Postulates and Dynamical Pictures

Enumerate the postulates of quantum mechanics as a unified axiomatic framework and identify how each postulate constrains the theory's structure.

Apply the Born rule to compute measurement probabilities and describe the projection postulate for sequential measurements.

Derive the generalized uncertainty relation for two arbitrary observables from the Cauchy-Schwarz inequality.

Derive the time-evolution operator from the Schrödinger equation and establish its unitarity.

Distinguish between the Schrödinger, Heisenberg, and interaction pictures and transform operators and states between them.

Derive the time-energy uncertainty relation and explain its distinct interpretation compared to the position-momentum relation.