Thursday, January 13, 2011

Blast from the past

Yesterday I received a very nice and welcome email from a faculty member who had been one of my best classroom instructors in graduate school. This email was, effectively, a reply to an email that I had sent him regarding Stanford's graduate physics curriculum. The amusing bit is that I had sent him that email 14 years ago, when I was a senior grad student representative to Stanford's physics graduate committee. At the time, there had been ongoing discussions about what topics should be in the first-year graduate curriculum, particularly the "mechanics" sequence, and my opinion had been asked for. It's interesting to look back now as a faculty member at what I'd suggested at the time. Here are the bullet point topics I'd suggested. Remember that Stanford is on the quarter system, meaning that there are three ten-week quarters during the regular academic year.

For "Mechanics of Particles" (basically graduate mechanics and dynamics), I'd said:
- Brief review of variational calculus
- Lagrangians and Hamiltonians, action principle
- Canonical transformations, phase space
- Symmetries and conservation laws (Noether's thm?)
- Normal modes, harmonic oscillator review
- Rigid body motion (numerical work?)
- Orbital mechanics review
- Classical perturbation theory (w/ orbits, rigid body dynamics, anharmonic oscillator)
- Action-angle variables
- Poisson brackets, symplectic structure (*definitions of 1-forms, tangent spaces, tangent bundles?)
- Chaos, nonlinear dynamics, ergodicity
- Brief review of Einstein summation convention
- Special relativity w/ Einstein summation convention, space-time diagrams

For "Continuum mechanics" (fairly unique, I now realize - many departments offer no such course), my suggestions reflected my undergrad engineering background to some degree. I now realize that what I list below is considerably too much for a 10 week course:
- Mechanics of solids:
+ Continuum mechanics version of Hooke's law; stress, strain, tension, compression, shear, bulk modulus, a few numbers about strength of materials, Young's modulus, shear modulus
+ Lagrangian/Hamiltonian densities, more variational calculus
+ *Flexure of beams, bending moments, areal moments of inertia (why I-beams are stiffer than rods of the same cross-sectional area)
+ *Torsion of members, polar "moments of inertia"
+ *Dynamics of beams: the wave equation, longitudinal and transverse sound, natural frequencies of cantilevers
+ Acoustics, idea of acoustic impedance and mismatch
- Fluid statics
+ Hydrostatics, Archimedes' principle, buoyancy
+ *Surface tension, capillary action, wetting
- Fluid mechanics
+ Euler and Lagrange pictures
+ "Convective derivatives", transport of momentum and energy
+ The energy equation, the momentum equation, the continuity equation, the Navier-Stokes equation
+ Inviscid, incompressible flow:
- Bernoulli's Eqn.
- Potential theory
- *Vorticity, circulation, Magnus' law, "lift"
+ Viscous, incompressible flow:
- Definition of viscosity, comparison w/ shear modulus, definition of Newtonian fluid
- Stoke's law
- Intro to dimensional analysis, Reynolds' number
- Laminar flow, parabolic velocity profile in a round pipe
- Turbulent flow, mention engineering approach to these problems (Moody chart, friction factor, Bernoulli w/ losses)
- Froud number, hydraulic jumps (example of a "shock" discontinuity that you can demonstrate in a sink)
+ Compressible flow
- Mention of shockwaves, scaling

For "Statistical Mechanics", the main challenge was dealing with the divergent backgrounds of incoming students - some people had very strong undergrad preparation in statistical and thermal physics, others much less so. This is an issue in graduate quantum mechanics to an even greater degree. Now that I've taught undergrad stat mech several times, I think what I listed below could use some additional advanced topics:
- Definition of entropy, why it's a log
- The equal prob. postulate/ergodic thm.
- The Boltzmann factor and the partition fn., Fermi and Dirac distributions
- *Mention of Feynman diagram methods, saddle-point integration to get Z in complicated systems
- The canonical and grand canonical ensembles, the chemical potential
- "Natural" variables, Legendre transforms, thermodynamic potentials, *the idea of a constrained maximization of S, the Maxwell relations, the "thermodynamic square"
- Gases
+ Ideal classical
+ Van der Waals, virial coefficients
+ Fermi gas at zero and finite T
+ Ideal Bose gas, BEC, phonons & photons *(incl. laser discussion!)
- Liquids - diagrammatic methods of treating interactions?
- Solids
+ Phonons
+ Concept of long-range order
- *Correlation functions, *connection w/ susceptibilities
- *Correlations and fluctuations, *how they're measured!
- Theories of phase transitions
+ Concept of order parameter
+ Ginsberg-Landau theory, diff. betw. 1st and 2nd order, extensions to include fluctuations
+ 1st order: Van der Waals reprise, Clausius-Clapeyron
+ Mean-field theory, example of magnetism
+ Ising model in 1-d
+ Renormalization group to solve Ising model, critical behavior, correlation length ideas
- *Transport
+ *Boltzmann equation
+ *Noise in transport: fluctuation/dissipation thm
It was definitely interesting to me to see how my thinking on this stuff has evolved now that I have to teach it.


Tahir said...

This is very useful and interesting!

I wonder ... did you have any ideas or thoughts on how to teach quantum mechanics, and especially, quantum field theory? The latter in particular does not seem very standardized, and I'd be interested in hearing your opinion on how best to pedagogically approach it.

Douglas Natelson said...

Hello Tahir - Graduate quantum mechanics faces some of the same issues as stat mech. Some people come into a physics graduate program with very strong undergrad quantum preparation. Undergrads from Rice, for example, have seen time-dependent perturbation theory, serious angular momentum addition (Clebsch-Gordon coefficients), etc. Other domestic students may be coming from programs where they don't see these topics. A good graduate quantum class that serves both audiences well is very tough to structure.

As for QFT, I'm not really the best person to ask. I had one quarter of QFT in grad school, taught out of Peskin and Schroeder. I also took a many-body class and found Mattuck's book to be useful. Any readers out there have a strong opinion on the right way to teach/learn QFT?