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).
What is the transformation caused by your outer product?
What is the determinant of your outer product?
What is the square of your outer product?
Bonus: What happens when you add the outer products for a complete orthonormal basis?
Bonus 2: How would you answer questions (2) and (4) staying purely in Dirac bra-ket notation?
group Small Group Activity
60 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
energy conservation mass conservation collision
Groups are asked to analyze the following standard problem:
Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?
group Small Group Activity
30 min.
Cartesian Basis $S_z$ basis completeness normalization orthogonality basis
Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.face Lecture
120 min.
Gibbs entropy information theory probability statistical mechanics
These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.group Small Group Activity
10 min.
group Small Group Activity
30 min.
keyboard Computational Activity
120 min.
inner product wave function quantum mechanics particle in a box
Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.assignment Homework