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.

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 the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.

Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

Students consider the dimensions of spin-state kets and position-basis kets.

The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}

1. Find the total differential of \(G\). As always, show your work.
2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).

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).

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.