Superposition of position eigenstates after measurement of position?

Question

In a discussion with someone recently, the person made the following statement:

Even when we observe a particle, it is still in superposition of position states within the resolution of the measurement

Now, according to what I've been taught, this is not true - a measurement will always yield a specific eigenstate of the system (eigenvector of the relevant self-adjoint operator corresponding to the observable), which the system will then be in. Yes, there will be some limit to the precision of the measurement, but this is not the same as the system still being in a superposition of position states.

It seems to me that this is obvious if we consider the conceptually simpler case of e.g. a Stern-Gerlach experiment, where we can unambiguously measure the particles as being in the spin up or down state (because we have a discrete spectrum of states).

I didn't want to disagree with the person because they are an expert on quantum information/compuation. Am I missing something here? Is the quote above true in some sense that I don't understand?

The broader context is that I took issue with the person stating that 'particles can be in two positions at the same time'. I take this to be an unfortunate and imprecise use of language - the notion seems semantically vacuous to me, unless we take it to be a trivial synonym for 'being in a linear combination of two position eigenstates at the same time'.

Answer 1

Describing a measurement as the application of an operator is an idealised case and real measurements are not this simple. For example measuring the energy of a state precisely would take an infinite time. Any real measurement taking a finite time always returns a superposition of energy eigenstates, though in practice it generally takes only picoseconds to achieve sufficient accuracy for most experiments.

The position eigenfunctions are somewhat pathological since they can't be normalised and have an infinite uncertainty in the momentum and therefore an infinite energy. No measurement can ever return a position eigenstate because they don't exist in nature. Any measurement can only ever return a superposition of position eigenfunctions, though we can make the measurement arbitrarily accurate by using enough energy.

Finally I agree with your comments on:

particles can be in two positions at the same time

but language like this is often used when describing the situation to beginners who may not have a grasp of linear algebra. In that context we can forgive it.

Answer 2

You seem to assume that a physical measurement of position results in projecting the state vector (ket) into eigenvector of position operator. This is possible for finite-dimensional description, like for spin states or for position states on a countable grid of points.

But for position that is continuous, the position operator has no eigenfunctions. So, the above simple modelling of the result of the process of measurement on the state vector is not possible.

One way to resolve this problem is to notice that any measurement of continuous variable is imprecise, there is some errorbar in the result. Correspondingly, one can introduce a $\psi$ function that is consistent with the result of the measurement but accounts also for the fact that the result is not certain. Thus, the effect of measurement of position should collapse the psi function into smaller region, corresponding to the errorbar.

Another way is to say the position is actually discrete, and then there are eigenfunctions. This may be attractive for computer simulations, but has obvious problem with special relativity, so it is not very popular.

What happens to the state vector as result of measurement is, in fact, a highly contested topic. According to some people (orthodox interpretation), a projection occurs, according to others, measurement can be described via Schroedinger equation too but in insanely big Hilbert space that describes the whole world, according to most practitioners, this is unresolved mystery that can be ignored since the theory has lots of uses that do not seem to require resolving it.

Answer 3

John Rennie’s answer applies to discrete eigenstates as well since they are also idealizations. There is no manifest physical existence of a mathematically ideal direction “up,” or any other direction that is a real-numbered angle. A spin state with a real-numbered angle would be a zero entropy state, which is strictly forbidden by the Nernst statement of the third law of thermodynamics, and would also violate the Bekenstein bound. So all discrete and continuous eigenstates must actualize in real physical experiments (and more generally, physical manifestations) as narrow distributions of states, ie, superpositions. All physically existing states, even at the end of any measurement, are superpositions.