8 October 2010http:// Eigenstates in Quantum Mechanics Corinne Manogue Tevian Dray

8 October 2010http://

8 October 2010http://

8 October 2010http:// Outline Geometric Interpretation of Eigenvectors Eigenstates on the Ring Spherical Harmonics Spins

8 October 2010http:// Schrödinger’s Equation The Hamiltonian is an operator representing the energy. In undergraduate courses, it will always be given to you.

8 October 2010http:// Separation of Variables

8 October 2010http:// Eigenvalue of the Hamiltonian In undergraduate courses, the professor will always show you how to find eigenstates of the Hamiltonian.

8 October 2010http:// Superpositions of Eigenstates Today, I will show you several quantum systems and tell you the eigenstates. We will explore together what you can learn once you know these eigenstates.

8 October 2010http:// Eigenstates on the Ring

8 October 2010http:// Time Dependence

8 October 2010http:// Time Dependence

8 October 2010http:// Eigenstates on the Ring

8 October 2010http:// Eigenstates on the Sphere

8 October 2010http:// Spins

8 October 2010http:// KetFunctionMatrix Hamil- tonian Eigen- state Coeff- icient

8 October 2010http:// Early Quantum Mechanics Spin & Quantum Measurement –Successive Stern-Gerlach Measurements 1-D Waves –cf. classical waves on string –1-d Schrödinger—particle-in-a-box Central Forces –cf. classical orbits –3-d Schrödinger—the hydrogen atom

8 October 2010http:// Eigenvectors Activity Draw the initial vectors below on a single graph Operate on the initial vectors with your group's matrix and graph the transformed vectors

8 October 2010http:// Eigenvectors Activity Note any differences between the initial and transformed vectors. Are there any vectors which are left unchanged by your transformation? Sketch your transformed vectors on the chalkboard.

8 October 2010http:// Effective Activities Are short, containing approximately 3 questions. Ask different groups to apply the same technique to different examples. Involve periodic lecture/discussion with the instructor.

8 October 2010http:// Spin & Quantum Measurement Uses sequential Stern-Gerlach experiments as a concrete context for exploring the postulates of quantum mechanics. Probability, eigenvalues, operators, measurement, state reduction, Dirac notation, matrix mechanics, time evolution, spin precession, spin resonance, neutrino oscillations, the EPR experiment. J. S. Townsend, A Modern Approach to Quantum Mechanics (McGraw-Hill, New York, 1992).

8 October 2010http:// Spin & Quantum Measurement Students infer wave function from “data.” Measurement based. D. V. Schroeder and T. A. Moore, "A computer-simulated Stern- Gerlach laboratory," Am. J. Phys. 61, (1993).

8 October 2010http:// 1-D Waves (Classical) Waves in electrical circuits, waves on ropes. (Quantum) Matter waves of quantum mechanics. (Math) Fourier analysis to begin the study of eigenstates.

8 October 2010http:// 1-D Waves Coax Cable: –Standing waves. –Traveling waves. –Wave packets. –Dispersion. –Energy. –Reflection. –Transmission. –Impedance.

8 October 2010http:// ODE’s vs. PDE’s

8 October 2010http:// Central Forces (Classical) Orbits. (Quantum) Unperturbed hydrogen atom. (Math) Special functions.

8 October 2010http:// Central Forces Classical Orbits & Quantum Hydrogen Atom –Use reduced mass –ODE’s PDE’s (interpretation of QM) –Use spherical symmetry to simplify equations –Conserved—Angular momentum & Energy –Effective potential –Symmetric potential but asymmetric solutions

8 October 2010http:// Effective Potential

8 October 2010http:// Central Forces—Activities Students draw potentials for 2-d air table Interactive orbits in Maple Ring (1d) Sphere (2d) Hydrogen (3d) Use color for value of probability density Time dependent superpositions

8 October 2010http:// Eigenstates on the Ring

8 October 2010http:// Using Color to Visualize Spherical Harmonics

8 October 2010http:// Using Color..\OSU\mathphys\mathphys\paradigm6\flatylm.mws

8 October 2010http:// Active Engagement Effective but Slow –Precious commodity –Use wisely Special Needs of Upper-Division Easily Over-Scheduled Can Get Out-of-Synch Short Activities Mid-Lecture Moving Rooms: awkward but possible

8 October 2010http:// Two Messages I.Plan for a concept to build over time. Within a single course. Across several courses.

8 October 2010http:// Eigenstates Preface –2-D eigenvectors in Bra-Ket notation Spin & Quantum Measurements –2 state systems 1-D Waves –Fourier series and 1-D Schrödinger Central Forces –Ring (1-D) Sphere (2-D) Hydrogen (3-D) Periodic Systems –Band Structure

8 October 2010http:// Two Messages I.Plan for a concept to build over time. II.Use an appropriate mixture of lecture and active engagement.

8 October 2010http:// Lecture vs. Activities The Instructor: –Paints big picture. –Inspires. –Covers lots fast. –Models speaking. –Models problem- solving. –Controls questions. –Makes connections. The Students: –Focus on subtleties. –Experience delight. –Slow, but in depth. –Practice speaking. –Practice problem- solving. –Control questions. –Make connections.

8 October 2010http:// Central Forces—Activities Students draw potentials for 2-d air table Interactive orbits in Maple Ring (1d) Sphere (2d) Hydrogen (3d) Use color for value of probability density Time dependent superpositions

8 October 2010http:// Physicists can’t change the problem. Physics involves the creative synthesis of multiple ideas. Physics problems may not be well-defined math problems. Physics problems don’t fit templates. Physics involves the interplay of multiple representations. –Dot product. –Words, graphs, symbols