Intuitive

A Hilbert space is the natural arena of quantum mechanics. Each point in it represents one specific configuration a given system can be in.

Concrete

Recommended Resources

• The best introduction can be found in chapter 2 of Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers by G. Fano and S. M. Blinder
• See the nice explanations at page 257ff in "The Emperors new Mind" by R. Penrose

The most fundamental property of a Hilbert space is that it is what is called a vector space in fact, a complex vector space. This means that we are allowed to add together any two elements of the space and obtain another such element; and we are also allowed to perform these additions with complex-number weightings. We must be able to do this because these are the operations of quantum linear superposition that we have just been considering, namely the operations which give us $\Psi_t +\Psi_b$, $\Psi_t - \Psi_b$, $\Psi_t + i\Psi_b$ etc., for the photon above. Essentially, all that we mean by the use of the phrase 'complex vector space', then, is that we are allowed to form weighted sums of this kind.

page 257 in "The Emperors new Mind" by R. Penrose

Abstract

Why is it interesting?

Recall that in Chapter 5 the concept of phase space was introduced for the description of a classical system. A single point of phase space would be used to represent the (classical) state of an entire physical system. In the quantum theory, the appropriate analogous concept is that of a Hilbert space. A single point of Hilbert space now represents the quantum state of an entire system.

page 257 in "The Emperors new Mind" by R. Penrose