Problem session 2

Solutions to problem set 1. Given hints on solving set 2 and getting pretty plots with MAPLE. See example at http://erwin.phys.virginia.edu/~jsf9k/hints.html. See also MAPLE short course notes, on reserve.

Lecture 4, Feb 2

Demos: shown "insides" of electrostatic air cleaner and laser printer.

Uniqueness of solution of Laplace's equation, , with assigned vaues of V on the boundary (in practice, V has assigned values on various conductors). How to proceed if the charges on the conductors are given, instead of the values of the potential. Simplest example: Q = C(V₁ - V₂) for a capacitor with charges +Q and -Q on the plates at potentials V₁ and V₂. C is the capacitance.

Started review of quantum theory:

What is quantized and when? In periodic motion, action is quantized in units of h; as a consequence, other quantities are quantized too, but not so simply.
What is action? It is a quantity with dimensions ML²T^-1; it can be energy × time, or momentum × length. Angular momentum has the same dimensions as action and is simply quantized in units of .
What is spin? It is something like the internal angular momentum of an elementary particle. However, electrons (and quarks) have spin /2, while orbital angular momentum must be an integer multiple of . Photons have spin . One often says that electrons have spin 1/2 and photons have spin 1.
What is the Pauli exclusion principle? Read about it in Serway, page 860, or Tipler, page 1206 and 1234, as it applies to electrons. Actually, it applies to all elementary particles with spin 1/2 (called fermions) and is mysteriously related to their unusual spin. It also applies to composite particles with half-integer spin (1/2, 3/2, etc.).

References:

• For capacitances: Tipler, chapter 21 ; PQRG, page 91; Serway, pages 480 - 495.
• For quantum mechanics: Tipler, chapter 35; Serway, chapter 29.
• A readable review of the early history and basic concepts of quantum theory is found in the sections on "Atoms" and on "Particles and Waves" of Fowler's notes for Phys 252.

Note however that these references use Planck's constant h instead of , so that some formulas look a little different. Physics articles today almost always use , unless they use "atomic units" in which is set equal to 1. See last semester's lecture on atomic units.
Phys 252 lectures are at www.phys.virginia.edu/classes/252.
The Phys 311 lecture is at www.phys.virginia.edu/classes/311/notes/units/node2.html

Lecture 5, Feb 4

Review of quantum mechanics: action, phase, sum over all possible paths with a phase factor, uncertainty principle, spin and the Pauli exclusion principle, plane wave solution of the time-dependent Schrödinger equation, E = hf = w and p = k.

Assigned problem set 2 (engineering of electrostatic precipitator, basic quantum theory).

Problem session 3

Assignment 2 solution: how to compute the gradient in cartesian and spherical coordinates. Using MAPLE: field plots using gradplot, integrals using int, range and properties of variables using assume and about. Given hints for problem set 3.

Lecture 6, Feb 9

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

where n is a positive integer. Plotted eigenfunctions sin(k_nx) 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.

- 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:
- Tipler Chapters 36 and 37, or Serway, chapter 29;
- PQRG, page 93;
- Fowler's notes for Phys 252, especially "Electron in a box" and the rest of the "Schrödinger equation" section.

Lecture 7, Feb 11

Assigned problem set 4 (complex numbers, coupled pendulums, two-level quantum system)

Demos: two and three coupled pendulums, array of coupled torsion bars, standing waves and traveling waves.

Reviewed complex numbers and the quantum phase factor exp(-iw t). 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). 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).

References:

• Tipler, Chapter 37
• For complex numbers, Feynman I, page 22-7 to 23-4.
• For two-level mixing, Feynman III, pages 8-11 to 8-14 (he applies exactly the same equations we used to the two possible states of NH₃, treats electron states for molecules in Ch. 10 in a more general way that leads far afield)
• For energy bands: Melissinos, pages 1 to 4

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.

Problem session 4

Gone over the solution of problem set 3. Shown how to use Maple to work with complex numbers. Given hints for problem set 4.

Lecture 8, Feb 16

Assigned reading: Melissinos, pages 3-7;

Filling of bands. Metals (good conductors), semiconductors and insulators. Reviewed band structure, valence and conduction band, energy gap. In insulators and semiconductors the conductivity is strongly affected by the concentration of carriers: both electrons added to the conduction band and holes left in the valence band act as carriers (of electrical current). Three mechanisms of creating carriers

1. Thermal excitation, dominated by Boltzmann factor where is the energy gap from the top of the valence band to the bottom of the conduction band. Orders of magnitude: eV is typical of a semiconductor; eV at room temperature,
2. Doping with impurities. Donor impurities supply extra electrons that go into the conduction band; acceptors create holes in the valence band, making the remaining valence electrons free to move.
3. Absorption of light (photoconductivity).

Reference: Tipler, Chapter 38, up to page 1291.

Lecture 9, Feb 18

Lecture 10, Feb 23

The conductivity $\sigma$ is by definition the proportionality constant in the microscopic form of Ohm's law, $J=\sigma E$, where J is the current density and E is the electric field. It is related to the microscopic properties of the conducting medium by Drude's formula

$$\sigma =\frac{N_{c}e^{2}\tau }{m^{*}}.$$

To obtain this formula, consider the effect of an electric field E on a particle of charge e and mass m^* in the presence of a frictional force: the drift velocity v_drift of the particle satisfies $m^{*}v_{drift}\mathbf{/}\tau =eE.$ If there N_c particles per unit volume, the current density is J=N_cev_drift and Drude's formula follows by comparing with $J=\sigma E$.

A more general formula is obtained when there are several groups of electrons (or holes), each with its m^* and $\tau :$just add the conductivities. The most refined quantum statistical treatments yield formulas of the same type and in addition relate the relaxation time $\tau$to the microscopic properties of the medium and to the temperature. A very useful formula is v$\tau$ = l, where l is the mean free path and v (notto be confused with v_drift) is an average speed of the charge carriers in the absence of the field E. For a classical gas, m^*v²=3k_BT gives a good estimate of vat temperature T, but in a metal quantum effects are important and v is the Fermi velocity, given by (1/2)m^*v² = E_F. (See below for the Fermi energy E_F.)

Density of states and occupation probabilities for electrons and holes in solids. Intrinsic semiconductors. The basic formulas are given by Melissinos in eqs. (1.4), (1.5), and (1.7).

Melissinos does not derive the last formula or explain what is. See the class notes in the file carriers.pdf

Assigned reading: Bloomfield, Section 6.3 (incandescent light bulbs).

Lecture 11, Feb 25