Evidence of Predictive Power and Experimental Relevance of Weak-Values Theory

Abstract

The concepts of Weak Values (WV) and Two-State Vector Formalism (TSVF) appear to motivate new experiments and to offer novel insights into dynamical processes in various materials of several scientific and technological fields. To support this view, here we consider the dynamics of hydrogen atoms and/or molecules in nanostructured materials like e.g., carbon nanotubes. The experimental method applied is incoherent scattering of thermal (i.e., non-relativistic) neutrons (INS). In short, the main finding consists in the following effect: the measured energy and momentum transfers are shown to contradict even qualitatively the associated expectations of conventional scattering theory. This effect was recently observed in INS experiments, e.g., in H${}_2$ adsorbed in carbon nanotubes, where a large momentum transfer deficit was found. Due to the broad abundance of hydrogen, these findings may be also of technological importance, since they indicate a considerably enhanced H mobility in specific structured material environments. A new INS experiment is proposed concerning the H mobility of an ultra-fast proton conductor (H${}_3$OSbTeO${}_6$) being of technological relevance. Further neutron scattering investigations on other systems (metallic hydrides and H${}_2$ encapsulated inside C${}_{60}$) are proposed. As concerns theoretical implications, the analysis of the experimental results strongly supports the view that the wavefunction (or state vector) represents an ontological physical entity of a single quantum system.

1. Introduction

The theoretical frame of Weak Values (WV) and Two-State-Vector Formalism (TSVF) sheds more light into quantum mechanics, and provides novel insights in various physical and technological fields. Moreover, based on this theory, new experiments were suggested and various novel quantum effects were discovered; see Refs. [1,2,3,4,5,6,7] and references cited therein.

In this paper, the focus is not on interpretational discussions and/or new theoretical derivations; rather, we try to concentrate our attention on the question of whether the WV-TSVF theory has predictive power beyond that of the well-established and widely used standard quantum mechanics.

Moreover, an associated question is whether this theory provides any “useful” and powerful new theoretical tools that enable us to make (or create new) direct applications in already existing broad experimental fields of physical sciences. For example: Does WV-TSVF provide not only some small quantitative improvements of measurement accuracy, but  a qualitatively new insight into the research (and technologically important) field of neutron scattering?

Additionally, does this new theoretical frame help our intuition and creativity in predicting new quantum effects? This point is also addressed in Ref. [8] dealing with the classical limit of quantum optics—which also stimulated our own neutron scattering investigations reported in the present paper. e.g., as Aharonov et al. put it: “An essential point to emphasize is that this whole discussion is not just restricted to interpretations. Quite the opposite. Allowing us to have a better intuition is essential for finding new and interesting quantum effects and may lead to new experiments and potential practical applications.” [8], p. 2.

In the following sections, as we believe, these (and similar) questions will be answered in the affirmative. The new effect of “momentum transfer deficit” in neutron scattering from H-containing materials represents such a quantum effect that standard theory cannot explain, as will be discussed in certain detail. Putting it in a broader context (also for the non-specialized readers), this effect and the associated presented experimental results, seem to be also of potentially technological relevance (e.g., for the technology of hydrogen storage materials, or proton mobility enhancement in fuel cells).

In the experiments under consideration in this paper, the well-established method of incoherent inelastic neutron scattering (INS, or IINS) of thermal neutrons is used. Note that INS, which uses thermal neutrons (with velocities up to a few km/s), clearly belongs to non-relativistic physics. The effects considered below and their theoretical treatment, especially the momentum transfer deficit, belong to non-relativistic quantum mechanics—and no claim is made about their validity in the relativistic regime.

Already here it should be pointed out that only open quantum systems are considered and/or investigated in the present paper. This “innocent” looking remark will be shown to have certain “dramatic” consequences. For instance, INS measurements from H${}_2$ molecules enclosed in C-nanotubes—if analyzed with standard, conventional INS theory—show that the H${}_2$ roto-translational motion along the C-nanotube axis is extremely enhanced; more precisely, the impinging neutron apparently scatters from a fictitious “molecule” having mass $M\approx 0.64$ atomic mass units (a.m.u.)—instead of the conventionally expected 2 a.m.u. for the mass of H${}_2$. Equivalently, the momentum conservation law seems to be strongly violated here. As we will discuss below, this effect finds a natural “from first principles” interpretation in the frame of WV-TSVF theory, combined with the fact that the neutron scatters from an open quantum system.

