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Wavefunction
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Wavefunction

In quantum mechanics the state of a system is represented by an object called a ket vector, which is an element of an abstract mathematical structure called a Hilbert space. For isolated systems, the dynamics (or time evolution) of the system can be described by a one-parameter family of unitary operators. In a wide class of systems this Hilbert space of ket vectors has one or more realizations as spaces of complex-valued functions on some space; in this case we refer to these functions as wavefunctions. Moreover, in some of these representations the time evolution of the system has the form of a partial differential equation, namely Schroedinger's equation.

Wavefunction representations

An 
orthonormal basis {ei}i in a Hilbert space H provides a representation of elements of H by finite or countable vectors of abstract Fourier coefficients
In fact, there is a far-reaching generalization of this representation, which gives an analogous representation with respect to what we could loosely call a continuously indexed orthonormal basis of a Hilbert space. In this representation, ket vectors are represented by functions on the continuous index set and the inner product of the Hilbert space corresponds to the integral of the product of two wavefunctions. In mathematical terms, such continuous orthonormal bases are referred to as diagonalizations, because mathematically they correspond to representing certain commutative algebras of operators as algebras of multiplication operators. The technical details of how this diagonalization is carried out is beyond the scope of this article, but it generalizes the result of linear algebra that a commutative algebra of operators closed under operator adjoint is diagonalized in some orthonormal basis.

Two common diagonalizations used in quantum mechanics are the configuration (position) space representation (which diagonalizes the position operators) and the momentum space representation (which diagonalizes the momentum operators). These are also called by physicists the 'r-space representation' and the 'k-space representation', respectively. Due to the commutation relationship of the position and momentum operators, for a system of spinless particles in euclidean space the r-space and k-space wavefunctions are Fourier transform pairs. The precise formulation of this last statement is rather subtle and is called the Stone-von Neumann theorem in the mathematical physics literature.

A more general diagonalization in which ket vectors are represented by Hilbert space valued functions on some space occurs naturally, for example, those which involve half-integer spin or systems in which the number of particles or quanta is variable, for example, most of nonlinear quantum optics or atom optics, and any treated by quantum electrodynamics or other quantized-field theories. This diagonal representation is usually called a direct integral of Hilbert spaces.

If the energy spectum of a system is (partly) discrete, such as for a particle in an infinite potential well or the bound states of the hydrogen atom, then the position representation is continuous while the momentum representation is partly discrete.  Wave mechanics are most often used when the number of particles is relatively small and knowledge of spatial configuration or 'shape' is important.  Additionally, the density-operator formalism is very awkward when expressed in the terms of wave mechanics. 
Because the wavefunction relative to the configuration representation has a (comparatively) simple interpretation as a probability in configuration space, many introductory treatments of quantum mechanics are very much wave mechanical.  Wave mechanics also dominated many of the more popular older standard textbooks, such as Messiah's Mecanique Quantique.  Hence the term wavefunction is sometimes used as a colloquialism for "state vector".  This use, however, is deprecated; not only are there systems which cannot be represented by wavefunctions, but the term wavefunction also leads to the belief that there is wave propagation in some medium.