Diatomic hydrogen

  • rigid rotor hamiltonian angular momentum ground state hydrogen diatomic probability
    • face Quantum Reference Sheet

      face Lecture

      5 min.

      Quantum Reference Sheet
      Central Forces 2021 (2 years)
    • assignment Find Force Law

      assignment Homework

      Find Force Law
      Central Forces 2021

      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.

    • assignment Spiral Orbit

      assignment Homework

      Spiral Orbit
      Central Forces 2021 A mass \(\mu\), under the influence of a central-force field, moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants. Determine the force law of this central-force field.
    • assignment Helix

      assignment Homework

      Helix

      Integration Sequence

      Static Fields 2022 (4 years)

      A helix with 17 turns has height \(H\) and radius \(R\). Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix. At the bottom of the helix the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the top of the helix, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the helix?

    • assignment Sphere in Cylindrical Coordinates

      assignment Homework

      Sphere in Cylindrical Coordinates
      Static Fields 2022 (3 years) Find the surface area of a sphere using cylindrical coordinates.
    • computer Effective Potentials

      computer Mathematica Activity

      30 min.

      Effective Potentials
      Central Forces 2021 Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
    • group Scalar Surface and Volume Elements

      group Small Group Activity

      30 min.

      Scalar Surface and Volume Elements
      Static Fields 2022 (4 years)

      Integration Sequence

      Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

      This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

    • face Phase transformations

      face Lecture

      120 min.

      Phase transformations
      Thermal and Statistical Physics 2020

      phase transformation Clausius-Clapeyron mean field theory thermodynamics

      These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
    • face Gibbs entropy approach

      face Lecture

      120 min.

      Gibbs entropy approach
      Thermal and Statistical Physics 2020

      Gibbs 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 Gibbs entropy is extensive

      assignment Homework

      Gibbs entropy is extensive
      Gibbs entropy Probability Thermal and Statistical Physics 2020

      Consider 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.

      1. 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.
      2. 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}

    1. 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.

    2. 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 \(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)

    3. At what temperature is the value of this ratio 1?

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