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).
1. << Dimensional Analysis of Kets | Completeness Relations |
Outer Product of a Vector on Itself
Your group will be given a pair (or triple) of vectors below, find the matrix that is the outer product of each 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*} 1)\qquad \left|{+}\right\rangle &\doteq \begin{bmatrix} 1\\0 \end{bmatrix} &\left|{-}\right\rangle &\doteq \begin{bmatrix} 0\\1 \end{bmatrix} \\[10pt] 2)\qquad\left|{+}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\1 \end{bmatrix} &\left|{-}\right\rangle _x &\doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1\\-1 \end{bmatrix}\\[10pt] 3)\qquad\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] 4)\qquad\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}\\[10pt] 5)\qquad\left|{v_9}\right\rangle &\doteq \begin{bmatrix} a\\be^{i\phi} \end{bmatrix} &\left|{v_{10}}\right\rangle &\doteq \begin{bmatrix} b\\-ae^{i\phi} \end{bmatrix}\\[10pt] 6)\qquad\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*}
What is the square of each of your outer products?
What is the product of each pair of your outer products?
For each row of vectors, add all of the outer products.
What is the determinant of each of your outer products?
What is the transformation caused by each of your outer products?
Bonus: How would you answer questions (2), (3), (4) staying purely in Dirac bra-ket notation?
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:
If students have done the https://paradigms.oregonstate.edu/act/2221, 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.