In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.

These lecture notes from the ninth week of https://paradigms.oregonstate.edu/courses/ph441 cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that \(\Delta V \approx V_g\). Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.

1. Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor and the latent heat \(L\) and the temperature.

2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

First complete the problem Diagonalization. In that notation:

1. Find the matrix \(S\) whose columns are \(|\alpha\rangle\) and \(|\beta\rangle\). Show that \(S^{\dagger}=S^{-1}\) by calculating \(S^{\dagger}\) and multiplying it by \(S\). (Does the order of multiplication matter?)
2. Calculate \(B=S^{-1} C S\). How is the matrix \(E\) related to \(B\) and \(C\)? The transformation that you have just done is an example of a “change of basis”, sometimes called a “similarity transformation.” When the result of a change of basis is a diagonal matrix, the process is called diagonalization.

Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.