IndeterminaxQuantum Mechanics / Uncertainty Principle

Indeterminax has a question mark for a head. Fair: some pairs of questions have no simultaneous answers.

Heisenberg's 1927 paper argued from a thought experiment: to see an electron sharply you must hit it with short-wavelength light, and the light kicks it. That picture of clumsy disturbance is what most people learn. It is not the principle physicists now use. Two years later Howard Percy Robertson proved the general statement on this page, and it says nothing about clumsiness.

Take two quantities, A and B. Prepare many copies of a system in the same state, measure A on some copies and B on others, and record the spread of each result, \(\sigma_A\) and \(\sigma_B\). Robertson showed that their product can never be less than half the size of the average of the commutator \([\hat A,\hat B]\), which measures how much doing A then B differs from doing B then A. For position and momentum the commutator is \(i\hbar\), a constant, so the bound is \(\hbar/2\) whatever the state. That is the familiar case.

Now take two components of an electron's spin, along x and along y. Their commutator is not a constant; it is proportional to the third component, along z. So the bound becomes \(\hbar/2\) times the average spin along z, and that depends on the state. Prepare the spin pointing along z and the trade-off is strict: neither x nor y can be sharp. Prepare it pointing along x and the bound falls to zero, and indeed the x spin is then perfectly definite. Uncertainty is not a universal fog. It belongs to the questions, and to the state.

Quantities whose commutator is zero, such as the positions of two different particles, have no trade-off at all.

Indeterminax flickers only when asked two incompatible things at once.