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group Small Group Activity

5 min.

Leibniz Notation
This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.

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.

group Small Group Activity

30 min.

Expectation Values for a Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence

Quantum Ring Sequence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.

assignment Homework

Series Notation 2

Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

  1. \[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]

  2. \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]

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