Home / Topics / Quantum Mechanics
Normalisation
Advanced
Total probability is one.
∫∣ψ∣²dx = 1Probabilities must add to one: ∫|ψ|²dx = 1. Multiply ψ by a constant to normalise it.
ψ = Ae^{−x²/2} needs A = π^{−1/4}.
Non-normalisable states like plane waves need special treatment.
In more detail
Normalisation requires the total probability of finding a particle somewhere to equal one, so the integral of |ψ|² over all space is one. It fixes the overall scale of the wavefunction.
Real-world uses
- Quantum calculations always begin by normalising wavefunctions.
- Numerical methods normalise states to keep probabilities consistent.
- Physicists normalise wavefunctions so that the total probability of finding a particle somewhere is one.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
Why must it equal one?
Because the particle must be found somewhere, so the total probability is one.
Can any wavefunction be normalised?
Only those that are square-integrable, meaning their total probability is finite.
Why must the integral be one?
The particle must be found somewhere, so the total probability over all space must add up to one.
Related topics in Quantum Mechanics
Energy–time uncertainty · Wavefunction · Born rule · Schrödinger equation (time-dependent) · Schrödinger equation (time-independent) · Operators and observables