Wednesday, January 21, 2009
What and why is normalization of wave function?
Quantum mechanics treats moving matter as a wave, called a 'matter wave'. A matter wave is always assigned a wave function, usually called a 'psi'. The wave function is usually complex and composed of two parts - an amplitude and a phase. The square of the modulus of the wave function gives the probability density of finding the particle somewhere in space. If this quantity is integrated around the whole space, it gives the probability of finding the particle in that space. Since the particle should exist somewhere in space, this probability should be a maximum. To fulfill this requirement, the wave function is normalized in such a way that the total probability in the whole space of our consideration is 1 (maximum). The fixing of the coefficient of the wave function by requiring to fulfill the above condition is called normalization. Therefore, to get a normalized wave function - (1) Write a suitable wave function with some coefficient to represent the quantum system of your consideration. (2) Integrate the modulus square of the wave function over the whole space of your consideration and get the value of the coefficient in terms of some known quantities. (3) Plug the coefficient back into the wave function, which now is a normalized wave function. It is more convenient to work with a normalized wave function than with a non-normalized one !