\input psfig \documentstyle[12pt]{article} \setlength{\textwidth}{6.5in} %5.8 \setlength{\textheight}{9.0in} %8.2 \setlength{\topmargin}{-0.5in} %-0.3 \setlength{\oddsidemargin}{-0.0in} %0.38 \begin{document} \centerline{\bf PHYSICS 354} \centerline{\bf Modern Physics } \centerline{\bf Spring 1997 -- Final Exam -- May 6, 1997} Instructions: You have 120 minutes to complete this exam. Put your name, ID\#, and section on the front of {\bf both} blue books you have at your seat. Please write {\bf all solutions and show all work in the blue books provided}. This exam is closed book and closed note. You may use a calculator, but do so for calculations only, not for formulas. If you don't have a calculator, reduce the final expression to the simplest terms and estimate. Please note the point numbers associated with each part and plan your time accordingly. The exam totals 115 points, and is much longer than the time allotted, so choose your problems and plan your time! Equations and formulas you might need are supplied on the last page. Good luck ! \begin{enumerate} \item {\bf Blackbody } (10 pts) Describe how the energy levels from Planck's quantization hypothesis depicted in the following plot can be used to explain the solution to the ultraviolet catastrophe in classical solutions to the blackbody problem. \newbox\figa \setbox\figa=\vtop{\kern0pt\psfig{figure=energy-dist1.eps,width=2.3in}} \centerline{\box\figa} \item {\bf Density of states in 2 dimensions} Assuming free-electrons, derive the density of states in 2 Dimensions. (a) (5 pts) Start by assuming free electron behavior (plane wave solutions) confined in a 2D area of sides of length $L$. What are the equations which determine the allowed wavevectors $k_x$ and $k_y$ for periodic boundary conditions? Thereby show what the volume'' in $k$-space is occupied by each point. (b) (5 pts) The number of quantum states is given by the product of the density of states times the volume in $k$-space between $k$ and $k+dk$ times 2 for the spin degeneracy. The number of quantum states can also be written as the density of states per unit $k$ times $dk$. Use this fact to determine the density of states in 2D in terms of the wavevector. (c) (5 pts) The density of states is often written in terms of the number of states per unit energy. Use the result from above and the relation between energy and wavevector to show that the density of states in 2D is constant in energy. (You are welcome to use methods other than those described above to arrive at this final answer.) \item {\bf Addition of angular momentum} (a) (5 pts) Consider an atom which has a single electron in the $n=3$ state. Determine the possible values of the total angular momentum. (b) (5 pts) Now consider an atom which has two electrons in the d-shell. Determine the possible values of the total angular momentum, being careful about the total symmetry of the wavefunction. \item {\bf Degeneracy} (15 pts) Consider a particle of mass $m$ in a rectangular three-dimensional box, where two sides have length $L/2$ and the remaining side has length $L$. Outside the box, the potential is infinite and inside the box it is zero. Find the energies, quantum numbers and degeneracies of the five lowest unique energy states. \item {\bf Distribution functions} (a) (7 pts) Describe in words the three different physical situations under which the Fermi-Dirac, Bose-Einstein, and Maxwell-Boltzmann distribution functions are appropriate descriptions of particle energy distribution. Be specific and give an example of each. (b) (8 pts) At high energy, the Bose-Einstein and Fermi-Dirac distributions both approach the Maxwell-Boltzmann form. Show this mathematically, writing down the high energy'' criteria, and then {\bf explain} physically why this occurs. \item {\bf Semiconductors} (a) (5 pts) Imagine I have a direct band-gap semiconductor, designed to form the active region of a semiconductor laser structure. Start with the zone center band diagram, that is draw an $E(\vec k)$ vs $\vec k$ for a direct band-gap semiconductor at $T=0$ temperature. Assume free electron behavior and that the effective mass for the holes is twice that of the electrons. (b) (5 pts) Now draw the same band diagram except with some $n-type$ doping, and indicate in the diagram the level of the Fermi energy on the vertical energy scale. Continue to assume $T=0$. Draw next to this a plot of the carrier density as a function of energy. (c) (5 pts) Next, raise the temperature, and indicate the change in the carrier density versus energy plot. Assume a temperature of $kT \sim {1\over 5}E_F$. (d) (5 pts) In the following two