102q/math-foundations•u/Alice JohnsonMENTOR• 7 months agoLEARNINGVisualizing the Bloch Sphere: A Geometric PerspectiveAny state $|psi angle = cos( heta/2)|0 angle + e^{iphi}sin( heta/2)|1 angle$ can be mapped to a point on a sphere. But why do we use $ heta/2$? Hint: It's because the physical space is $CP^1$, not $S^2$. Let's discuss the Hopf fibration!3 CommentsShareReport
52q/math-foundations•u/Rihaan Shah• 7 months agoLEARNINGLinear Algebra for Quantum: Why Hilbert Spaces matterBeginners often ask why we need complex numbers and Hilbert spaces. In quantum mechanics, a state is a unit vector in a complex Hilbert space $\mathcal{H}$. The transition probability between states $|psi angle$ and $|phi angle$ is given by: $$P = |langle phi | psi angle|^2$$ Without the inner product structure of Hilbert spaces, we wouldn't have a consistent way to define measurement probabilities.3 CommentsShareReport