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 \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}
Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
- Given a vector written in the polar basis \[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \] where \(a\) and \(b\) are known. Find coefficients \(c\) and \(d\) such that \[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \] Do this by using the completeness relation: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
- Using a completeness relation, change the basis of the spin-1/2 state \[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \] into the \(S_y\) basis. In otherwords, find \(j\) and \(k\) such that \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]
Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
- (4pts) Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
- (4pts) Find a formula for the charge density that creates this electric field.
- (2pts) Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
Students work in small groups to use completeness relations to change the basis of quantum states.
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 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 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.