Review of Fortin, S., Lombardi, O. Is the problem of molecular structure just the quantum measurement problem? Found Chem 23, 379–395 (2021). https://doi.org/10.1007/s10698-021-09402-x

Summary

The authors (FL) examine the question of whether or not the problem of molecular structure is solved if the quantum measurement problem were to be solved. Essentially, this paper is largely a review and comments on an earlier paper [1] by Franklin and Seifert (FS) that we have previously reviewed. FL argue against the claims made by FS solution of the measurement problem is not sufficient to account for the particular geometrical features of the molecule. In Sec. 2, the problem of quantum measurement is reexamined. The authors state, “the quantum measurement problem consists in explaining why the pointer acquires a definite value even if the post experimental state is a superposition.” The three problems as presented in FS’s paper are then considered. In Sec. 3 Hund’s paradox of chirality is examined, in Sec. 4 isomers and the inability of resultant Hamiltonians to determine molecular structure (Section 4), and the question of symmetry breaking is considered in Sec. 5.

Main Results

FL note that the quantum measurement problem defined by Mauldin includes the problem of definite outcomes as opposed to the Schrödinger prediction of entanglement. However, an important second problem they state “is as relevant as the first one, and is logically previous. The problem of the preferred basis derives from the fact that the expansion of the post-experimental state is in general not unique and, therefore, neither the measured observable nor the apparatus’ pointer are uniquely defined.” This additional preferred basis problem is not directly addressed by Maudlin’s definition of the QMP in [2].
FL note that the problem of chirality would be solved by SL if the measurement process provides a rationale to explain why states such as (|L>+|R>)/Sqrt(2) reduce to one of the two definite outcomes |L> or |R>.
FL in Sec. 5 consider the problem of symmetry breaking. Let us denote the molecule of ammonia where the nitrogen is at the top of the hydrogen as |N+> and |N-> when the nitrogen is found at the bottom of hydrogen. Two of the eigenstates of the configuration Hamiltonian are given by superpositions i.e. (|N+> +|N->)/Sqrt(2) and (|N+> -|N->)/Sqrt(2). Similar to the case of chirality, FL agree that, “Therefore, also in this case, solving the measurement problem explains the breaking of parity symmetry.”
In the case of why certain isomeric structures are obtained in the laboratory as opposed to the superposition of all the states composing the ground state, FL make several points:
- FL argue, similar to as in [3,4] that the very general Coulombic Hamiltonian or the resultant Hamiltonian does not fix the positions of the nuclei. The derivation of the eigenstates in molecular theory utilizes the Born-Oppenheimer approximation which fixes the positions. However, if one starts with the resultant Hamiltonian, particular configurations of molecules such as the particular isomers observed, do not result.
- If one is to believe the resolution of the QMP also offers some usefulness towards a solution to molecular structure of isomers, they argue that the problem of the preferred basis must also be taken into account. Since Hilbert space has an infinite basis, and the ground state can be expressed as a superposition of the members of any of them then why does the measurement outcome corresponds to the basis constituted by the observed isomers and not to any other basis? If one is to believe the QMP as a potential solution to molecular structure of isomers, FL argue that the ground state should contain superpositions of all possible nuclear configurations. They then state, “The problem is that this is not the case: given the number and the types of the elements composing a molecule, in general only few nuclear configurations are experimentally measured, that is, only few isomers are considered effectively ‘real’. But, from the viewpoint of quantum mechanics the question is: Why certain isomeric structures are obtained in the laboratory, and not all those necessary to reconstruct the symmetry of the ground state?” Here FL are saying that some of the isomers that should compose a superposition are not experimentally seen, only a few are seen.
