1. SWBQ: Compute the outer product of the general quantum state vector $$\vert\psi \rangle\,=\, \left(\begin{array}{c} sin\left(\frac{\theta}{2}\right)\\ cos\left(\frac{\theta}{2}\right)e^{i\phi}\\ \end{array}\right) \; \; .$$
2. Students find that the outer product of any quantum state vector with itself is Hermitian.
3. SWBQ: Take the determinant of the 2×2 matrix yielded from the outer product.
4. Students find that in any case, the determinant of the outer product of $\vert\psi \rangle$ with itself is zero and the trace of the outer product is one.

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