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Projection Operators: Instructor's Guide

Main Ideas

  • Outer products are projection operators
  • Projection operators have determinant zero
  • Projection operators are idempotes - they square to themselves

Students' Task

Estimated Time: 10 minutes

  • Find the outer product of a vector on itself
  • Determine the linear transformation created by the outer product
  • Dind the determinant of the outer product
  • Dind the square of an outer product

Prerequisite Knowledge

  • Matrix Multiplication
  • Complex number arithmetic
  • How to take a determinant


Activity: Introduction

We've been talking about inner products as an operation that you can do with kets. We interpret the result of an inner product as being related to probabilities and as expansion coefficients when writing a vector.

$\bra{+}\psi\rangle \rightarrow$ scalar

$\ket{\psi} = \bra{+}{\psi}\rangle \ket{+} + \bra{-}{\psi}\rangle \ket{-}$

Probabilitity($S_z = \hbar/2$) = $|\bra{+}{\psi}\rangle|^2$

An outer product is another kind of operation we can do with vectors.

$\ket{\psi}\bra{\psi} \rightarrow$ operator/square matrix

Activity: Student Conversations

  • Students may have trouble identifying the transformation.
    • For real vectors, have them plot the vectors on the $\ket{\pm}$ axes.
    • For imaginary vectors, try factoring out a common factor from both components.
  • Students will be curious is the matrices are projections or scrinches. You can point out that the scaling factor on their transformed vector is the inner product between their original vector and the untransformed vector:

$(\ket{v_i}\bra{v_i})\ket{\psi} = \bra{v_i}\psi\rangle \ket{v_i} = \ket{\psi'}$

Activity: Wrap-up

Estimate Time: 30 min. This activity works well if different groups are assigned different vectors and the different results are reported at the end. Wrap-up should emphasize that:

  • an outer product of two vectors produces a matrix
  • an outer product of a vector $|a\rangle$ on itself produces an operator that projects vectors onto the line with the same slope as the $|a\rangle$.
  • the determinant of a projection operator is zero
  • A projection operator squares to itself

If students have done the SPINS Lab 1 , the facilitator can point out that a projection (and renormalization) operation is consistent with the transformation that occurs when a Stern-Gerlach measurement is made.


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