Wave function
Index
Quantum mechanics introduced the concept of the duality of the particle and wave descriptions of matter. A particle with a momentum, p, has an associated wavelength, l, and wavenumber, k, such that: p = (h/2p)k , where h is Planck's constant. Also, l = 2p/k.

The quantum state of the particle is described by a Wave function, y(r, t), which depends on time, t, and position, r. The wave function is related to the probability, dP(r, t), of the particle being in an element of volume, d3r, viz: 
dP(r, t) = C [ y(r, t) ]2 d3r, where C is a constant that ensures that the integral of the probability over all space is unity.

In many systems it is the time independent electron distribution that is of interest. In this case the time independent wave function describes the system. For the hydrogen atom, the lowest electronic state has a wavefunctionof the form, y1S = (Z3/pa3)0.5 exp( -Zr/a), where a is the location of the maximum in the probability distribution.