Hilbert space
Originally, classical mechanics was described by Newton's three laws, and these were geometrically expressed in the three-dimensional Cartesian coordinate system.
Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system.
First Dirac, and then Von Neumann set quantum theory in an infinite-dimensional complex ray space called Hilbert space. In this space, a ray, or single direction, represents a quantum state, as does the wave function in Schrödinger's formulation.
Each dimension in Hilbert space represents a possible state for a quantum system, so an unmeasured electron exists as a very complicated pattern.
A particular state is selected by projecting the total quantum ray on to one of the basic rays. The set of basic rays forms a frame of reference to observe some property. Each basic ray represents a possible choice or result of a measurement. The resulting coordinates are quantum probability amplitudes that are points in a complex plane, one plane for each basic ray. Points in the complex plane are two real dimensional vectors with a magnitude and also a direction known as the relative phase. The relative phase is observable when we change the experiment and measure a property that is incompatible with the original experiment.
The wave function description in Hilbert space cannot by itself tell us which state will be selected for reality. The projection, of the total ray onto a basic ray, multiplied by its complex conjugate projection is the real number probability for the transformation of a quantum potentiality into a classical actuality. The different quantum potentialities, which coherently co-exist in quantum reality, are mutually exclusive in the classical reality of our consciousness. A particle cannot be in two places at the same time in classical reality, but it is so in quantum reality. This point is basic to the so-called measurement problem and to our understanding of what underlies our three-dimensional world.
The wave function is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which, give the probability distribution that the system will be in any of the possible states.
Quantum mechanics is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically-observable quantities and the state space represents the possible states of the system under study. Furthermore, each observable corresponds, to the classical idea of a degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators [math]\hat{x}[/math] and [math]\hat{p}[/math].
The momentum and position wave functions are Fourier transform pairs to within a factor of Planck's constant. Energy and time are another Fourier pair because a state which only exists for a short time cannot have a definite energy. In order to have a definite energy, the frequency of the state needs to be accurately defined, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. This was clear to many early founders such as Einstein and Niels Bohr because they saw that energy bears the same relation to time as momentum does to space in special relativity.
The probability amplitudes are complex numbers whose modulus squares represent probability or probability density. For example, the values taken by a normalised wave function ψ are amplitudes, since |ψ(x)|2 gives the probability density at position x. Probability amplitudes may also correspond to probabilities of discrete outcomes.
