- rigid rotor hamiltonian angular momentum ground state hydrogen diatomic probability
*assignment*Carbon monoxide poisoning*assignment*Homework##### Carbon monoxide poisoning

Equilibrium Absorbtion Thermal and Statistical Physics 2020In carbon monoxide poisoning the CO replaces the \(\textsf{O}_{2}\) adsorbed on hemoglobin (\(\text{Hb}\)) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or may be occupied either with energy \(\varepsilon_A\) by one molecule \(\textsf{O}_{2}\) or with energy \(\varepsilon_B\) by one molecule CO. Let \(N\) fixed heme sites be in equilibrium with \(\textsf{O}_{2}\) and CO in the gas phases at concentrations such that the activities are \(\lambda(\text{O}_2) = 1\times 10^{-5}\) and \(\lambda(\text{CO}) = 1\times 10^{-7}\), all at body temperature \(37^\circ\text{C}\). Neglect any spin multiplicity factors.

First consider the system in the absence of CO. Evaluate \(\varepsilon_A\) such that 90 percent of the \(\text{Hb}\) sites are occupied by \(\textsf{O}_{2}\). Express the answer in eV per \(\textsf{O}_{2}\).

Now admit the CO under the specified conditions. Fine \(\varepsilon_B\) such that only 10% of the Hb sites are occupied by \(\textsf{O}_{2}\).

*face*Quantum Reference Sheet*computer*Visualization of Quantum Probabilities for the Hydrogen Atom*computer*Mathematica Activity30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom

Central Forces 2023 (3 years) Students use*Mathematica*to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).*computer*Visualizing Combinations of Spherical Harmonics*computer*Mathematica Activity30 min.

##### Visualizing Combinations of Spherical Harmonics

Central Forces 2023 (3 years) Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.*group*Hydrogen Probabilities in Matrix Notation*group*Small Group Activity30 min.

##### Hydrogen Probabilities in Matrix Notation

Central Forces 2023 (2 years)*group*Matrix Representation of Angular Momentum*group*Small Group Activity10 min.

##### Matrix Representation of Angular Momentum

Central Forces 2023 (2 years)*assignment*Find Force Law: Spiral Orbit*assignment*Homework##### Find Force Law: Spiral Orbit

Central Forces 2023 (3 years)In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a central-force field that allows a particle to move in a spiral orbit given by \(r=k\phi^2\), where \(k\) is a constant.

*face*Gibbs entropy approach*face*Lecture120 min.

##### Gibbs entropy approach

Thermal and Statistical Physics 2020Gibbs entropy information theory probability statistical mechanics

These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.*assignment*Find Force Law: Logarithmic Spiral Orbit*assignment*Homework##### Find Force Law: Logarithmic Spiral Orbit

Central Forces 2023 (3 years)In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a mass \(\mu\), under the influence of a central-force field, that moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants.

*assignment*Gibbs entropy is extensive*assignment*Homework##### Gibbs entropy is extensive

Gibbs entropy Probability Thermal and Statistical Physics 2020Consider two

*noninteracting*systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.- Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
- Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

##### Note

In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.-
Energy and Entropy 2021 (2 years)
At low temperatures, a diatomic molecule can be well described as a

*rigid rotor*. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}What is the energy of the ground state and the first and second excited states of the \(H_2\) molecule? i.e. the lowest three distinct energy eigenvalues.

At room temperature, what is the relative probability of finding a hydrogen molecule in the \(\ell=0\) state versus finding it in any one of the \(\ell=1\) states?

i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)At what temperature is the value of this ratio 1?

- At room temperature, what is the probability of
finding a hydrogen molecule in any one of the \(\ell=2\) states versus
that of finding it in the ground state?

i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)