The following short remarks regarding some common nomenclature of INS may prevent unnecessary confusion and/or misunderstandings:

(1)

“Incoherent” means that each neutron collides with, and scatters from, a single particle (nucleus, atom, etc.). In the case of neutron scattering from protons (conventionally often referred to as H-atoms), the scattering is incoherent in a very good approximation, due to the spin-flip mechanism accompanying the neutron-proton collision [9]. This effectuates a “which-path information”, thus destroying coherence between paths of the neutron originating from scattering off different nuclei. A first-principles explanation of this spin-flip effect, and  of coherent versus incoherent scattering, may be found in the “Feynman Lectures, Vol. III” [10], particularly in Section 3.3 therein.

(2)

“Inelastic” may be rather confusing in the context of a strict neutron-proton scattering. (since here the total kinetic energy of the two-body system is conserved, i.e., the collisional process is clearly elastic). This commonly used term, however, only means that the neutron’s kinetic energy—i.e., the quantity recorded by the spectrometer—has changed after the collision. The same remark applies also to the neutron-nucleus scattering at the much higher energies of so-called epithermal neutrons (say, $E_0>10$ eV), a process belonging to deep-inelastic neutron scattering (DINS), which is synonymous with neutron Compton scattering (NCS).

(3)

The INS from H using thermal neutrons represents s-wave scattering [9] which is isotropic.

(4)

The term effective mass of atom A is exclusively used in the frame of, and with respect to, a non-relativistic neutron-A collision of conventional theory [9].

The paper is structured as follows:

In Section 2 we shortly consider a remarkable theoretical result of Ref. [8]—a striking exchange of momentum between a photon and a mirror of a Mach-Zehnder interferometer (MZI). This result motivated (at least intuitively) our INS investigations. A careful consideration of the derivation of this effect shows that both WV and TSVF are used in the derivation; thus they extend standard QM and can make qualitatively new predictions—which contradicts various beliefs and/or claims of the opposite.

Section 3 and the associated Appendix B mention a few remarkable theoretical (e.g., the well-known Elitzur-Vaidman effect) and technological (e.g., the GOOGLE quantum processor) advances that support the view that the wavefunction, or state vector, is a basic, i.e., an ontic, quantity that refers to single quantum systems, and not to ensembles of systems. This view is in line with the WV-TSVF theory, which constitutes the theoretical frame for the interpretation of the striking INS effect of “momentum transfer deficit”—equivalently, a measured strong reduction of the effective mass of H—presented in the experimental Section 5.

Section 4, together with the associated more extended Appendix A, provide a self-contained presentation of the experimental and instrumental details of a TOF spectrometer, being needed in order to understand and to assess the implications of the striking experimental INS observations. This self-contained part is and essential in order to explain in full detail why the observations are “anomalous” (i.e., they have no interpretation) in the frame of standard INS theory (and, even more generally, in non-relativistic incoherent scattering theory). Here, the quantity effective mass in the considered specific INS experimental context is defined, and its meaning is explained with the aid of two real INS experiments (scattering from liquid ${}^4$He and from protons of ice).

The theoretical interpretation of the INS experiments, based on the WV-TSVF theory, is presented in Section 6 in a concise form only, since the full theoretical treatment was very recently published in Refs. [11,12,13].

Section 7 deals with some proposed new experiments in order to learn more about the physics of the “momentum transfer deficit” effect, and also its potential relevance for some technological (or industrial) fields (e.g., for fast proton conductors in fuel cells).

Finally, Section 8 provides some additional discussion of various (theoretical and experimental) issues of the paper’s contents, as well as short concluding remarks supporting the new predictive power of WV-TSVF and its broader applicability in various fields.

