Activity: Outer Product of a Vector on Itself

Quantum Fundamentals 2023 (2 years)
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 students learn
  • 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)
Outer Product of a Vector on Itself
  1. For one of the vectors below, what matrix is the outer product of the vector on itself (i.e., \(\left|{v_1}\right\rangle \left\langle {v_1}\right|\))? All the vectors are written in the \(S_z\) basis. \begin{align*} \left|{+}\right\rangle &\doteq \begin{bmatrix} 1\\0 \end{bmatrix} &\left|{-}\right\rangle &\doteq \begin{bmatrix} 0\\1 \end{bmatrix} &\left|{+}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\1 \end{bmatrix} \\[10pt] \left|{-}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\-1 \end{bmatrix} &\left|{+}\right\rangle _y &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\i \end{bmatrix} &\left|{-}\right\rangle _y &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\-i \end{bmatrix}\\[10pt] \left|{v_7}\right\rangle &\doteq \frac{1}{5}\begin{bmatrix} 3\\4 \end{bmatrix} &\left|{v_8}\right\rangle &\doteq \frac{1}{5}\begin{bmatrix} 4\\-3 \end{bmatrix} &\left|{v_9}\right\rangle &\doteq \begin{bmatrix} a\\be^{i\phi} \end{bmatrix}\\[10pt] \left|{1}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} \frac{1}{\sqrt{2}}\\1\\\frac{1}{\sqrt{2}} \end{bmatrix} &\left|{0}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\0\\-1 \end{bmatrix} &\left|{-1}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} \frac{1}{\sqrt{2}}\\-1\\\frac{1}{\sqrt{2}} \end{bmatrix} \end{align*}

  2. What is the transformation caused by your outer product?

  3. What is the determinant of your outer product?

  4. 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?

Prompt:

I recommend doing the activity as a Compare & Contrast. Make sure students who get the first two do a second one. Have groups record their results on a public table.

Student Conservations

  • Students may have trouble identifying the transformation.
    • For real vectors, have them plot the vectors on the \(\left|{\pm}\right\rangle \) axes.
    • For imaginary vectors, try factoring out a common factor from both components.
  • Students will be curious is the matrices are projections or scrinches/smooshes. You can point out that the scaling factor on their transformed vector is the inner product between their original vector and the untransformed vector: \((\left|{v_i}\right\rangle \left\langle {v_i}\right|)\left|{\psi}\right\rangle = \left\langle {v_i}\right|\psi\rangle \left|{v_i}\right\rangle = \left|{\psi'}\right\rangle \)

Activity: Wrap-up

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 Spin 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.

This activity works well as a follow-up to the Linear Transformations activity.
Learning Outcomes