Activities
Problem
The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
- Find the total differential of \(G\). As always, show your work.
- Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
- From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.
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Students work in small groups to use completeness relations to change the basis of quantum states.
Students use a completeness relations to write hydrogen atoms states in the energy and position bases.
Students use completeness relations to write a matrix element of a spin component in a different basis.
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
- Outer products yield projection operators
- Projection operators are idempotes (they square to themselves)
- A complete set of outer products of an orthonormal basis is the identity (a completeness relation)
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
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.
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.