Postulate 3

If we measure a Quantum state $\psi$ with result $\lambda \in \mathbb{R}$, Then immediately after the Measurement, the system will be in a state ${\psi }^{\prime }$ satisfying $\hat{f}{\psi }^{\prime }=\lambda {\psi }^{\prime }$ Where $\hat{f}$ is the Self-Adjoint Operator associated with $f$.

Properties

If a quantum system is described by a Quantum state $\psi \in H$, the probability distribution of the Classical observable $f$ is given by $\mathbb{E}[f^m]=⟨\psi ,(\hat{f}{)}^m\psi ⟩$

Where

• $f:{\mathbb{R}}^{2n}\to \mathbb{R}$ is a Classical observable
• $\hat{f}:{\mathbb{C}}^n\to {\mathbb{C}}^n$ is the corresponding Quantum observable
• $\psi$ is a unit vector

Although we are in a Quantum Hilbert Space, we still care about measuring Classical observables. Rather than having a deterministic Classical observable value, a quantum system state gives us a probability distribution for the classical counterpart.