Quantum logistic map considered as discrete-time Heisenberg equation

We represent scalar logistic iterates as multiplication operators on $L^2([0,1])$ and study their fixed matrix elements in the normalized shifted-Legendre basis. Three parameter regimes permit analytical control. At $r=5/2$, every fixed matrix element converges to $3\delta_{kl}/5$, where $\delta_{kl}$ is the Kronecker delta. At $r=16/5$, the attracting period-two orbit yields phase-resolved limits for the even and odd subsequences. At $r=4$, an exact Chebyshev-moment representation gives an $O(4^{-n})$ approach of every fixed matrix element at iteration $n$ to $\delta_{kl}/2$. The case $r=37/10$ is treated only by a controlled finite numerical refinement study. Complementary finite diagnostics comprise matrix-element time dependence, a bifurcation-style plot, a time-averaged mean intensity, a normalized second-order intensity moment, and normalized scalar OTOC-type commutator correlation matrices. We also examine the separate finite-dimensional recursion $X_{k+1}=R X_k(I-X_k)R^\dagger$, with fixed matrix $R$, using diagonal and tridiagonal amplitude profiles. These operator-valued calculations are exploratory finite-time numerics. The analytical statements concern fixed matrix elements for the specified basis indices; they are distinct from the finite-resolution observations and do not imply operator-norm convergence. A regularized phase-space lift is included as a controlled visualization.