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Sequences

Sequence

Completeness Relations

This set of activities explores completeness relations in quantum mechanics and how they eventually support understanding of wavefunctions and what it means to be in the position or momentum representatations.

Activities

Small Group Activity

10 min.

Changing Spin Bases with a Completeness Relation
Students work in small groups to use completeness relations to change the basis of quantum states.

Small Group Activity

30 min.

Completeness Relations
Students use a completeness relations to write hydrogen atoms states in the energy and position bases.

Problem

5 min.

Matrix Elements and Completeness Relations
Students use completeness relations to write a matrix element of a spin component in a different basis.

Problem

30 min.

Completeness Relation Change of Basis
None

Problem

10 min.

Coefficients in Wavefunction notation
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.

Small Group Activity

30 min.

\(|\pm\rangle\) Forms an Orthonormal Basis
Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.

Lecture

30 min.

Compare & Contrast Kets & Wavefunctions
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

Problem

5 min.

Dimensional Analysis of Kets
Students consider the dimensions of spin-state kets and position-basis kets.

Small Group Activity

30 min.

Outer Product of a Vector on Itself
  • Outer products yield projection operators
  • Projection operators are idempotes (they square to themselves)
  • A complete set of outer products of an orthonormal basis is the identity (a completeness relation)