Measurement and uncertainty

For an ideal projective measurement, the outcome $a$ corresponds to an orthogonal projector $P_a$. Its probability is $\langle\psi|P_a|\psi\rangle$. If that outcome is recorded, the conditional state is the projected component divided by its norm. This is the state used to predict subsequent measurements on that selected subensemble. The division is essential: projection alone generally produces a vector with norm less than one.

A degenerate outcome may contain several independent states. An ideal minimally disturbing projective measurement keeps the part inside that entire eigenspace; it need not select one arbitrarily chosen eigenvector. Real instruments can introduce additional disturbance, inefficiency, or unrecorded outcomes, requiring a more general measurement model. Neither the projector rule nor its laboratory use requires consciousness. The relevant ingredients are physical interactions, recorded alternatives, and which data the prediction is conditioned on.

The Robertson inequality bounds the product of standard deviations of two observables in the same preparation. For position and momentum it becomes $\Delta x\,\Delta p\ge\hbar/2$. Imagine preparing many equivalent systems and measuring position on one collection and momentum on another. Even ideal instruments cannot make both distributions arbitrarily narrow for that state. This statement is not merely a claim that clumsy equipment kicks the particle.

The more general bound contains the expectation of a commutator and can depend on the state. A zero right-hand side does not establish that both spreads vanish; it may simply make that particular lower bound uninformative. Standard deviations must also exist for the stated distributions. Energy–time relations require their own operational definitions because time is usually the evolution parameter here, not an operator interchangeable with position.

For a two-level system, define the $Z$ basis as $|0\rangle,|1\rangle$ and the $X$ basis as $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$. An ideal $Z$ measurement of $|0\rangle$ returns zero with certainty and leaves that state unchanged. A later identical ideal measurement remains certain if no intervening dynamics changes the preparation. Repeatability is thus conditional on both the measurement model and what happens between readings.

Insert an $X$ measurement between those two $Z$ measurements. The intermediate outcome is plus or minus with equal probability, and each corresponding state yields either $Z$ outcome with equal probability. Even when the intermediate result is ignored, the final zero probability falls to one half. This is a change in the physical measurement sequence, not a contradiction or a dependence on whether a person chooses to inspect the intermediate display.