- It is argued that, “The stability of molecules that are not in the ground state of the resultant Hamiltonian cannot be easily explained exclusively in quantum terms: this is a central aspect of the problem of molecular structure that is not related with the quantum measurement problem, at least in Maudlin’s version.” That is, they are saying that if the eigenstate of the Hamiltonian were a superposition, and the resolution of the QMP were to project the superposition to one of the many terms, this term would not be an eigenstate and hence would be non-stationary. Hence it should not be stable, but rather oscillate. Yet FL argue that isomers observed are in-fact stable.
FL argue that the quantum measurement problem consists in explaining why a macroscopic pointer acquires a definite value even if the post experimental state is a superposition, However, they claim that this does not mean that the pointer projected statedefines the values of all the microscopic observables of the measuring apparatus. In fact, in practice the apparatus is a macroscopic system, with a huge number of quantum degrees of freedom represented in his Hilbert space, so the pointer cannot define a basis of measurement. In short, they quote Omnes 1994 that “the pointer is a collective observable that is highly degenerate and introduces a “coarse-graining”. As a consequence, the definite value of P does not imply the definite-valuedness of all the microscopic observables of the apparatus.” This has several consequences regarding the issue of molecular structure:
- Although quantum measurement might explain the projection of a superposition of chiral molecules (|L>+|R>)/Sqrt(2) to one of |L> or |R>, FL believe that a resolution of the QMP still cannot explain the details of how precise molecular structures occur in isomers. They state, “Broadly speaking, there are many different ways to being a chiral molecule: many different geometrical configurations of the nuclei can realize, say, a left-handed molecule. This means that solving the measurement problem involved in Hund’s paradox amounts to explaining the chiral nature of the molecule, but is still far from solving the problem of the molecular structure as a whole.”
- Similarly, when one considers the resultant Hamiltonian of a number of atoms, there is no clear path to yield particular specific isomers. The ground state wavefunction of the resultant Hamiltonian is claimed, “corresponds to a superposition of all the different isomers.” If this is the case, then the resolution of the QMP could provide a measurement projection process under an assumed set of isomers that result from the BO approximation, but it still does not explain the particular set of atomic positions starting from the resultant Hamiltonian.
- In regard to the issue of symmetry breaking, FL state “But, as in the case of optical isomerism, in the Hilbert space Mcorresponding to the molecule is also a highly degenerate ‘collective observable’, whose definite value ‒explained by solving the quantum measurement problem‒ have no relevance regarding the values of the nuclei’s positions. Therefore, also in this case, solving the measurement problem explains the breaking of parity symmetry, but does not offer a complete solution of the problem of the molecular structure.”
Hence for all three problems of the detailed molecular structure encountered in the molecular problems of definite chirality, precise configuration of isomers observed, and in the case of the precise molecular structures encountered in the breaking of parity symmetry, FL claim that the solution of the QMP cannot offer a complete solution to the detailed molecular structure questions. On the other hand, if a priori a given set of configurations of particular molecular structures that are either chiral molecules, isomers, or molecules of definite parity, FL agree that the reduction from a superposition, of the a priori given molecules, to a single molecule, can be solved by resolution of the QMP.