plots, the zero temperature response of a semiconductor with and without high optical pumping are displayed. Draw the carrier density versus energy for both cases, and explain why and how lasing can occur in the one to the right. Be sure to describe the elements of the two plots. \newbox\figa \setbox\figa=\vtop{\kern0pt\psfig{figure=abs-gain1.eps,width=2.3in}} \newbox\figb \setbox\figb=\vtop{\kern0pt\psfig{figure=abs-gain2.eps,width=2.3in}} \centerline{\box\figa\hfil\box\figb} \item {\bf Schr\"odinger Equation: quantum mechanical tunneling} In this problem, you will derive the approximate transmission probability of a particle with energy E incident on a potential barrier of energy U, $U>E$. In the figure below the wavefunction in region $I$ is shown split into two parts, one for the incoming wave and one for the reflected wave. The number of particles per unit time which impinge on the barrier is given by $S = |\Psi_{I+}|^2 v_{I+}$ where $v_{I+}$ is the velocity (group velocity of the wave). The transmission probability is given by the ratio of the outgoing flux to the incoming flux, or $$T = { {|\Psi_{III}|^2 v_{III}\over{|\Psi_{I+}|^2 v_{I+}}}}$$ For this problem, since the energy is the same on either side of the barrier (same zero potential), the group velocity is the same, so we need only derive the ratio of the probability amplitudes. \newbox\figa \setbox\figa=\vtop{\kern0pt\psfig{figure=tunnel1.eps,width=3.3in}} \centerline{\box\figa} (a) (2 pts) Write down the Schr\"odinger equation and the trial wavefunctions for the three regions. (b) (2 pts) Write down the boundary conditions for this problem. (c) (8 pts) Apply the boundary conditions and solve for the ratio of the incoming to outgoing amplitude. (d) (8 pts) Finally, make the following approximations to find the transmission probability: Assume that the height of the barrier is high relative to the incident energy E of the particles, and that the barrier is relatively wide such that $k_{II}L >>1$, where $k_{II}$ is the wavevector (real) in the barrier region. Show, finally, that the transmission probability is proportional to $\sim e^{-2kL}$. \item {\bf Optical selection rules} (a) (6 pts) The general form of the transition probability and hence selection rules for optical transitions is proportional to: $$\int_{-\infty}^{+\infty} x \Psi_n \Psi_m^* dx$$ Determine the selection rules for a particle in a one-dimensional box using the parity of the wavefunctions from the symmetry of the potential. (b) (4 pts) A Hydrogen atom is excited to a 4$p$ state. To what state or states can it go by radiating a photon in an allowed transition? \end{enumerate} \newpage \small \centerline{\bf Formulas and expressions} $$N_A = 6.02\times 10^{23} \hskip 1.5in h=6.63\times 10^{-34} {\rm J\cdot s} = 4.14\times 10^{-15}{\rm eV\cdot s}$$ $$c=3.00\times 10^8{\rm m/s} \hskip 1.5in \hbar = 1.06\times 10^{-34} {\rm J\cdot s} = 6.58\times 10^{-16}{\rm eV\cdot s}$$ $$e = 1.60 \times 10^{-19}{\rm C} \hskip 1.5in G = 6.67\times 10^{-11}{\rm m^3\cdot kg^{-1}\cdot s^{-2}}$$ $$k = 8.62\times 10^{-5}{\rm eV/K}$$ \smallskip $$ke^2 = 1.44{\rm eV\cdot nm} \hskip 1.5in kT = 0.02885\ {\rm eV\ at\ } T=300K$$ $$hc = 1240{\rm Ev\cdot nm} \hskip 1.5in \hbar c = 197{\rm eV\cdot nm}$$ \smallskip $$a_0 = {\hbar c\over{\alpha m c^2}} = 0.0529\ {\rm nm} \hskip 1.0in \lambda_c = {h c\over{m c^2}} = 2.43\ {\rm pm}$$ $$\mu_{\rm B} = {e\hbar\over{2m}} = 5.79\times 10^{-5}\ {\rm Ev\cdot T} \hskip 1.0in \sigma = {\pi^5 k^4\over{15 h^3 c^2}} = 5.67\times 10^{-8}\ {\rm W\cdot m^{-2}\cdot K^{-4}}$$ \smallskip $${\rm Energy\ and\ temperature\ } = {3\over 2}kT$$ $${\rm Maxwell-Boltzmann\ Distribution\ } f_{\rm MB} = Ce^{-E/kT}$$ $${\rm Fermi-Dirac \ Distribution\ } f_{\rm FD} = {1\over{Ae^{E/kT} + 1}} \ \rm{or} \ f_{\rm FD} = {1\over{e^{(E-E_F)/kT} + 1}}$$ $${\rm Bose-Einstein \ Distribution\ } f_{\rm BE} = {1\over{Ae^{E/kT} - 1}}$$ $${\rm Thermal\ radiation\ (power\ per\ unit\ area\ per\ unit\ wavelength)\ } {dR\over{d\lambda}} = {2\pi h c^2\over{\lambda^5(e^{hc/ \lambda k T} - 1)}}$$ $$\lambda_m = {250\ {\rm Ev\cdot nm}\over{kT}}$$ $$\beta = {v\over c} \hskip 1.0in \gamma = {1\over{\sqrt{1-\beta^2}}}$$ $${\rm Energy,\ mass,\ momentum\ } E = \sqrt{(mc^2)^2 + (pc)^2} \hskip 0.25in \vec P = \gamma m\vec v \hskip 0.25in E = \gamma mc^2 = E_K + mc^2$$ $${\rm Wavelength\ and\ momentum\ } \lambda = {h\over p}$$ $${\rm Schrodinger\ Equation\ } {-\hbar^2\over{2m}}{d^2\Psi(x)\over{dx^2}} + V(x)\Psi(x) = E\Psi(x)$$ $${\rm Particle\ in\ a\ 1D\ box\ energy\ solutions\ } E = {(n^2 h^2)\over{8mL^2}}$$ {\rm Hydrogen\ atom\ } E_n = {-{13.6eV\over{n^2}}}; L = \sqrt{l(l+1)}\hbar \ l