http://sites.science.oregonstate.edu/portfolioswiki/ 2020-01-26T18:56:45-08:00 http://sites.science.oregonstate.edu/portfolioswiki/ http://sites.science.oregonstate.edu/portfolioswiki/lib/images/favicon.ico text/html 2011-07-15T16:38:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecatten?rev=1310773082 Lecture (20 minutes) Notes on and illustrations of attenuation in waves The students have studied attenuation in the context of oscillators, and the intent here is to have them work out the details as an independent exercise. The purpose is to enable them to confront their experimental results of reflection of waves at the boundary of two coaxial cables. In the experiment, the amplitude of the reflected pulse is different not only because of impedance mismatch, but also because of energy l… text/html 2011-07-18T22:15:41-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecbarrtunn?rev=1311052541 Barriers and tunneling (xx minutes) Slides: [Barriers and tunneling] The discussion of the eigenstates of a particle in a barrier potential follows exactly the same method as the bound particle in the finite well, only the exponential form of the solutions is the most convenient in all cases, not the sine and cosine forms. (The case of the particle with energy greater than the depth of a finite potential energy well is also easy). text/html 2012-07-05T11:14:03-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecbasiclanguage?rev=1341512043 Lecture (10 minutes) Notes on & illustrations of basic language of functions that are harmonically varying in space The students are thoroughly familiar with descriptions of quantities that oscillate in time \[\psi \left( t \right)=A\sin \left( \omega t+\varphi \right)\] Introduce the corresponding spatial analogs: text/html 2012-07-16T16:08:45-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecfinitewell?rev=1342480125 The finite potential energy well(xx minutes) Slides: [The finite well] This lecture is interspersed into the activity. The main point is to solve the energy eigenvalue equation in a potential energy well that is piecewise continuous, matching the boundary conditions. text/html 2012-07-19T17:55:49-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecfourier?rev=1342745749 Lecture (xx minutes) Notes on and illustrations of Fourier superposition of discrete frequency sinusoids and of continuous frequency sinusoids The first part of the notes are the introduction to the activity above, superposing waves of frequencies that are an integer multiple of some fundamental frequency. The second part relates to the creation of a pulse by the superposition of waves with all possible frequencies. Often the activity on the discrete superposition eats up all the time availa… text/html 2012-07-20T08:47:13-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecheisuncertprinc?rev=1342799233 Lecture ( minutes) Slides: [Heisenberg uncertainty principle and wavepacket evolution] Discuss the two “extremes” of wave functions, namely the exponential function (extending to $ \pm \infty$ in the spatial dimension) $A{e^{ikx}}$ and the delta function $A\delta \left( x \right)$. The exponential function represents a state where the momentum (wavelength) is known with certainty (students need to be reminded that it is that the infinite extent that gives this certainty) and the position… text/html 2012-07-05T09:58:33-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecinitcond2var?rev=1341507513 Lecture (5 minutes) This is really just an introduction to the activity. The students know that the position and velocity of an oscillator at, say, t = 0, determine the two arbitrary constants that appear in the equation of motion for the oscillator: \(B_{p}\cos \omega t+B_{q}\sin \omega t\) or \(A\cos \left( \omega t+\varphi \right)\) . They also know that this is the solution to a 2nd order ordinary differential equation. text/html 2012-07-05T16:08:18-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecnewtonlawrope?rev=1341529698 Lecture (xx minutes) Notes on the application of Newton's law to small transverse displacements of a rope under tension Show that the non-dispersive wave equation results when Newton's law is applied to small transverse displacements of a rope under tension. text/html 2012-07-05T09:54:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecnondispwaveeq?rev=1341507242 Lecture (30 minutes) This lecture demonstrates how to solve the non-dispersive wave equation \(\frac{\partial ^{2}\psi \left( x,t \right)}{\partial t^{2}}=v^{2}\frac{\partial ^{2}\psi \left( x,t \right)}{\partial x^{2}}\) by the technique of separation of variables. Tell the students that this equation results from application of Newton's law to a rope under tension or the Maxwell equations to fields in free space, and that they will shortly study this, but that at present, we want to show t… text/html 2012-07-16T15:19:27-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecoperators?rev=1342477167 Lecture (xx minutes) Slides: In the case of continuous observables where continuous wave functions represent the quantum state, operators take the form of differential operators (momentum, energy) or the variable itself (position). At this stage, it is too early to discuss the differences between the position and momentum representations; we implicitly use the position representation. Relate to matrices of the Spins course. text/html 2012-07-16T10:20:13-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecsprobdens?rev=1342459213 Lecture (xx minutes) Slides: [Probability density and probability] Having established the connection between the wave function and the column vector as representations of the quantum state, for the continuous and discrete observables, respectively, it now remains to make some of the other connections that will be familiar to students who have studied the spin-1/2 system. text/html 2012-07-05T16:04:49-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecsreftran?rev=1341529489 Lecture (30 minutes) Notes on & illustrations of reflection and transmission & impedance: Set up the problem of an abrupt boundary in a rope, e.g. a seamless transition from a rope of one density to another, that cause the wave speed to change at a position called $x = 0$. Ask the students if they can think of a situation where they have encountered something analogous. The previous day's discussion usually prompts a response of a light wave incident on glass or water. If not, you can elic… text/html 2012-07-05T12:21:35-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecstandtravel?rev=1341516095 Lecture (30 minutes) Notes on standing and traveling waves, and phase, (group,) and “material” velocity [standtravel_wiki] Present the continuous “wave” first as a series of discrete oscillators at different locations, each of whose phase has a definite relationship to the phase of the others. Show animations of standing waves and traveling waves. Point out that standing waves can be viewed as in-phase oscillators with amplitudes that depend on position and how the mathematical form … text/html 2015-10-07T13:04:05-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecsuperpos?rev=1444248245 Lecture (xx minutes) Slides: [Superposition, measurement, normalization, //etc//]. Discussion of the concept of superposition of wave functions and the implications for measurement, projection, probability of measuring a particular eigenvalue, normalization, expectation values. The most tractable example is the infinite square well potential energy. text/html 2011-07-15T16:01:54-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecswave_energy?rev=1310770914 Lecture (12 minutes) Notes on energy density in waves Discuss the concept of energy density, which is more relevant for a continuum system. There is energy transmitted when a harmonic wave propagates in a rope. What is its origin? The kinetic energy in a rope comes from its motion, of course, but where does the potential energy come from? (It's stored in the stretch). Must then find an expression for how much the rope is stretched at each point. $\ell -\Delta x\approx \frac{1}{2}\left[ \fr… text/html 2012-07-16T16:09:37-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlectimeevo?rev=1342480177 Lecture (xx minutes) Slides: [Time-dependent Schroedinger equation] This discussion goes over the solution to the TDSE, $\hat{H}\psi \left( x,t \right)=i\hbar \frac{\partial \psi \left( x,t \right)}{\partial t}$, as discussed in the Spins paradigm, but now in wave function language. The students generally have much less recall of this particular topic than they do of others encountered in Spins, for example, the idea of projection (which they know well). text/html 2012-07-11T09:10:10-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:wvlec:wvlecwavefunction?rev=1342023010 The wave function (xx minutes) Notes:[Basics of wave functions] The lecture makes a transition from a classical wave function to the idea of a wave function applied to a quantum system, where its modulus squared is an indication of the particle density in the quantum system. It is important to connect the students' previous experience with quantum systems, namely the spin-1/2 system, with the wave function, and to motivate the utility of the wave function.