Wavefunctions as Amplitudes

Normalization Over Space

all probabilities add to one

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=13∣L⟩+23∣M⟩+23∣R⟩\psi = \frac{1}{3}\lvert L\rangle + \frac{2}{3}\lvert M\rangle + \frac{2}{3}\lvert R\rangle
Three-bin wavefunctionThree bins make one normalized finite state.left1/31/9middle2/34/9right2/34/9

The middle and right bins carry equal probability

The middle and right amplitudes match, so their probability rows match too.

binaPleft1319middle2349right2349\begin{array}{c|c|c}\text{bin}&a&P\\\text{left}&\frac{1}{3}&\frac{1}{9}\\\text{middle}&\frac{2}{3}&\frac{4}{9}\\\text{right}&\frac{2}{3}&\frac{4}{9}\\\end{array}
Three-bin probability scanEach probability row is squared from its amplitude.left1/31/9middle2/34/9right2/34/9

The probability budget closes

The probabilities are 1/9, 4/9, and 4/9. They add to 1.

19+49+49=1\frac{1}{9}+\frac{4}{9}+\frac{4}{9}=1
Closed probability budgetAll displayed probabilities come from the amplitudes.left1/31/9middle2/34/9right2/34/9