# Wave Function

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## wave function

[′wāv ‚fəŋk·shən]
(quantum mechanics)

## Wave Function

in quantum mechanics, a quantity that completely describes the state of a microscopic object (for example, an electron, proton, atom, or molecule) and of any quantum system (for example, a crystal) in general.

A description of the state of a microscopic object by means of the wave function is statistical, or probabilistic, in character: the square of the absolute value (modulus) of a wave function indicates the probability of those quantities on which the wave function depends. For example, if the dependence of the wave function of a particle on the coordinates x, y, and z and on time t is given, then the square of the absolute value of this wave function defines the probability of finding the particle at time t at a point with coordinates jc, y, z. Insofar as the probability of the state is defined by the square of the wave function, the latter is also called the amplitude of probability.

At the same time, a wave function also reflects the presence of wave characteristics in microscopic objects. Thus, for a free particle with given momentum p and energy δ. to which a de Broglie wave with a frequency v = δ/h and a wavelength λ = h/p (where h is Planck’s constant) is compared, the wave function must be periodic in space and time, with the corresponding value of X and a period T = l/v.

The superposition principle is valid for wave functions. If a system may be found in various states with wave functions ψ1, ψ22, .… , then a state with a wave function equal to the sum—and in general, to any linear combination—of these wave functions is also possible. The addition of wave functions (amplitudes of probability), but not of probabilities (the squares of wave functions), fundamentally distinguishes quantum theory from any classical statistical theory in which the theorem of the addition of probabilities is valid.

The properties of the symmetry of wave functions, which define the statistics of the aggregate of particles, are essential to systems consisting of many identical microparticles.

V. I. GRIGOR’EV

References in periodicals archive ?
The Canadian scientists claim that their new method of probing molecular structure has yielded the most revealing image so far of an electron's wave function.
Harmonic spectra obtained at 19 different angles were analyzed and the 3D shape of the molecule's wave function was derived.
In Dirac's bracket notation, the upper line defines a matrix element specifying quantum numbers k, l, m of amplitude functions of two particular combining states in the bra and ket quantities, one being the ground electronic state; the integral in the lower line expresses a general calculation directly in terms of the wave functions associated with any two states.
Solutions of the Maxwell Equations and Photon Wave Functions.
Since the developments of the Faddeev (33) and the Faddeev-Yakubovsky (34) methods to solve the 3- and the 4-body Schrodinger equations, respectively, in the form of integral equations, they have been successfully applied in many regions of few body physics because of their nature to give rigorous wave functions under assumed two-body interactions.
By using the method of separation of variables for the vector wave functions in the spheroidal coordinate system, Asano and Yamamoto studied the scattering of a linearly polarized plane electromagnetic wave by a homogeneous isotropic spheroid with any size and refractive index [1,2].
Formulae (11), (12) are not difficult for computations due to the program developed in [15] for Coulomb wave functions.
According to the Iran Nanotechnology Initiative Council, the researchers derived electron and hole wave functions in a finite potential well of nanometric structures analytically by Schrodinger equation.
Mathematical appendices cover wave functions and complex numbers, Brag diffraction, and Fourier transforms and Fourier holograms.
The q-analogs of boson operators have been studied by various authors and the corresponding wave functions were constructed in terms of the continuous q-Hermite polynomials of Rogers [42]-[44] by Atakishiyev and Suslov [15] and by Floreanini and Vinet [29]; in terms of the Stieltjes-Wigert polynomials [46], [60] by Atakishiyev and Suslov [17]; and in terms of q-Charlier polynomials of Al-Salam and Carlitz [3] by Askey and Suslov [12], [13] and by Zhedanov [61].
A1] between the partial wave functions can be superimposed.
The wave functions provide exact values of energy transfers but only hint at how transfers happen.

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