2. Theoretical Motivation

The central issue of this paper concerns $\left(i\right)$ results of momentum and energy transfer measurements in real incoherent neutron scattering (INS) experiments, $\left(ii\right)$ the associated predictions of conventional (classical or quantum) theory and, in particular, $\left(iii\right)$ the comparison of the experimental INS results with a new theoretical expectation based on, and motivated by, the theory of Weak Values (WV) and Two-State-Vector Formalism (TSVF). Point $\left(iii\right)$ has been intuitively motivated by intriguing results presented in a recent theoretical paper entitled “The classical limit of quantum optics: Not what it seems at first sight” by Aharonov et al. [8].

A particularly motivating result of [8] concerns a striking non-conventional momentum exchange between two quantum objects—a photon impinging on a small mirror of a MZI—combined with suitable post-selection of the final neutron’s state, cf. Figure 1. This bears some analogy, at least intuitively, with various neutron-proton collision (or scattering) experiments considered recently [11,12,14] and in the present paper.

What is also remarkable in the derivations of Ref. [8] is that they require absolutely no previous knowledge of WV and TSVF theory, yet they succeed to demonstrate straightforwardly how the concept of WV appears naturally in the analysis of the physical problem under consideration.

Let us outline a particularly striking theoretical result of [8], i.e., a striking momentum exchange between the photon and one of the mirrors of MZI.

Single photons (or particles) enter a device similar to a usual MZI, with the exception that one reflecting mirror is sufficiently small (say, a meso- or nanoscopic quantum object M) in order that its momentum distribution may be detectable by a suitable non-demolition measurement. With the aid of a fully self-contained and remarkably short derivation, the authors derived from first principles the following astonishing result:

Let us select those single-photon experiments, in which the outgoing photons are measured with detector $D_2$ and ask for the action of these photons on the (say, mesoscopic) mirror M, which should be subject to quantum mechanics. It turns out that, although the post-selected photons—as all photons do, of course—collide with the mirror M only from the inside of the MZI, they do not push M outwards, but rather pull it in! Undoubtedly, this theoretical result does not have any classical interpretation. As Aharonov et al. put it: “This is realized by a superposition of giving the mirror zero momentum and positive momentum—the superposition results in the mirror gaining negative momentum.” (A concise derivation is given in [12]).

It is instructive to realize how the theory of WV (and TSVF, as we will see below,) appears naturally in this derivation, and therefore, it is necessary to point out a few formal details and/or analytic steps:

When a photon impinges from the left on the beam splitter BS${}_1$, see Figure 1, the effect of the beam splitter is to produce inside the interferometer the state $|\Psi \rangle =ir|A\rangle +t|B\rangle$ where $|A\rangle$ and $|B\rangle$ denote the photon propagating along the A and B arms of the interferometer, respectively. See caption of Figure 1 for further notations.

The second beam splitter, BS${}_2$, is identical to the first, and so one can readily check that a photon in the state

$$|{\Phi }_2\rangle =-ir|A\rangle +t|B\rangle$$

emerges towards detector D${}_2$ only; see [8], p. 3. The explanation in italics will be of crucial importance in the following Section 2.1.

Let ${\mathbf{P}}_B=|B\rangle \langle B|$ be the projection operator on the photon’s state to follow the path B of the MZI, see Figure 1.

If $\varphi \left(p\right)$ is the initial quantum state of the mirror M and $|\Psi \rangle$ is the quantum state of the photon before reaching M, the reflection on M results in the transition

$$|\Psi \rangle \varphi (p)=(ir|A\rangle +t|B\rangle \left)\varphi \right(p)\to ir|A\rangle \varphi \left(p\right)+t|B\rangle \varphi (p-\delta )$$

(1)

Given $\delta >0$ to be sufficiently small, one may make a perturbative approximation of the total state of the photon and M mirror just before the photon reaches the output beam splitter BS${}_2$:

$$|\Psi \rangle \varphi \left(p\right)\approx ir|A\rangle \varphi \left(p\right)+t|B\rangle \left(\varphi \left(p\right)-\frac{d\varphi \left(p\right)}{dp}\delta \right)=|\Psi \rangle \varphi \left(p\right)-t|B\rangle \frac{d\varphi \left(p\right)}{dp}\delta$$

(2)

