Hilbert space

A Hilbert space is the mathematical home for quantum states. The beginner mental model is: a quantum state is a vector, and the Hilbert space is the vector space containing all allowed states of the system.

For a spin-1/2 system, the Hilbert space is two-dimensional. For a particle moving along a line, the state can be represented by a Wavefunction, so the space is infinite-dimensional.

Quantum notation often writes states as kets:

$$|\psi ⟩=a|0⟩+b|1⟩.$$

The inner product gives amplitudes and probabilities. A normalised state obeys

$$⟨\psi |\psi ⟩=1.$$

Why it matters

• Superposition is ordinary vector addition, with physical interpretation.
• Observables are operators acting on the space.
• Orthogonal states can be perfectly distinguished; non-orthogonal states cannot.
• Entanglement lives in tensor-product Hilbert spaces for composite systems.

Common pitfalls

• The vector itself is not a little arrow in ordinary space.
• Overall phase does not change measurement probabilities, but relative phase does.
• Normalisation matters: probabilities must add or integrate to one.

Maths bridge: Vector space. Source trail: Susskind The Theoretical Minimum index; reference book: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman.