Relativistic
Theory''
\begin{center}
\section*{Schr{\"o}dinger equation in momentum representation; Time-Independent Perturbation Theory}
\end{center}
\begin{enumerate}
\item Consider a particle in a shallow two-dimensional potential well, $U({\bf r})$, with the typical radius, $a$, and the typical strength, $U_0$, such that
$U_0 \ll \hbar^2/\left( ma^2 \right)$. Such a shallow potential well can be faithfully represented by the $\delta$-functional potential, $U({\bf r}) \approx - U_0 a^2 \delta^{(2)} ({\bf r})$. Following the lecture notes, use the Fourier transform to derive the corresponding Schr{\"o}dinger equation in momentum representation and use it to obtain a self-consistent equation for the bound state in the $\delta$-well, $E_0 < 0$. Use this equation to determine the energy of the state, $E_0$, within logarithmic accuracy ({\em hint:}~you may encounter logarithmically divergent integrals over momenta in your calculation; if this indeed happens, notice that large momenta or equivalently large wave-vectors correspond to small distances, so you may replace the corresponding ``ultraviolet'' limit of integration over ${\bf k}$ by the inverse radius of the potential, $k_{\rm max} \sim a^{-1}$).
Also extend your derivation to three dimensions and determine whether their exists a bound state in a three-dimensional delta-potential well (or equivalently, whether there always exists a bound state in an arbitrarily weak potential well in 3D).
\item Consider a {two-dimensional} harmonic oscillator,
\begin{equation}
\label{harm}
\hat{H}_0 = {\hat{\bf p}^2 \over 2m} + {m \omega^2 \hat{\bf r}^2 \over 2},
\end{equation}
with $\hat{\bf r} = \left(\hat{x}, \hat{y} \right)$ and $\hat{\bf p} = \left(\hat{p}_x, \hat{p}_y \right)$, and
a perturbation
\begin{equation}
\label{harm}
\hat{V} = \alpha \hat{x} \hat{y}^3,
\end{equation}
where $\alpha$ is small (in some sense). Calculate the first non-vanishing correction to the energy of the ground state, the first and the second excited energy level. Determine the domain of applicability of perturbation theory in terms of the parameters of the model.
\item The operator corresponding to the relativistic kinetic energy can be na{\"\i}vely written as $\hat{K} = \sqrt{ \hat{\bf p}^2 c^2 + m^2 c^4} - mc^2$. Consider the relativistic {three-dimensional} harmonic oscillator, $\hat{H} = \hat{K} + {m \omega^2 \hat{\bf r}^2 / 2}$ and use perturbative expansion in $1/c$ to derive a first relativistic correction, $\hat{V}$, to the harmonic oscillator Hamiltonian, $\hat{H}_0$ as in Eq.~(\ref{harm}) with $\hat{\bf r} = \left(\hat{x}, \hat{y}, \hat{z} \right)$ and $\hat{\bf p} = \left(\hat{p}_x, \hat{p}_y, \hat{p}_z \right)$. Then use quantum-mechanical perturbation theory to determine the first non-zero correction to the energy of the ground state and the first excited state. Determine explicitly the domain of validity of both the non-relativistic expansion and perturbation theory.
\item{} Consider a quantum particle with mass, $m$, inside a three-dimensional ellipsoidal cavity; i.e., the quantum well
\begin{equation}
\label{ellipsoid}
U(x,y,z) =
\left\{
\begin{array}{ll}
0, & {x^2 + y^2 \over a^2} + {z^2 \over b^2} < 1;\\
\infty, & {x^2 + y^2 \over a^2} + {z^2 \over b^2} \ge 1.
\end{array}
\right.
\end{equation}
Assume that $|a - b| \ll a$. Find the shift of the energy of the state, $|n_r,l,m\rangle_0$ in first-order perturbation theory, $E_{n_r,l,m}^{(1)}$ compared with the eigenenergy of the particle in a spherical well, $E_{n_r,l}^{(0)}$. Here, $n_r$, $l$, and $m$, label a radial quantum number, angular momentum, and its projection onto the $z$-axis, correspondingly.
\begin{enumerate}
\item{} Following your lecture notes, reduce the problem to that of a particle in a spherical quantum well with radius $a$, but with a slightly anisotropic mass.
\item{} Determine the unperturbed wave-functions $\psi_{n_r,l,m}^{(0)} ({\bf r}) = \langle {\bf r} |n_r,l,m\rangle_0$ and the unperturbed eigenenergies $E_{n_r,l}^{(0)}$ in the spherical well (the energy spectrum may involve quantities parametrized by zeros of some special functions).
\item{} Find an expression for $E_{n_r,l,m}^{(1)}$ in terms of an integral (matrix element) involving $\psi_{n_r,l,m}^{(0)} ({\bf r})$ and its complex conjugate.
\item{} {\em Difficult problem:} Calculate the integrals and determine explicitly the energy shift, $E_{n_r,l,m}^{(1)}$, for arbitrary, $n_r$, $l$, and $m$. Is the degeneracy of the levels lifted by the perturbation (weak ellipsoidal deformation of the spherical quantum well)?
\end{enumerate}
\end{enumerate}
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