Complex Phase and Spin

Spin Basis Choice

different questions, different probabilities

Spin measurement depends on which basis asks the question, so the same state can split in one basis and be definite in another. The probability rows are exact only for the stated ideal state and basis choice; laboratory measurements add preparation error, analyzer misalignment, decoherence, calibration drift, and finite-sample uncertainty.

Spin measurement depends on the basis used to ask the question. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The same state can be asked a z-basis question

For the plus x state, a z-basis measurement gives 1/2 and 1/2.

P↑=12,P↓=12P_{\uparrow}=\frac{1}{2},\quad P_{\downarrow}=\frac{1}{2}
Spin z-basis questionThe z-basis bars are computed from the state.state1/sqrt(2) up1/sqrt(2) downz basis1/2up1/2down

The basis choice changes the probability rows

The z-basis row splits evenly; the x-basis row is definite for the same state.

basisplusminustotalz12121x101\begin{array}{c|c|c|c}\text{basis}&\text{plus}&\text{minus}&\text{total}\\\text{z}&\frac{1}{2}&\frac{1}{2}&1\\\text{x}&1&0&1\\\end{array}
Spin-basis scanThe same state is read with a different basis.state1/sqrt(2) up1/sqrt(2) downx basis1plusx0minusx

The x-basis question can be definite

The x-basis probabilities are 1 and 0. A basis choice is part of the measurement.

P+x=1,P−x=0P_{+x}=1,\quad P_{-x}=0
Spin x-basis questionThe x-basis bars are computed from the same state.state1/sqrt(2) up1/sqrt(2) downx basis1plusx0minusx