Quantum Mechanics: Postulates
States as Hilbert-space vectors, observables as Hermitian operators.
- States $|\psi\rangle$ are unit vectors in a complex Hilbert space.
- Observables are Hermitian operators $\hat A = \hat A^\dagger$.
- Born rule: $P(a_n) = |\langle n|\psi\rangle|^2$.
- Collapse onto the measured eigenspace.
- Unitary evolution $i\hbar\partial_t |\psi\rangle = \hat H |\psi\rangle$.
The canonical commutator $[\hat x, \hat p] = i\hbar$ implies $\Delta x\, \Delta p \geq \hbar/2$.
Interactive: free Gaussian wavepacket
$|\psi(x,t)|^2$ for a free particle — initially Gaussian, spreads with time, and translates with group velocity $p_0/m$.