Phase Can Hide - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

A phase can change the state while leaving direct squared-magnitude probabilities unchanged, so the ledger separates amplitude information from what one basis immediately shows. The direct-probability equality is exact only for the stated complex amplitudes and measurement basis; real interference tests add phase drift, decoherence, alignment error, and readout uncertainty.

A complex phase may not change a direct measurement even though it changes later interference. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Phase can hide from a direct measurement

The down amplitude carries an imaginary phase, but the direct basis probabilities are still 9/25 and 16/25.

P_{\uparrow}=\frac{9}{25},\quad P_{\downarrow}=\frac{16}{25}

Direct rows ignore the phase signpost but not the magnitude

The direct measurement reads the squared magnitudes, so the imaginary branch keeps the same probability row.

\begin{array}{c|c|c}\text{branch}&P&\text{phase role}\\\text{up}&\frac{9}{25}&\text{real branch}\\\text{down}&\frac{16}{25}&\text{imaginary branch}\\\text{total}&1&\text{direct budget}\\\end{array}

Phase matters when amplitudes recombine

The phase is not visible as a new direct probability. It matters when a later gate or basis choice recombines amplitudes.

\text{phase changes later interference}