Wavefunction Normalization Scan - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

Normalization becomes easier to see when several exact rows square into closed budgets. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One normalized pair is a full probability budget

The first row uses amplitudes 3/5 and 4/5. The diagram squares them into two probability bars that close at one.

\frac{9}{25}+\frac{16}{25}=1

Three rows show which number changes

The scan repeats the same rule with two more exact pairs. Each row changes amplitudes, then checks the squared probabilities and the total.

\begin{array}{c|c|c|c}\text{amps}&P_L&P_R&\text{sum}\\\frac{3}{5},\frac{4}{5}&\frac{9}{25}&\frac{16}{25}&1\\\frac{12}{13},\frac{5}{13}&\frac{144}{169}&\frac{25}{169}&1\\\frac{5}{13},\frac{12}{13}&\frac{25}{169}&\frac{144}{169}&1\\\end{array}

Swapping amplitudes swaps probabilities

The third row swaps the two amplitudes. Nothing else is hidden: the left and right probability bars swap because the squared amplitudes swap.

\left(\frac{5}{13}\right)^2=\frac{25}{169},\quad \left(\frac{12}{13}\right)^2=\frac{144}{169}