Does the collapse axiom predict non-physical states in the case of measurement of continuous-spectrum quantities?

Question

Let us assume a continuous wave function.

Suppose we make a measurement of the position of the particle and obtain as a result the interval $(x_0-\delta, x_0+\delta)$. $\delta>0$ stays for the accuracy of the instrument.

If I understand correctly, following the measurement, the collapse axiom (Luders-von Neumann's axiom) predicts that the new wave function (up to normalization) is the same as the previous function within the interval and is zero outside the interval (See for example Valter Moretti's answer https://physics.stackexchange.com/q/247521).

So the wave function is discontinuous at the extremes of the interval, at the two points $x_0-\delta$ and $x_0+\delta$.

The discontinuous functions have the trouble that the expectation value of the kinetic energy is infinite (See for example Luboš Motl's answer https://physics.stackexchange.com/q/38183), so it is a non-physical state.

What is the origin of this apparent contradiction? Does the collapse axiom predict non-physical states in the case of continuous-spectrum measurement? Is it possible to predict a valid shape of the wave function after the collapse?

Answer 1

Since my name directly appears in the question, I think that I should try to answer. Also because I substantially changed some ideas about some of these issues in the last years.

First of all, the right approach when considering the measurement problem is the one of "quantum measurement theory" based on POVMs (Positive Operator Valued Measures), Kraus decompositions and, more generally, the notion of quantum measurement scheme.

An important theoretical reference is the book by P. Busch and collaborators "Quantum Measurement".

When dealing with observables -- in the general sense either described in terms of POVMs or PVMs (Projection-Valued- Measures, i.e. selfadjoint operators)-- which attain discrete values, then the Lueders-vonNeumann postulate projection can be assumed among a plethora of other possibilities however..

Instead, when the values are continuous, the projection postulate is physically untenable for many reasons (failure of additivity of the post measurement state, Ozawa theorem,...).

In this case, but also in the general case, the post measurement state is described by a suitable measurement scheme.

There are infinitely many measurement schemes for a given observable (also if described by a PVM, i.e. a selfadjoint operator). In general, the post measurement state turns out to be mixed even if the premeasurement state is pure.

So, assigning an observable in terms of a POVM or a selfadjoint operator (PVM) says nothing about the postmeasurement state associated to an outcome and a premeasurement state.

If we make discrete a continuous observable by brute force, using approximations, then there are processes which agrees with the projection postulate state.

However, even making discrete some observables like the non-relativistic position, e.e.g, by using a sequence of contiguous instruments each capable to measure a segment $I_n:= (x_n-\delta, x_n+\delta]$, it is difficult to assert that $\psi_n= \chi_{I_n}(x) \psi(x)$ is the post measurement state if $I_n$ is the outcome for the premeasuerment state $\psi$.

That is because $\psi_n$ has infinite energy or suffers from some similar pathologies. There are better descriptions of measurements schemes for the position observable which take advantage of probe particles with their own wavefuntions which, in turn, enter in the description of the post measurement state.

A similar measurement procedure already appears in von Neumann's book about mathematical foundations of QM.

Answer 2

Suppose you have an electron interacting with a Geiger counter. There is a description of what is happening in terms of some quantum field theory such as QED. Those equations describe what is happening and make predictions about correlations among observables and that sort of thing. I looked in the index of the following five textbooks on QFT and collapse didn't appear in the index in any of them:

"A modern introduction to quantum field theory" by Maggiore

"Quantum field theory for the gifted amateur" by Lancaster and Blundell

"Quantum field theory in a nutshell" by Zee

"Quantum theory of fields, Volume I: Foundations" by Weinberg

"The conceptual framework of quantum field theory" by Duncan

Collapse doesn't appear in any of these books because it is incompatible with the equations of motion of quantum theory. It appears only in books that abstract away from those equations of motion. A real measurement is an interaction that takes place according to those equations of motion and produces a record of some attribute of a system. When you describe a measurement according to the actual laws of physics that have survived testing in reality what happens is that having information copied out of a record suppresses interference between some selected set of roughly orthonormal states - decoherence:

https://arxiv.org/abs/1911.06282

You don't measure a function that cuts of sharply at the ends of an interval because that is incompatible with real physics. In some situations collapse might be an adequate approximation for the purposes of making a prediction but many real measurements can't be treated that way:

https://arxiv.org/abs/1604.05973

If you want to predict and understand what will happen in a real measurement according to quantum theory you have to use quantum theory not unphysical rules of thumb like collapse.