face Lecture

10 min.

Introduction to Central Forces
Central Forces 2023

assignment Homework

Diatomic hydrogen
rigid rotor hamiltonian angular momentum ground state hydrogen diatomic probability 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 \(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)

group Small Group Activity

10 min.

Matrix Representation of Angular Momentum
Central Forces 2023 (2 years)

group Small Group Activity

30 min.

Energy and Angular Momentum for a Quantum Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.