Lecture 1, Jan 16, 97

Review of electrostatics: Coulomb's law, (or ), . Relation of field to potential, , (or ). In charge-free space (Laplace's equation). Analogy of electric field to fluid velocity for an incompressible, non-viscous fluid.

Assigned reading: Bloomfield, section 11.1

Assigned problem set 1 (field enhancement at a bump)

References: Serway, chapters 16 and 17 or equivalent (review); PDR, page 14 (Differential operators) and 29 (Electrostatics); Dorsey 311 notes, chapter 2

Lecture 2, Jan 21

Demos: discharges from domes and a sharp point; Jacob's ladder (effect of previous discharges)

Review of quantum mechanics: action, sum over paths, Schrödinger equation, plane wave solution ; eigenstates.

References: Serway, chapter 29 or equivalent.

Problem session 1

Given hint on solving problem set 1. (See also Dorsey 311 notes, 2.4.5, but note that the problem there is for a cylinder, not a sphere). Reviewed electrostatics in more detail. Shown ''insides'' of Sears air cleaner.

Lecture 3, Jan 23

Demos: precipitating smoke particles, speed of air flow in electrostatic precipitator.

Assigned problem set 2 (engineering of electrostatic precipitator; energy and wavelength of emitted photon)

Review of quantum mechanics:

- Particle in a box (one dimensional); obtained energy eigenvalues

where n is a positive integer. Plotted eigenfunctions for n=1,2,3. One can write where is the ground state energy.

- Harmonic oscillator; eigenvalues

where n is a positive integer or zero. One can write where is the ground state energy, also called the zero-point energy.

- Classical Kepler problem (hydrogen atom, neglecting spin and relativistic correction); there are bound states (E<0) that correspond classically to elliptic orbits, and scattering states (E>0) that correspond classically to hyperbolic orbits. The bound state eigenvalues are given by Balmer's formula

where n is a positive integer and is the ground state energy. For hydrogen, eV.

- Schrödinger equation for a many-electron atom:

is the potential due to the nucleus:

and is the coulomb potential of interaction between the electrons:

where and so on. For the time-independent equation, replace by .

The presence of makes the problem intractable analytically. Numerical solutions are obtained by starting with the approximation that each electron moves in the average potential of the others. In this way accurate results can be obtained not only for atoms, but also for molecules and chunks of matter.

References: Serway, chapter 29; PDR, page 34 (item D) and chapter 5; Fowler notes, especially ``Electron in a box'' (pages 14 - 16), and ``Multiparticle wavefunctions'' (pages 38-42)

Lecture 4, Jan 28

Hydrogen atom orbits and degeneracy; atomic orbitals, periodic table. Demonstrated energy transfer and eigenmodes in coupled pendulums, as a model of two coupled quantum states. The wavefunction of the coupled quantum states is

and the time evolution is given by

Considered the case when (identical atoms, for instance). To be continued next lecture.

References: Dorsey notes, lectures 22 and 23 (handed out). For atoms, Serway, Ch. 30, especially 30.4 and 30.6. For two-level mixing, Feynman III, pages 8-11 - 8-14 (he applies exactly the same equations we used to the two possible states of NH , treats electron states for molecules in Ch. 10 in a more general way that leads far afield)

Problem session 2

Used Heisenberg's uncertainty relation to derive and the energy for the ground state of the harmonic oscillator. Noted that and in this case. Assumed that has the minimum possible value which is true. Used the equipartition relation Found and compared with interatomic spacing. Guessed that the ground state wave function is

Gone to the computer room 109, shown how to use Scientific Workplace..

Lecture 5, Jan 30

Assigned problem set 3 (harmonic oscillator eigenfunctions, coupled pendulums)

Demos: three and four coupled pendulums, array of coupled torsion bars, standing waves and travelling waves.

Shown that for two identical atoms (see last time) there are two eigenstates: one with (bonding orbital, energy E+V), and one with (antibonding orbital, energy E-V). For N atoms, get a band of N eigenstates of the whole system from each atomic level (eigenstate of one atom). Filling of bands. Metals (good conductors), semiconductors and insulators. Typical resistivities. Brief explanation of superconductivity (electrons move in concert with each other and with the vibrations of the atomic nuclei, not indipendently as in normal metals).

Question: In the equations for two coupled quantum states (such as atomic orbitals), what is V and does it have dimensions of energy?

Answer: In this case V denotes a number (not a function of x,y,z as it did in electrostatics) and represents an energy of interaction, suitably weighted over the atomic orbitals. For example, consider two hydrogen atoms that share one electron (hydrogen molecular ion). Then, in gaussian units,

where denotes the position of the electron, and those of the nuclei (just protons in this case); and are the atomic eigenfunctions that were denoted as and for short.

Question: How can current run forever in a superconductor? Does this mean that perpetual motion is possible?

Answer: all quantum systems are in perpetual motion, usually on a scale too small for us to perceive. Superconductors can exhibit quantum behavior on a large scale. A superconductor does not carry a net current in its ground state, although the electrons move back and forth a lot even then. It is possible to set up an excited state of a superconductor that carries a net current and is similar to for a single electron, except that now all the electrons move in step in a concerted way and collectively resist disruptions of the current-carrying state.