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

An imaginary phase changes the state without changing direct measurement 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

An imaginary phase preserves direct probabilities

The phase gate turns the down branch into an imaginary branch. The direct probabilities remain 1/2 and 1/2.

P_{\text{zero}}=\frac{1}{2},\quad P_{\text{one}}=\frac{1}{2}

Both direct rows keep one half

The phase changes the branch relation, while the direct probability rows stay equal.

\begin{array}{c|c|c}\text{branch}&P_{\text{before}}&P_{\text{after}}\\\text{zero}&\frac{1}{2}&\frac{1}{2}\\\text{one}&\frac{1}{2}&\frac{1}{2}\\\text{total}&1&1\\\end{array}

The phase is stored for later

The diagram does not show a new direct chance. It shows a state that can behave differently when measured in another basis.

S\lvert +x\rangle = \lvert +y\rangle