Physics 155 Outline

Here is an outline of our topics and goals for this quarter. Some parts have been changed to be how we should have done it, or how we would like to see it now, in retrospect.

1. Lattice vibrations in spatially periodic systems.  We used a mass-spring type approach to look at the vibrational eigenmodes of a 1-dimensional chain of nucleii. This didn't really fit in with anything else, and it turned out to be confusing, so it is not a point of emphasis.

2. Electron states in a spatially periodic system. For a lattice of atoms, we showed that each atom bound state evolves into a band of N crystal states. This association between atom states and crystal bands is a critical starting point for solid state physics.  With "projection techniques" (appendix A) and approximations, we showed how band state energies depend on an atom state energy, an overlap integral, \(I_{11}\), and a crystal quantum number q (or k).

3. Bands arising from a 3 bound state square well. We looked explicitly at the bands that form for a lattice composed of square wells. We saw that bandwidth depends critically (exponentially) on how far the atomic state wave-function extends outside the well and that the bandwidth of the highest energy band is largest because its atomic state extends furthest.

4. Conductivity of a metal. We discussed how conductivity can be described, for a half-filled band at T=0, as a shifting of occupation boundaries in k-space leading to an imbalance of electron velocities and thus a net current.

5. Specific heat of a metal. We calculated the specific heat of a metal at low T, \(C_v = dE/dT\), using an integral expression for E and taking the derivative of the Fermi function inside the integral and making approximations.

6. Spontaneous magnetization. We created a simplified model for electron-electron interaction and showed that, in the case of a narrow bandwidth, e-e interaction can lead to spontaneous magnetization below a critical temperature, \(T_c\). The driving force for this phase transition and instability is the electron-electron repulsion, which is diminished when electrons spins align. The e-e interaction has to compete with both the band energy and entropy, both of which favor a state with no spin alignment, known as the normal state.

7. Magnetic susceptibility. Using a free energy minimization approach, we showed that magnetic susceptibility is enhanced by e-e interaction even when that interaction is too weak to establish spontaneous magnetization.

8. Matrix approach to band theory. The states of a single band can be obtained by expressing the quantum Hamiltonian as a Hermitian matrix with \( E_o\) on the diagonal and \( I_{11}\) on the off-diagonal. This matrix is written in the "local basis", however, its eigenstates are Bloch states and its eigenvalues are the Bloch state energies that we got before (Appendix A). The matrix approach allows the introduction of disorder and other variation into the crystal.

9. Matrix approach generalized to include electron-electron repulsion. Bloch states extend throughout the crystal and are very useful, but they are basically one-electron states. Sometimes that is not good enough. For example, anti-ferromagnetism and superconductivity cannot be explained in any way using one-electron states. We introduce a model for 2 electrons in a 2 site lattice that includes on-site electron-electron repulsion, U.

10. The nature of superconductivity. ... I get the impression that people really want to learn about superconductivity, even though it might be difficult. Is that true?

11. "More is Different"  "The Theory of Everything" (see attached papers)

Appendix A: Electron eigenstates can be written in "Bloch form", i.e., as a sum involving the particular atomic state with a shifted peak position and systematically varying "phase factor" coefficients. These states are completely itinerant, i.e, the probability density is the same at every site in the crystal. The Bloch form is:
\(\psi_k (x) = \Sigma_n e^{inak} \psi_o (x-na)\).
The projection involves multiplying the expression \( E_k \psi_k (x) = H \psi_k (x)\) on the left by \(\psi_o(x)\) and integrating to get: \(E_k = E_o + 2 I_{11} cos (ak)\), where \( I_{11}\) is a nearest neighbor overlap integral with units of energy.