Normalization is the check that all position-bin probabilities share one closed budget. The sum to one is exact only for the stated amplitudes and complete finite basis; laboratory estimates depend on calibration, binning choices, missed probability outside the window, noise subtraction, and counting uncertainty.
A wavefunction can spread over several positions while keeping one probability budget. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A state can spread over more bins
This finite model has amplitudes 1/3, 2/3, and 2/3 across three positions.
ψ=31∣L⟩+32∣M⟩+32∣R⟩
The middle and right bins carry equal probability
The middle and right amplitudes match, so their probability rows match too.
binleftmiddlerighta313232P919494
The probability budget closes
The probabilities are 1/9, 4/9, and 4/9. They add to 1.