normalization

Normalization is the act of scaling a wavefunction so that the total probability of finding the particle somewhere comes out to exactly one. Since |ψ|² is the probability density, adding it up over all of space must give one: the particle is certainly somewhere. If a raw wavefunction integrates to some other number, you simply divide ψ by the square root of that number, and the probabilities then fit neatly between zero and one.

The need for normalization comes straight from the meaning of probability. A density that summed to two would claim the particle is found 'twice over'; one that summed to a half would lose half the particle. Only the value one is honest. Because |ψ|² is what carries physical meaning, multiplying ψ by any overall constant of magnitude one — a pure phase — changes nothing measurable; normalization fixes the magnitude but leaves that harmless freedom alone.

Not every solution of the equations can be normalized. A wavefunction that does not fall off at large distances may enclose infinite total probability, and then it cannot describe a single localized particle. Such non-normalizable functions, like a perfectly sharp plane wave, are still useful as idealizations and building blocks, but a genuine physical state must be normalizable — its probability cloud must add up to one.