Wave Function
A wave function in a quantum physics is a mathematical description of the quantum state of isolated quantum system. The wave function is a complex valued probability amplitude and the probability for the possible results of measurements made on the system can be derived from it. The most common symbols for wave function are the Greek letter 'Ψ'. It is represented by-
`\psi_{(r,t)}=<r/\psi_{(t)}>`
Ψ(r, t)is called wave function associated with moving particle. The wave function corresponds to physical system.
It must be satisfied following systems-
(i) Ψ(r, t) are physically acceptable.
(ii) coordinate derivative dΨ/dx must be finite continuous and single valued.
(iii) Square of its norm has meaning. According to born probability interpretation square of the norm represent a position probability density.
`P_{(r,t)}=\|\psi_{(r,t)}\|^2`
i.e. |Ψ(r, t)|2 d3r represent the probability of finding the particle at time "t" is the volume element d3r therefore the total probability of finding the system somewhere in space is = 1
`\int_{-\infty}^\infty\|\psi_{(r,t)}\|^2d^3r=1`
wave function Ψ(r, t) which satisfy this relation is called normalized wave function.
(iv) wave function must be square integrable.
Physical importance of the Wave Function
According to this view Ψ*Ψ = |Ψ|2 represents probability density of the particle in the state Ψ. Then the probability of finding the particle in a volume element dτ = dx dy dz about any point r(x, y, z) at time t is expressed as
`P_{(r,t)}d\tau=\|\psi_{(r,t)}\|^2d\tau`
The wave function Ψ is sometimes called probability amplitude for the position of the particle. The postulate suggested by Born show that the quantum mechanical laws and the results of their measurements can be interpreted on the basis of probability considerations.