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wave function[′wāv ‚fəŋk·shən]
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