Wavefunction

Vectors in a vector space are most often expressed with respect to some basis--a designated set of vectors that "span" the space, that is, from which any other vector in the space may be constructed by taking a weighted sum. When this basis is indexed by a discrete (finite or countable) set, the vector representation is just the column of numbers familiar to high-school students. Vector bases may also be continuously indexed. When a quantum mechanical state vector is represented with respect to such a continuous basis it is called a wavefunction.

Extending the vector treatment, one can easily define a continuous-basis inner product, namely, the so-called overlap integral, or integral of the product of two wavefunctions. Functions for which this product is well-defined are said (just like discrete vectors) to form an Hilbert space. Using this product we can perform quantum mechanical calculations just as we do for more abstract vectors. The dual or adjoint vector is given by the complex conjugate. Under this treatment, the interpretation of the absolute value of the square of the wavefuction as a probability density is a direct and clear consequence of the postulates of quantum mechanics, and the 'wave mechanics' in which observables are given as linear differential operators falls right out.

Most often, the vectors making up the basis correspond to precise positions or momenta and are not realizable quantum states. The two most common continuous bases are the configuration (position) space basis and the momentum space basis, sometimes called by physicists the 'r-space basis' and the 'k-space basis', respectively. Due to the commutation relationship of the position and momentum operators, the r-space and k-space wavefunctions are Fourier transform pairs.

If the energy spectum of a system is discrete, such as for the particle in an infinitely deep box or the bound states of the hydrogen atom, there can actually be both continuous and discrete bases for the same system, and thus either wave mechanics or matrix mechanics may be used to study the system. Wave mechanics are most often used when the number of particles is relatively small and knowledge of spatial configuration or 'shape' is important. Many quantum-mechanical systems cannot be described at all using the wave mechanics, for example, those which involve half-integer spin or systems in which the number of particles or quanta change with respect to time, for example, most of nonlinear quantum optics or atom optics, and any treated by quantum electrodynamics or other quantized-field theories. Additionally, the density-operator formalism is very awkward when expressed in the terms of wave mechanics.

Because of the concrete relationship between the wavefunction and location of a particle in configuration space, many treatments of quantum mechanics at the high-school or early undergraduate level are very 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 some medium is waving in the mechanical sense.