*group*Raising and Lowering Operators for Spin*group*Small Group Activity60 min.

##### Raising and Lowering Operators for Spin

Central Forces 2023 (2 years)*face*Quantum Reference Sheet*group*Applying the equipartition theorem*group*Small Group Activity30 min.

##### Applying the equipartition theorem

Contemporary Challenges 2021 (4 years) Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature \(T\).*keyboard*Position operator*keyboard*Computational Activity120 min.

##### Position operator

Computational Physics Lab II 2022quantum mechanics operator matrix element particle in a box eigenfunction

Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.*face*Angular Momentum Commutation Relations: Lecture*assignment*Magnetic susceptibility*assignment*Homework##### Magnetic susceptibility

Paramagnet Magnetic susceptibility Thermal and Statistical Physics 2020Consider a paramagnet, which is a material with \(n\) spins per unit volume each of which may each be either “up” or “down”. The spins have energy \(\pm mB\) where \(m\) is the magnetic dipole moment of a single spin, and there is no interaction between spins. The magnetization \(M\) is defined as the total magnetic moment divided by the total volume.

*Hint:*each individual spin may be treated as a two-state system, which you have already worked with above.Find the Helmholtz free energy of a paramagnetic system (assume \(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of only the ratio \(x\equiv \frac{mB}{kT}\).

Use the canonical ensemble (i.e. partition function and probabilities) to find an exact expression for the total magentization \(M\) (which is the total dipole moment per unit volume) and the susceptibility \begin{align} \chi\equiv\left(\frac{\partial M}{\partial B}\right)_T \end{align} as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization is \begin{align} M=nm\tanh\left(\frac{mB}{kT}\right) \end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.

Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the limit \(mB\ll kT\).

*group*Hydrogen Probabilities in Matrix Notation*group*Small Group Activity30 min.

##### Hydrogen Probabilities in Matrix Notation

Central Forces 2023 (2 years)*assignment*Isothermal/Adiabatic Compressibility*assignment*Homework##### Isothermal/Adiabatic Compressibility

Energy and Entropy 2021 (2 years)The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} \(K_T\) is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

*assignment*Bottle in a Bottle 2*assignment*Homework##### Bottle in a Bottle 2

heat entropy ideal gas Energy and Entropy 2021 (2 years)Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.The volume of the small bottle is 0.001 m

^{23}and the volume of the big bottle is 0.01 m^{3}. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

- Discuss your results.

*computer*Blackbody PhET*computer*Computer Simulation30 min.

##### Blackbody PhET

Contemporary Challenges 2021 (4 years) Students use a PhET to explore properties of the Planck distribution.-
Central Forces 2023 (3 years)
You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length
\(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D
box, then you just
*multiply*together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)- Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L_x\) in the \(x\)-direction and length \(L_y\) in the \(y\)-direction.
- Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a

**double**sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

- Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.