Physics 451 Quantum mechanics I Fall 2012 Sep 7, 2012 Karine Chesnel

Phys 451 Announcements Help sessions: T Th 3-6pm Homework 3: F Sep 7th by 7pm Pb 1.4, 1.5, 1.7, 1.8 Homework 4: T Sep 11 by 7pm Pb 1.9, 1.14, 2.1, 2.2 Homework 5: Th Sep 13 by 7pm Pb 2.4, 2.5, 2.7, 2.8 Homework Please don’t forget to submit your homework on time!

Phys 451 Remarks from the TA after grading the first homework 1.Simplify your answers to their simplest forms. Don't leave it like x=(1/3-1/5)^(1/2) or x=1-Sigma, while you already have a value for Sigma. 2. Don't make your "rough sketch" too rough. Label your axes, and draw the curve nicely. Be a little more professional than the Physics 121 students. 3. Some simple calculus and graphs can be done by hand, such as a standard Gaussian. Don't rely entirely on Mathematica. 4. Don't write too compactly. Leaving enough space in your writing not only benefits the TA but also helps yourself when you go back and check. 5. Write your CID instead of your name.

Quiz 3a “If the wave function is normalized at a time t, it is then normalized at any time.” A. True B. False Quantum mechanics Evolution of  in time?

Quantum mechanics Probabilities & Wave function Density of probability (now function of space and time): Normalization: Solutions have to be normalizable: - needs to be square-integrable

Quantum mechanics Expectation values Density of probability: Average position x: Probabilities Average value for f(x): Quantum Mec.

Expectation values The expectation value is the average of all the measurements of the quantity f on a ensemble of identically prepared particles Quantum mechanics Differentiation between expectation value and most probable value See pb 1.4 and pb 1.5

Expectation values Evolution of in time? Quantum mechanics Expectation value for the momentum Expectation value for the velocity Schroedinger equation

Expectation values Generalization Quantum mechanics “Operator” x “Operator” p

Expectation values Examples Quantum mechanics Kinetic energy: Angular momentum: and so on..

Ehrenfest’s theorem Quantum mechanics Equivalent to Newton’s second law See pb 1.7

Quantum mechanics Effect of potential offset? V V+V0V+V0  ? (picking an extra phase) ? (no effect) See pb 1.8