Repeated Position Measurements

Question

I'm reading Griffith's Introduction to Quantum Mechanics and came across the extract below:

"What if I made a second measurement, immediately after the first? Would I get C again, or does the act of measurement cough up some completely new number each time? On this question everyone is in agreement: A repeated measurement (on the same particle) must return the same value. Indeed, it would be tough to prove that the particle was really found at C in the first instance, if this could not be confirmed by immediate repetition of the measurement."

So the text implies that a second position measurement on a particle will return the same value as the first. However if position is known, then the momentum (and hence velocity) should be uncertain according to the uncertainty principle, and therefore shouldn't the position of the particle have changed between the first and second measurement (due the particle having an uncertain and likely non-zero velocity)?

Answer 1

The key idea here is that, just after measurement, a quantum system is always in an eigenstate of the observable (operator) that represents the measurement. Therefore, if the same measurement is applied again straight away, before significant quantum state evolution governed by the Schrödinger equation has happenned, then the probability of observing the system in that eigenstate is unity, because the projection of the state onto the eigenstate has no effect, whereas there is zero projection onto any other eigenstate, because eigenstates of self adjoint operators (which observables all are) are orthogonal.

You seem worried that the uncertain momentum implies the particle has to move between measurements and indeed you are partially correct. But the assertion is being made with an assumption of negligible time between measurements. No, or negligible, quantum state evolution can happen in between the measurements to allow the assertion claimed.

It can be shown that the evolution projection of the position measurement onto the other position eigenstates after the measurement is quadratic in time, so the situation changes only relatively slowly. Normally projections change linearly with time over short timescales: this unusual situation in the position eigenstate arises because the projection onto the eigenstate corresponding to the measurement has reached a maximum, and therefore linear terms in the evolution equations vanish. This unusual quadratic dependence gives rise to the quantum Zeno effect, whereby one can keep a quantum system frozen in an eigenstate of an observable by repeatedly making the observation, often enough so that significant quantum state evolution has no time to happen before the state is "reset" back (almost certainly) to the nearest eigenstate.