## 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.
##### 1. Activity: $|\pm\rangle$ Forms an Orthonormal Basis
Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.
##### 2. Activity: Outer Product of a Vector on Itself
Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
##### 3. Activity: Changing Spin Bases with a Completeness Relation
Students work in small groups to use completeness relations to change the basis of quantum states.
##### 4. Homework Problem: Completeness Relation Change of Basis
Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.
##### 5. Homework Problem: Matrix Elements and Completeness Relations
Students use completeness relations to write a matrix element of a spin component in a different basis.
##### 6. Activity: Going from Spin States to Wavefunctions
Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.
##### 7. Activity: Compare \& Contrast Kets \& Wavefunctions
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
##### 8. Homework Problem: Dimensional Analysis of Kets
Students consider the dimensions of spin-state kets and position-basis kets.