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
- 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*}
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?
- Keywords
- Projection Operators Outer Products Matrices
- Learning Outcomes
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