Wave Function - Normalization Invariance

Normalization Invariance

It is important that the properties associated with the wave function are invariant under normalization. If normalization of a wave function changed the properties, the process becomes pointless as we still cannot yield any information about the particle associated with the non-normalized wave function.

All properties of the particle, such as momentum, energy, expectation value of position, associated probability distributions etc., are solved from the Schrödinger equation (or other relativistic wave equations). The Schrödinger equation is a linear differential equation, so if Ψ is normalized and becomes AΨ (A is the normalization constant), then the equation reads:

which is the original Schrödinger equation. That is to say, the Schrödinger equation is invariant under normalization, and consequently associated properties are unchanged.