http://sites.science.oregonstate.edu/portfolioswiki/ 2020-01-27T04:35:13-08:00 http://sites.science.oregonstate.edu/portfolioswiki/ http://sites.science.oregonstate.edu/portfolioswiki/lib/images/favicon.ico text/html 2012-07-16T16:29:08-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwbarrtunn?rev=1342481348 Homework for Waves Reflection of a quantum mechanical particle from a potential step The potential energy part of the Hamiltonian operator (a potential step of height $V_0$) for a system is depicted as a solid line in the energy/position diagram. This is an example of an unbounded system, so there is no condition on the energy eigenvalue. A particle of mass $m$ and energy $E$ is incident from the left. There are 2 cases: $E >V_0$, and $E <V_0$. (As you work through this problem, think caref… text/html 2012-07-05T13:27:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwbasiclanguage?rev=1341520032 Homework for Basic Language of Waves 1. (a) for A = 1 unit; $k$ = 2$\pi$ m$^{-1}$; $\omega = \pi $ rad/s. What are the wavelength, period and amplitude of the disturbance? Discuss the dimensions of A. (b) Plot in Mathematica two spatial cycles of the waveform and animate for two time periods. text/html 2012-07-06T09:15:32-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwfinitewell?rev=1341591332 Homework for Waves McIntyre 5.18: Energies for a particular finite well. Hints: Use Mathematica to plot Eqs. 5.88 as a function of $z$ (or the alternative formulation of the same equations that we used in class – you’ll get the same result). Zoom in to find intersection points and hence find energies. Show your working. text/html 2015-06-19T12:59:05-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwfourier?rev=1434743945 Homework for Waves (Fourier series) Analysis of normal modes of a string (start in class): A string (mass per unit length $\mu$, under tension $T$) is anchored at $x$ = 0 and $x = L$. It is then displaced so that it has the following profile at $t$ = 0, and the transverse velocity at all points is zero at $t$ = 0. Give a complete mathematical description of the motion of the string, including a Mathematica animation. text/html 2012-07-19T17:51:57-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwheisuncertprinc?rev=1342745517 Homework for Waves McIntyre 5.5: Expectation values, uncertainties for infinite well Use Mathematica to do non-trivial integrals. Having found $\Delta x$ and $\Delta p$ , also calculate the uncertainty product $\Delta x\Delta p$ for the first two states and demonstrate that $\Delta x\Delta p > \frac{\hbar }{2}$. text/html 2012-07-19T09:51:43-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwreftranatten?rev=1342716703 Homework for Waves This homework assignment is designed to take you through the experiment you conducted with pulses in a coaxial cable, and to work through each step again, thinking through the physical model and the experimental data. You are expected to write good prose, valid equations, and to pay attention to clarity. Be succinct, but make sure you explain clearly and demonstrate steps in important derivations. Your audience is incoming juniors majoring in physics who do not know the e… text/html 2012-07-05T12:24:35-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwstandtravel?rev=1341516275 Homework for Waves Traveling and Standing Waves: (a) “A standing wave is the sum of two traveling waves propagating at the same speed in opposite directions”. Prove analytically that this statement is true or untrue by explicitly adding the waves ${{\psi }_{1}}={{A}_{1}}\sin \left( kx-\omega t \right)$ and ${{\psi }_{2}}={{A}_{2}}\sin \left( -kx-\omega t \right)$. You could look at an animated function to try some possibilities, but you also need an analytical proof. text/html 2012-05-30T23:21:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwstringwaves?rev=1338445262 Homework for Standing Waves Standing Waves in a rope: Download from the class website the Excel sheet that has the results of the standing wave experiment we did in class. (a) Tabulate the raw data (# nodes, frequency, wavelength, for several standing waves) and also do the necessary conversions (explain, please!) to plot the “dispersion relation” (ω vs. k) for the rope and obtain the phase velocity of the waves in the rope from the graph. text/html 2012-07-16T16:30:34-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwsubtopic1?rev=1342481434 Homework for Waves Below are two eigenfunctions of different Hamiltonian operators. Consider the Hamiltonian operators corresponding to the 5 potential wells drawn below the functions. For each function, decide whether or not it could be the eigenfunction of each the 5 Hamiltonian operators and say why. Eigenfunctions: text/html 2012-07-05T14:36:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwsubtopic2?rev=1341524172 Homework for Waves (wave equation) Describe the following waveforms in words (waveform, period, phase angle, direction & speed of travel … $etc$.). Demonstrate whether they are, or are not, solutions to the non-dispersive wave equation \[\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\psi (x,t)={{v}^{2}}\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}\psi (x,t)\]. text/html 2012-07-06T09:55:18-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwsuperpos?rev=1341593718 Homework for Waves McIntyre 5.2 A particle in an infinite square well potental has an initial state vector \[\left| \psi (t=0) \right\rangle =A\left( \left| {{\varphi }_{1}} \right\rangle -\left| {{\varphi }_{2}} \right\rangle +i\left| {{\varphi }_{3}} \right\rangle \right)\] text/html 2012-07-06T09:57:37-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwtimeevo?rev=1341593857 Homework for Waves McIntyre 5.2, (d) and (e): Time evolution of superposition state McIntyre 5.8: Time evolution of superposition state; expectation value and time evolution text/html 2012-07-05T16:17:45-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:hw:wvhw:wvhwwaveenergy?rev=1341530265 Homework for Waves (From $Vibrations$ & $Waves$, I.G. Main problem 9.11) Show that the average power carried along a stretched string by a sinusoidal traveling wave is ${{P}_{av}}=\frac{1}{2}{{Z}_{0}}{{\omega }^{2}}{{A}^{2}}$, where the symbols have their usual meanings