[the concept of a Hilbert space]… offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections.
Wikipedia, Hilbert space, as seen in 2009 on April the 6th, but transformed to use links that'll work from here.
Quantum mechanics has enough weirdness to it that its formalisation requires a significantly richer structure than the intuitively tractable three-dimensional space of real displacements that serves us so well in the description of the macroscopic world we inhabit. The Hilbert space is a structure rich enough to support the full panoply of quantum complications, yet retains as much as can be hoped for of the intuitive tractability of our familiar real three-dimensional geometry. Let me start with a very brief over-view of all the technical jargon used in Wikipedia's account; then I can devote sections to the parts thereof. Along the way, I'll get the chance to restate orthodoxy in my preferred forms.
A Hilbert space is a continuum which is also a vector space
over the complex numbers, equipped
with the most sensible length-like notion a complex vector space can have, a
positive-definite hermitian product. This last can be
encoded as an invertible mapping, called the metric, from the Hilbert space to
its dual, which enables us to divide
any other hermitian product by the
metric; the result, known as a hermitian operator, is a linear map from the
Hilbert space to itself. The natural equivalent of an isometry
(length-preserving transformation) in this context is a unitary operator.
For any given hermitian operator, the Hilbert space can be decomposed into orthogonal sub-spaces on each of which the operator acts simply as a scaling; each such sub-space is described as an eigenspace of the operator, the associated scaling is known as the eigenvalue for that space and each non-zero vector in the space is termed an eigenvector of the operator. This eigenspace decomposition lets us write any vector in the space as a sum of eigenvectors; one can define an orthogonal projection onto any eigenspace as a mapping which decomposes its input vector in this way, discards eigenvectors not in the selected eigenspace and returns what remains.
In quantum mechanics, the Hilbert space generaly arises as a sub-space of a more general space of wave functions of a system, namely the span of those that are solutions of the dynamical equation – archetypically Schroedinger's equation – governing the system. Solutions of the system's dynamical equations are identified with unit vectors in the Hilbert space. Superpositions of possible solutions are represented by linear combination of their corresponding unit vectors, followed by re-scaling to obtain a unit vector to represent the superposition.
Each real-valued quantity one can measure (e.g. total energy, or a single
component of its momentum) on the system corresponds to a hermitian operator on
the Hilbert space. Actually measuring such a quantity forces the system into an
state represented by an eigenvector of the measured quantity's associated
operator – such a state is called an eigenstate
of the operator,
associated with the same eigenvalue as the eigenvector in question – and
yields the associated eigenvalue as measured value. The action of observing
such a measurement projects the prior state vector orthogonally onto the
eigenspace for the observed value; this selects the component of the prior
state's unit vector, when decomposed into a sum of eigenvectors, in the given
eigenspace. The probability of observing any given value is simply the squared
magnitude of this component. A positive real scaling can then be applied to
this coponent to make it a unit vector once more.
I shall deliberately gloss over the topological details that make a Hilbert
space behave relatively sanely, compared to infinite-dimensional vector spaces
in general; in calling it a continuum I tacitly assert that it
is complete
in the necessary sense, but all that actually matters is that
it behaves enough like a finite-dimensional vector space to be
intelligible.
Written by Eddy.