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Activities

Problem

##### Diagonalization Part II
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.
• Found in: Quantum Fundamentals course(s)

Lecture

30 min.

##### Lorentz Transformation (Geometric)
In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
• Found in: Theoretical Mechanics course(s)

Lecture

120 min.

##### Phase transformations
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.
• Found in: Thermal and Statistical Physics course(s)

Problem

##### Vapor pressure equation
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).

• Found in: Thermal and Statistical Physics course(s)

Small Group Activity

60 min.

##### Linear Transformations
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.
• Found in: Quantum Fundamentals course(s) Found in: Matrices & Operators sequence(s)