http://sites.science.oregonstate.edu/portfolioswiki/ 2020-01-27T01:37:07-08:00 http://sites.science.oregonstate.edu/portfolioswiki/ http://sites.science.oregonstate.edu/portfolioswiki/lib/images/favicon.ico text/html 2011-08-11T11:19:22-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplec2wellsys?rev=1313086762 Finding the Energy Eigenstates of a 2-Well System (30 minutes) Add image: pplec2wellsys1 The function representation of this system's Hamiltonian will look like: $$H=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}} + V_{atom}\left(x-a\right) + V_{atom}\left(x-2a\right) \; \; . $$ text/html 2011-08-10T14:51:41-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecaandb?rev=1313013101 Solving for $\alpha$ and $\beta$ for the 2-well Case (10 minutes) $$H \; = \; \left[\begin{array}{cc} \alpha & \beta \\ \beta & \alpha \\ \end{array}\right] \; \; . $$ But, how do we explicitly find what $\alpha$ and $\beta$ are? First, let's find alpha. We will start by looking at the operation text/html 2011-08-26T10:32:41-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecapproxbe1d?rev=1314379961 Approximating the Internal Energy for a 1-Dimensional Chain (30 minutes) Note: Time permitting, this lecture would also work well as an activity for students to tackle in small groups. A 1-dimensional chain contains N atoms and has total length L. Find the total energy stored in the lattice when: text/html 2011-08-26T10:33:39-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecapproxbe3dlat?rev=1314380019 Approximating the Internal Energy for a 3-D Lattice (30 minutes) Time permitting, this lecture also converts well into an activity for students to tackle in small groups. * Before performing this lecture, it is recommended that students first Approximate the Internal Energy for a 1-D Chain. text/html 2011-08-26T10:35:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecdifferences?rev=1314380112 Differences Between Vibrating Atoms and Electrons in a Crystal (5 minutes) [(Add image: pplecdifferences1)][(Add image:pplecdifferences2)] text/html 2011-08-26T10:50:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:ppleceffmass?rev=1314381012 Effective mass (10 minutes) Let's first consider the mass of a classical object. Some important characteristics: Now, for a classical object, the kinetic energy of the object is equal to $$E \; = \; \frac{1}{2}mv^{2} \; = \; \frac{p^{2}}{2m} \; \; . $$ text/html 2011-08-26T10:49:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecfullband?rev=1314380966 Filled Electron Band in a Crystal (10 minutes) Let's now consider a crystal that has a filled band. Equivalently, let's consider a crystal that has an electron filling each possible eigenstate the crystal has to offer. [Add image: pplecfullband1] text/html 2011-08-26T10:48:25-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplechalfband?rev=1314380905 Half-Filled Electron Band in a Crystal (15 minutes) [Add image: pplechalfband] Some things to notice: Now, let's apply an electric field across the crystal and see what happens to the electrons. [Add image: pplechalfband2] Just like the single electron band, each electron is accelerated in the presence of an electric field. Now, there are more right movers than there are left movers in the system (i.e. there are more electrons with positive wave vectors). What does this tell us about … text/html 2011-08-09T10:24:56-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecheatcapintenergy?rev=1312910696 Using the Equipartition Theorem to Estimate Heat Capacity (10 minutes) Example: solids at room temperature and above. $$C_{\alpha} \, = \, \left(\frac{dU_{tot}}{dT}\right)_{\alpha} \; \; , $$ Where $\alpha$ is the variable of the system being held constant (volume, pressure, etc.). text/html 2011-08-26T10:24:16-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecinf1dchain?rev=1314379456 Infinite Chain of One-Dimensional Atoms (30 minutes) Consider an infinite chain of one-dimensional atoms, as seen below. $$x_{n}=\sin{kna} \; \; $$ where $na$ is the location of the nth atom and $k$ is the wave vector of the envelope function. The position function can be generalized by writing it as text/html 2011-08-26T10:23:46-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecinf1ddiachain?rev=1314379426 Infinite Chain of One-Dimensional Diatomic Molecules (40 minutes) $$m\ddot{x}_{n} \, = \, -\kappa\left(x_{n}-x_{n-1}\right) - \kappa \left(x_{n}-x_{n+1}\right) \; \; .