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Results: time dependence

group Small Group Activity

30 min.

Time Evolution of a Spin-1/2 System

quantum mechanics spin precession time evolution

In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.

accessibility_new Kinesthetic

10 min.

Using Arms to Represent Time Dependence in Spin 1/2 Systems

arms quantum states time dependence

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

face Lecture

30 min.

Time Evolution Refresher (Mini-Lecture)

schrodinger equation time dependence stationary states

Quantum Ring Sequence

The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.

group Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

Quantum Ring Sequence

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.

computer Mathematica Activity

30 min.

Visualization of Quantum Probabilities for a Particle Confined to a Ring

central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

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.
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