Measure a quantity and then measure it again straight away: the second result repeats the first, because the first measurement left the system in the eigenvector it found. Measure something incompatible in between and the memory is wiped: the first quantity is random again. Three polarisers show it vividly — horizontal and vertical together block everything, but a diagonal one slipped between them lets light through.

Asking someone the same question twice gets the same answer. Asking an unrelated question in between should not matter — but in quantum mechanics it can, and the second answer to the first question becomes a coin toss.

The axes of three polarisers in a row. Horizontal followed by vertical passes nothing. With the diagonal one between them, a photon that passed the first passes the second with probability $\tfrac12$, is left diagonal, and passes the third with probability $\tfrac12$: a quarter get through.

Because a measurement leaves the system in the eigenvector of the result found, the probability of a sequence of results is a product: the probability of the first result, times the probability of the second given the state the first one left, and so on.

• Repeating a measurement immediately gives the same result, since $|\braket{a|a}|^{2}=1$.
• An incompatible measurement in between erases the first result: from $\ket{0}$, measuring $$ leaves , and is then with probability each.

The first measurement gives $a$ with the Born probability and leaves the state $\ket{a}$. The second measurement is then made on $\ket{a}$, so the Born rule applies again with $\ket{a}$ in place of $$. The chance of both is the product, as for any two stages in succession.

The Born rule for the first measurement.

The result $a$ leaves the eigenvector $\ket{a}$ behind.

The Born rule for the second measurement, made on $\ket{a}$.

The probability of both is the first times the second given the first.

Each measurement leaves the system in the state it found, so the probability of a sequence of results is the product of one Born probability per step.

A measurement leaves the eigenvector of its result, so repeating it at once gives the same result with certainty.

Measuring a quantity whose eigenvectors are not those of the first leaves one of its own eigenvectors. Measured again, the first quantity is uncertain once more.

Unpolarised light is a random mixture of polarisations. An ideal polariser passes half of it, and what emerges is polarised along the polariser’s axis.

Great work! You now know how to:

• multiply Born probabilities along a sequence of measurements
• explain why repeating a measurement gives the same result
• explain how an incompatible measurement erases an earlier one