Is wave function measurable?

Question

I apologize for the length of this naive question. I am not sure it is appropriate for this community.

Is wave function measurable?

This is really a question in Atomic and Molecular Optics.

I hear some people say it is not.

There seems to be some experiments that measure at least some magnitude and phase information about the wave function, see refs below.

First, narrow down only to wave functions of electrons, not wave functions of atoms.

I think the following experimental situations have to be clearly distinguished.

1. Measurement of "free" electron, as in an electron beam or in geonium-like experiments ( e.g. in Prof. Gabrielse's experiments on precise electron g-2 measurements) It is not actually free because there is a macroscopic field that controls it. But it is not bound to an atom.

2. Measurement of electron in a solid ( we will not consider this situation any further)

3. Measurement of electron wave function in a bound state i.e. electron in a multi electron atom ( we will not consider molecules here)

4. If the wave function we attempt to measure is the ground state or excited state. It is not clear what should be the wave function of excited state as we know that the state will decay and will become entangled with the gamma. We therefore must consider higher Fock state with a non factorizable mixed wave function component. I believe we can still talk of the electron wave function, but it will make sense only with precision $\alpha$, where $\alpha$ is the fine structure constant.

The answer may also depend on which transitions from this state are possible - dipole, quadrupole,...

5. Are we measuring wave function of individual electron in the atom or of the whole electron system ( i.e. if we measure only its projection to one electron space of the tensor product of Hilbert spaces). If we are allowed to destroy the state of the rest of electrons if we attempt to measure only one of them.

6. Are we allowed to destroy the atom in the measurement, e.g. ionize it, and reconstruct the wave function from the ionization amplitudes.

6.1) Are we attempting a non-demolition measurement, i.e., we require that we do not disturb the atomic state.

7. Are we only using electromagnetic probes or we can also use electron or even atomic beams. It is very hard to make these beams, especially with energies below ionization threshold, but lets consider thought experiments here.

8. If external field is present, such as in Coherent Population trapping, and we are actually attempting to measure the wave function of a dark state.

9. We can use attosecond pulses. These pulses are very strong, and they actually tear away the electron, only later ( if tuned accurately) returning it to the atom (like in Keldysh mechanism). But I believe it is possible to design such a wave packet shape that the electron returns to the same state with probability 1, even if the electron was in an excited state ( this seems a difficult problem, I do not know any calculation that does this; it also does not mean that it is possible to realize this experimentally at this stage).

It seems that the situations above correspond to different experimental setups. In particular, in electromagnetic measurements we can focus on photons only, while in ionization and in electron scattering measurements we also have to consider detection of electrons.

Mathematically, we measure \begin{equation} |\int d^3\vec{x} \psi_i(t,\vec{x}) A_i(\vec{x},\xi)|^2 \end{equation} for some integral kernel $A$. There is no reason for $\psi$ being not reconstructible, up to overall phase, if $\xi$ varies across large enough space.

It should be noted that wave function seems to be an artefact of Dirac's interaction representation. At least at the level of partition function, it is possible to integrate out photon field, leaving non local electron interaction. (Likewise it is possible to integrate out electron field, leaving non local photon interaction, and see e.g. hydrogen spectrum in the singularities of the resulting 4-particle photon amplitudes). But it is not clear if this is true for bound state calculations, especially beyond 1loop.

Refs:

@article{weinacht1998measurement, title={Measurement of the amplitude and phase of a sculpted Rydberg wave packet}, author={Weinacht, TC and Ahn, Jaewook and Bucksbaum, PH}, journal={Physical Review Letters}, volume={80}, number={25}, pages={5508}, year={1998}, publisher={APS} }

@article{weinacht1999controlling, title={Controlling the shape of a quantum wavefunction}, author={Weinacht, TC and Ahn, Jaewook and Bucksbaum, Phil H}, journal={Nature}, volume={397}, number={6716}, pages={233--235}, year={1999}, publisher={Nature Publishing Group UK London} }

@article{nakajima2022high, title={High-resolution attosecond imaging of an atomic electron wave function in momentum space}, author={Nakajima, Takashi and Shinoda, Tasuku and Villeneuve, DM and Niikura, Hiromichi}, journal={Physical Review A}, volume={106}, number={6}, pages={063513}, year={2022}, publisher={APS} }

@article{goulielmakis2010real, title={Real-time observation of valence electron motion}, author={Goulielmakis, Eleftherios and Loh, Zhi-Heng and Wirth, Adrian and Santra, Robin and Rohringer, Nina and Yakovlev, Vladislav S and Zherebtsov, Sergey and Pfeifer, Thomas and Azzeer, Abdallah M and Kling, Matthias F and others}, journal={Nature}, volume={466}, number={7307}, pages={739--743}, year={2010}, publisher={Nature Publishing Group UK London} }