$$ $$m_{A}\ddot{x}_{n}^{A} \, = \, -\kappa\left(x_{n}^{A}-x_{n-1}^{B}\right) - \kappa \left(x_{n}^{A}-x_{n}^{B}\right) \; \; $$ text/html 2011-08-26T10:21:08-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecnmode?rev=1314379268 Approximating the N-th Normal Mode Frequency for an N-chain Oscillator (10 minutes) $$m\ddot{x}=-2\kappa x \; - \; 2\kappa x \; \; . $$ Assuming that the equation describing the particle's motion has the form $$x(t)=Ae^{i \omega t} \; \; , $$ text/html 2011-08-26T10:43:20-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecnwellhamil?rev=1314380600 Hamiltonian for the n-well System (20 minutes) the analysis of a 2-well system [ Add image: pplecnwellhamil] Recall that the Hamiltonian for our 2-well system had the form $$H \; = \; \left[\begin{array}{cc} \alpha & \beta \\ \beta & \alpha \\ \end{array}\right] \; = \; \left[\begin{array}{cc} \langle 1 \vert H \vert 1 \rangle & \langle 1 \vert H \vert 2 \rangle \\ \langle 2 \vert H \vert 1 \rangle & \langle 2 \vert H \vert 2 \rangle \\ \end{array}\right] \; \; . $$ text/html 2011-08-26T10:44:11-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecnwellsys?rev=1314380651 Finding the Energy Eigenstates of an N-Well System (40 minutes) how to find the Hamiltonian for the n-well system $$H \; \dot{=} \left[\begin{array}{ccccc} \alpha & \beta & 0 & 0 & \dots\\ \beta & \alpha & \beta & 0 & \\ 0 & \beta & \alpha & \beta & \ddots \\ 0 & 0 & \beta & \alpha & \ddots \\ \vdots & & \ddots & \ddots & \ddots \\ \end{array}\right] \; \; . $$ text/html 2011-08-26T10:46:23-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecpbc?rev=1314380783 Periodic Boundary Conditions on Long Chains of Atoms (15 minutes) periodic [Add image: pplecpbc1] The two important conditions that must be satisfied for periodicity: The first condition essentially tells us that if we continued the envelope function past our unit length, it must be continuous and smooth. Let's see if $k=\frac{\pi}{L}$ or $k=\frac{2\pi}{L}$ satisfy this condition. text/html 2011-08-26T10:17:24-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecperiodicdim?rev=1314379044 Periodic Systems in Different Dimensions (5 minutes) 3-D periodic system example: Below we can also see a cartoon interpretation of the unit cells in Opal. A 2-D periodic system can be acquired by taking only a single cross-section of unit cells in the pyramid. text/html 2011-08-26T11:11:24-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecphenomena?rev=1314382284 Common Phenomena in Periodic Systems (10 minutes) Sound waves in a crystal. [Add image: pplecphenomena1] Light waves in opal and photonic crystals. [Add image: pplecphenomena2] Electron waves in an atomic crystal. [Add image: pplecphenomena3 ] text/html 2011-08-09T16:57:07-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecquantenergy?rev=1312934227 Quantization of Energy in Mechanical Oscillators (15 minutes) Draw Some Bound States Add image: pplecquanenergy1 Monatomic Chain Lab Add image:pplecquanenergy2 In essence, each normal mode for a multi-particle oscillator can be thought of as a collective mass bound by an effective potential. text/html 2011-08-26T10:19:33-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecrevcircuitde?rev=1314379173 Review of Differential Equations in Circuits (5 minutes) $$V_{L}=L\frac{dI}{dt} \; \; ,$$ and $$V_{C}=\frac{Q}{C} \; \; .$$ Now, as an example of Kirchoff's Voltage Law, if we combine a capacitor and an inductor in series as shown below, we find that text/html 2011-08-26T10:47:29-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecsingleelec?rev=1314380849 Single Electron in a Crystal (15 minutes) $$E = \alpha + 2\beta \, \cos{ka} \; \; , $$ where we recall that $\alpha$ is a positive value and $\beta$ is a negative value. Let's plot this energy graph. [Add image: pplecsingleelec1] Recall that each allowed energy value is directly related to a particular wave vector $k$. The circles in the graph represent these discrete allowed energies. text/html 2011-08-26T10:20:33-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplectwocoupledosc?rev=1314379233 Two Coupled Oscillators (30 minutes) Before beginning this activity, it is recommended that the instructor ask the Single Simple Harmonic Oscillator small whiteboard question. This is useful for reviewing Hooke's Law and how to use the law to find the equations of motion for an oscillating system. text/html 2011-08-26T11:10:30-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecunit1terms?rev=1314382230 Definitions of Important Terms for This Unit (10 minutes) Presenting these definitions in between students Emulating a Wave in a Periodic System is highly recommended. Doing so will help solidify the connection between the verbal and physical representations. text/html 2011-08-26T10:42:03-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:pplec:pplecwavestates?rev=1314380523 Energy Eigenstates of a Single Potential Well (5 minutes) [{{courses:lecture:pplec:pplecwavestatesfig1.png|Add image: pplecwavestates1] [Add image: pplecwavestates2] Rather than write the exact equation for this function repeatedly throughout the course, let's represent this state as