Why normalization in quantum mechanics?

Answer 1

Take a wavefunction; call it psi.

Take another wavefunction; call it psi two.

These wavefunctions mus clearly both satisfy some sort of wave equation (say the Schrodinger Wave Equation 1926).

It turns out (if you do some maths) that if you addthese wavefunctions, psi+psiTwo is also a solution of the wave equation.

HOWEVER: SINCE THE SQUARE OF THE WAVE EQUATION IS THE PROBABILITY, THE TOTAL PROBABLILITY OF FINDING THIS PARTICLE ANYWHERE IN THE UNIVERSE IS NOW 1+1 = 2!!!!! How can the probability be two? It clearly can't. And so the new wave function has to be halved (normalisation) to give: 1/2 (psi+psiTwo) which satisfies this condition that the total probablility of finding the particle must be equal to one.

This condition is called the "Normalisation Condition" and is written mathematically thus:

Integral( psi^2 ) d(x^3) = 1.

Answer 2

Normalization in quantum mechanics is important because it ensures that the wavefunction describing the state of a system has a well-defined probability interpretation. The wavefunction must be normalized, meaning that the integral of the squared magnitude of the wavefunction over all space is equal to 1. This allows us to interpret the square of the wavefunction as the probability density of finding the particle in a particular state.