Hilbert-Space Quantum Mechanics

The Hilbert-space formalism distinguishes a physical pure state from any one vector used to represent it. This chapter focuses on the precise projective structure needed by symmetry theorems, while Core Formalism retains the first encounter with state vectors, global phase, and interference.

Enter the chapter

Complete the current probability on-ramp before entering the projective spine: State Vectors supplies normalized Hilbert-space representatives; Projectors and Probability Amplitudes prepare the Born Rule; and Transition Probabilities supplies the operational meaning of squared overlaps used below.

Physical States as Rays establishes three equivalent descriptions:

$$\text{complex ray} \quad\longleftrightarrow\quad \text{normalized vectors modulo }U(1) \quad\longleftrightarrow\quad \text{rank-one orthogonal projector}.$$

It then derives the intrinsic transition probability

$$\operatorname{Tr}(P_\psi P_\phi) = \frac{|\langle\psi|\phi\rangle|^2} {\|\psi\|^2\|\phi\|^2},$$

which is the invariant preserved in Wigner’s theorem.

What this chapter separates

• A normalized ket is a representative; the ray is the physical pure state.
• Global phase is quotient redundancy; relative phase can change the ray.
• Rank-one projectors are pure states; general density operators need not be.
• An entangled composite ray can induce a mixed subsystem state.
• Superselection can restrict physical pure states to sectorwise projective spaces.

After this page, continue to Transition-Probability Preserving Maps and then Wigner’s Theorem.

Further topics

Physical States as Rays supplies the projective-state foundation required by the symmetry sequence. The other Hilbert-space entries marked Planned identify articles still to be written; they do not supply additional prerequisite material.