Strengths

The authors have brought to attention important resolved problems in molecular structure.
A number of statements and issues with the paper by FS are been brought up and discussed in this paper.
The authors have included the problem of basis determination to the problem of definite outcomes that were utilized in the paper by FS.
The authors indicate that based on a macroscopic model of measurement, the resolution of the QMP will not be able to resolve the detailed problem of molecular structure.
- The major conclusion of this paper is that the detailed molecular structure that is found in problems of chirality, isomers, and symmetry breaking cannot be related through resolution of the QMP. This is based mainly on several observations, but mainly rests on the following:
  - That the detailed molecular structure with specific positions and configuration does not come about from the resultant Hamiltonian, but rather by making the BO approximation and using the configuration Hamiltonian.
  - The act of coupling a macroscopic pointer to a quantum system in the quantum measurement process introduces coarse graining for which the pointer device can act on the net dipole of a molecule, but this does not explain the specific configuration of the molecule which does not follow from resultant Hamiltonian.

Weaknesses

The conclusions that FL reach are based on assumptions that may or may not be correct. Because of this, the conclusions appear to be logically falsifiable without more justification.
- While it is true that the detailed molecular structure with specific positions and configuration does not come about from the resultant Hamiltonian, but rather by making the BO approximation and using the configuration Hamiltonian, this does not logically preclude a resolution of the QMP from providing additional constraints, imposed under the process of measurement, to resolving this latter problem.
- The measurement process is described as an act of coupling a macroscopic pointer to a quantum system in the quantum measurement process. While this model certainly represents Bohr’s and the Copenhagen interpretation of measurement, it is not logically a necessary model of measurement. The measurement problem and the word “entangled” was originally conceived by Schrödinger (Einstein also provided a letter to Schrödinger):
  - When two systems of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described the same way as before, viz., by endowing each of them with a representative state of its own. I would not call that one but the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or psi-functions) have become entangled
    - Note that in the original measurement problem as explained by Schrödinger, it isn’t specified where the measurement occurs nor the mechanism of the measurement. One is only left with the understanding that after interaction, quantum systems become entangled and therefore at some point after the interaction, perhaps still in the microscopic regime, perhaps in the mesoscopic regime, or perhaps in ultimately in the macroscopic regime, the entangled predicted state does not correspond to reality where product states are observed.
    - The Copenhagen interpretation takes the further stance that the interaction occurs between microscopic quantum phenomenon and macroscopic devices. However, this may or may not be true. Nobody knows at this time. The main conclusion drawn by FL is in fact logically falsifiable.
  - Suppose that it is found in the future that at some mesoscopic regime above a single atom but below multiple atoms, the measurement process acts. Then it may very well be that the resolution to the QMP does resolve the detailed problems of molecular structure in that it presents additional constraints on the evolution of the particles than only Schrödinger’s equation provides. Moreover, FL utilize a macroscopic device in their definition of the measurement problem, but this does not imply, from a purely logical standpoint, that the solution to this macroscopic coupling measurement problem does not occur at a mesoscopic regime above a single atom but below multiple atoms. It could very well still occur at the mesoscopic regime, but be amplified to move a pointer at the macroscopic regime. In the case that a measurement process occurs within the molecular regime, the argument of FL fails.
The authors state that: in Franklin and Seifert’s argument, the ground state psi is a non-trivial superposition of the states of the real isomers that are observed. “As a consequence, they are not eigenstates of the resultant Hamiltonian of the molecule. This means that, even if immediately after measurement the molecule is, say, in the particular state … at subsequent times it will evolve to different states according to the time-dependent Schrödinger equation”. FL claim that the stability of isomers that is observed by chemists cannot be explained if the state is projected to a particular term in the superposition. However, the argument used by FL is not yet convincing to us. In order for FL claim to be correct, the ground state would have to be non-degenerate, such that only the superposition state possesses the minimum ground state energy. However, it is at least logically possible for the ground state to be degenerate for which more than the superposition state has the ground state energy. If some or all of the terms composing the superposition state also have the same energy as the ground state, then it is possible that these individual terms are also eigenstates. For example, the DC Stark shift is an example where the degeneracy of multiple states initially exists but is lifted by the action of the electric field. In cases where degeneracy exists and is not lifted, it is conceivable that both superposition states and individual terms of the superposition will be stationary. Hence FL needs to establish that the ground state is non-degenerate. We don’t know ourselves here, perhaps it is, perhaps not, at least a reference to prior work that establishes non-degeneracy of the ground state seems to be needed here.

Conclusions

Several issues regarding claims in the paper by FS have been examined by FL. It is noted by FL that the claims made by FS are based on the definition of the measurement problem by Maudlin which does not include the problem of the measurement basis determination. The authors also agree that particular issues of chirality, isomers, and symmetry breaking might be resolved via a resolution of the QMP. Those issues however are claimed to not include the problem of obtaining the detailed molecular structures that are seen with particular configurations of the atoms. This later claim is based on a macroscopic model of measurement. However, such a model, which Bohr believed was related to measurement and the Copenhagen interpretation is based around, may or may not be indicative of the actual solution. That is, one cannot use a macroscopic model of measurement to draw arbitrary conclusions about the solution, unless such a model is shown to be correct, and that measurement does not occur at the molecular level.

Due to this, the major claim made by FL can be logically falsified, depending on the ultimate determination of the physics of measurement. Is the measurement process acting at the microscopic, mesoscopic, or macroscopic regime? Many might argue that experimental evidence indicates that it is not acting at the microscopic regime. However, this is not proof that a measurement projection process does not act in some particular microscopic processes and/or in mesoscopic processes that have not been experimentally verified as obeying Schrödinger’s equation. We simply do not know at this time, and these authors have essentially presupposed that macroscopic interaction is necessary for measurement which is used to justify their claims.

Therefore, all that can be logically deduced (not induced) at present is that the resolution of the QMP may or may not resolve the issue of how detailed molecular structures are obtained in Nature.

References

[1] A. Franklin and V. A. Seifert, The Problem of Molecular Structure Just Is the Measurement Problem, Br J Philos Sci 75, 31 (2021).

[2] T. Maudlin, Three Measurement Problems, Topoi 14, 7 (1995).

[3] B. T. Sutcliffe and R. G. Woolley, On the Quantum Theory of Molecules, J Chem Phys 137, 22A544 (2012).

[4] B. T. Sutcliffe and R. G. Woolley, Comment on “On the Quantum Theory of Molecules” [J. Chem. Phys. 137, 22A544 (2012)], J Chem Phys 140, 037101 (2014).