Amplitude Over Positions - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

A finite wavefunction assigns amplitudes to positions before it gives probabilities. 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 wavefunction assigns amplitudes

The left bin has amplitude 3/5 and the right bin has amplitude 4/5. The amplitudes, not the bars, are the state.

\psi = \frac{3}{5}\lvert L\rangle + \frac{4}{5}\lvert R\rangle

Each amplitude row squares into a probability row

The scan keeps the amplitude value and its squared probability in the same row.

\begin{array}{c|c|c}\text{bin}&P&\text{source}\\\text{left}&\frac{9}{25}&\text{squared amplitude}\\\text{right}&\frac{16}{25}&\text{squared amplitude}\\\text{total}&1&\text{probability budget}\\\end{array}

Probabilities come from squared amplitudes

Squaring gives probabilities 9/25 and 16/25.

P_L = \frac{9}{25},\quad P_R = \frac{16}{25}