Hilbert Space

Prerequisites: Linear Algebra.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

see also Configuration Space and Phase space

Intuitive

Start with ordinary arrows in space. We can add two arrows, stretch an arrow, measure its length, and measure the angle between two arrows. Every arrow can be built from three basic arrows pointing along the axes, and its three components say how much of each it contains.

A Hilbert space keeps all of these operations but allows many more directions, possibly infinitely many, and allows the components to be complex numbers. The "arrows" can then be functions. A wave function of an electron is one such arrow, and the basic directions can be, for example, the energy levels of an atom. The component of the wave function along one energy level says how much of that level the state contains.

Quantum mechanics needs exactly this structure. Adding arrows is the superposition principle: if two states are possible, so is any combination of them. The angle between two arrows tells us how likely it is to find a system prepared in one state when we test for the other. Two arrows at right angles describe states that can be distinguished with certainty. Stretching an arrow by a number does not change the physical state, so each state of the system corresponds to a direction in the Hilbert space rather than to a single arrow.

A Hilbert space also has a technical property called completeness: if we approach a state by a sequence of better and better approximations, the limit is again in the space. This is what makes it safe to expand states in infinitely many basis functions, as in a Fourier series.

A Hilbert space is the natural arena of quantum mechanics. Each nonzero vector in it, up to an overall complex factor, 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

• No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, chapter "Mathematical Arenas"

Abstract

A Hilbert space $\mathcal{H}$ is a complex vector space with an inner product $\langle \cdot, \cdot \rangle$ (in physics, antilinear in the first and linear in the second argument, with $\langle \psi, \psi \rangle > 0$ for $\psi \neq 0$) that is complete in the norm $\|\psi\| = \sqrt{\langle \psi, \psi\rangle}$: every Cauchy sequence converges. Standard examples are $\mathbb{C}^n$, the sequence space $\ell^2(\mathbb{N})$ of square-summable sequences, and $L^2(\mathbb{R}^3)$, the square-integrable functions modulo functions that vanish almost everywhere, with $\langle \psi, \phi \rangle = \int \overline{\psi(x)}\, \phi(x)\, d^3x$.

Every Hilbert space has an orthonormal basis $\{e_i\}$, and every vector has the expansion $\psi = \sum_i \langle e_i, \psi\rangle e_i$, with $\|\psi\|^2 = \sum_i |\langle e_i, \psi\rangle|^2$. Hilbert spaces are classified up to unitary isomorphism by the cardinality of such a basis. In particular every infinite-dimensional separable Hilbert space is unitarily isomorphic to $\ell^2(\mathbb{N})$, and $L^2(\mathbb{R}^3)$ is separable. This is the mathematical content behind the equivalence of Heisenberg's matrix mechanics and Schrödinger's wave mechanics. The Riesz representation theorem identifies $\mathcal{H}$ antilinearly with its continuous dual by $\phi \mapsto \langle \phi, \cdot \rangle$, which is what makes the bra-ket notation consistent for normalizable states.

Quantum mechanics. Pure states are rays, that is, points of the projective space $P(\mathcal{H}) = (\mathcal{H} \setminus \{0\})/\mathbb{C}^\times$. The transition probability between states $[\psi]$ and $[\phi]$ is $|\langle \psi, \phi\rangle|^2/(\|\psi\|^2 \|\phi\|^2)$. Wigner's theorem states that every bijection of $P(\mathcal{H})$ that preserves transition probabilities is induced by a unitary or an antiunitary operator on $\mathcal{H}$, unique up to a phase. Observables are self-adjoint operators $A = A^\dagger$, which in general are unbounded and only defined on a dense domain $D(A) \subset \mathcal{H}$ (position and momentum on $L^2(\mathbb{R})$ cannot both be bounded, since $[x, p] = i\hbar$ has no solution in bounded operators). The spectral theorem writes each self-adjoint operator as $A = \int \lambda\, dP(\lambda)$ with a projection-valued measure $P$, and the Born rule gives the probability of a measurement result in a set $B$ as $\langle \psi, P(B) \psi\rangle$ for normalized $\psi$. Composite systems are described by the tensor product $\mathcal{H}_1 \otimes \mathcal{H}_2$, and systems of identical particles by the symmetric or antisymmetric subspaces of the Fock space $\bigoplus_n \mathcal{H}^{\otimes n}$.

Position and momentum eigenstates $|x\rangle$, $|p\rangle$ are not elements of $L^2$. They are made rigorous in a rigged Hilbert space (Gelfand triple) $\mathcal{S}(\mathbb{R}) \subset L^2(\mathbb{R}) \subset \mathcal{S}'(\mathbb{R})$, where they are tempered distributions such as $\delta(x - x_0)$ and $e^{ipx/\hbar}$.

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

Leads to: Axiomatic Quantum Field Theory, Canonical Quantum Mechanics, Dirac Notation, Koopman-von-Neumann Mechanics, Stone-von Neumann Theorem.