Mathematical foundations of quantum mechanics

A qubit is in the state $\psi = \frac35 e_0 + \frac{4i}{5} e_1$. Measuring $\sigma_z = \operatorname{diag}(1, -1)$, what is the probability of the outcome $-1$?

A three-level system has an observable $A$ with orthonormal eigenvectors $e_1, e_2, e_3$ and eigenvalues $1, 2, 6$. In the state $\psi = (e_1 + e_2 + e_3)/\sqrt 3$, what is the expectation $\langle \psi, A\psi \rangle$?

In the state $e_0$ (spin up along $z$), compare $\sigma_{\sigma_x} \, \sigma_{\sigma_y}$ with $\frac12 |\langle [\sigma_x, \sigma_y] \rangle|$, using $[\sigma_x, \sigma_y] = 2i\sigma_z$.

On $L^2(0, 1)$ let $P_0 = -i\,d/dx$ with domain $\{\psi \in H^1(0, 1) : \psi(0) = \psi(1) = 0\}$. Which statement is true?

The proof of the Hellinger–Toeplitz theorem shows that the graph of an everywhere-defined symmetric operator $A$ is closed. Which theorem then gives that $A$ is bounded?

With $[a, a^\dagger] = 1$, $a h_0 = 0$ and $\|h_0\| = 1$, compute $\|(a^\dagger)^3 h_0\|^2$.

Which Lean statement correctly formalises no $n \times n$ complex matrices satisfy $XP - PX = c \cdot 1$ with $c \ne 0$ (and is provable)?

In Wielandt's proof that $XP - PX = cI$ ($c \ne 0$) is impossible for bounded operators, one needs $P^n \ne 0$ for every $n$. Why does no power of $P$ vanish?

Which normalised states $\psi \in L^2(\R)$ satisfy $\sigma_X \sigma_P = \hbar/2$ exactly? Select all that apply.

Write expectation(A, psi) returning $\langle \psi, A\psi \rangle = \sum_{i,j} \overline{\psi_i} A_{ij} \psi_j$ for a square matrix A (list of rows) and a vector psi (list of complex numbers), and std_dev(A, psi) returning $\sigma_A = \sqrt{\langle A^2 \rangle - \langle A \rangle^2}$ for a normalised psi and Hermitian A. The tests check Robertson's inequality for Pauli matrices.