Now one considers a photon emerging towards D${}_2$, after passing BS${}_2$. The state of the mirror M is then obtained (up to normalization) by projecting the joint state onto the state of the photon corresponding to the aforementioned selected final state ${\Phi }_2$, i.e.,

$$\langle {\Phi }_2|\left(|\Psi \rangle \varphi \left(p\right)-t|B\rangle \frac{d\varphi \left(p\right)}{dp}\delta \right)$$

(3)

A short derivation(see [8] or [12]) yields the result

$$\langle {\Phi }_2|\left(|\Psi \rangle \varphi \left(p\right)-t|B\rangle \frac{d\varphi \left(p\right)}{dp}\delta \right)=\langle {\Phi }_2|\Psi \rangle \varphi (p-P_B^w\delta )$$

(4)

Here appears the term

$$P_B^w=\frac{\langle {\Phi }_2\left|{\mathbf{P}}_B\right|\Psi \rangle }{\langle {\Phi }_2|\Psi \rangle }$$

(5)

which represents the WV of ${\mathbf{P}}_B$ taken between the initial state $|\Psi \rangle$ and the selected final state $|{\Phi }_2\rangle$.

The numerical value of $P_B^w$ is straightforwardly calculated to be

$$P_B^w=\frac{\langle {\Phi }_2\left|{\mathbf{P}}_B\right|\Psi \rangle }{\langle {\Phi }_2|\Psi \rangle }=-\frac{t^2}{r^2-t^2}$$

(6)

Furthermore, the momentum kick ($\delta p_M$) received by the mirror due to a photon emerging towards D${}_2$ is calculated to be

$$\delta p_M=P_B^w\delta =-\frac{t^2}{r^2-t^2}\delta <0$$

(7)

The last inequality represents the novel result under consideration, and it holds due to $r>t$.

The appearance of the WV of the projector ${\mathbf{P}}_B$ in the expressions of the derivation is a characteristic feature of the WV theory. Namely, the quantum mirror M is a device “measuring” whether or not the photon (here: the system) is in arm B. The momentum of the mirror acts as a so-called pointer variable (no kick: the photon is in arm A; kick: the photon is in arm B). However, since the photon can only change the momentum of the mirror (pointer) by far less than its spread, we are in the so called “weak measurement” regime.

Furthermore, M and the photon are both quantum systems, and their interaction creates quantum correlations between them which survive their spatial separation until detector $D_2$ makes a strong (projective) measurement and produces a classical record (i.e., a “klick”).

Concluding, the quantum-theoretical insight obtained here—the “anomalous” negative momentum transfer on M—is dramatically different from the classical (or conventionally expected) one.

2.1. A Digression: Is the Result Based on WV Theory Alone, or on Both WV and TSVF?

In this paper, WV and TSVF are considered to go beyond standard QM. However, there are differing opinions too. For instance, on may argue as follows. The WM framework is entirely contained within standard QM, and so the WV is considered to be an application of standard perturbation theory of QM. On the other hand, TSVF appears to be only a specific interpretation of pre- and post-selected system states (which of course is different from the standard Copenhagen interpretation of QM). Therefore, WV and TSVF should be distinguished and their possible significance separately investigated.

Therefore, it is revealing to consider here explicitly the following question: Is the derived striking effect, inequality (7), due to the quantum theory including WVs, but  without making use of the concepts of TSVF?

From first sight, the derivation of the striking result (7) in the original work [8] uses only perturbation theory (here: a simple Taylor expansion and keeping the linear term only) and concepts of standard QM.

However one should observe the following: A photon starting at D${}_2$, after entering the MZI at BS${}_2$ has the wavefunction ${|\Psi \rangle }^{{}^{\prime }}=+ir|A\rangle +t|B\rangle$. But this is equal to the wavefunction $|{\Phi }_2\rangle$ running backward in time, (the time is not explicitly shown and thus only i is replaced with $-i$). Thus the WV, Equation (1), uses the backward-in-time evolving final-state wavefunction starting at D${}_2$. In other words, we realize that the TSVF has been used here.

Going a little further, one may observe that, according to the orthodox (or standard) interpretation of QM, the state ${\Phi }_2$ does not exist—despite the innocent looking justification that it describes a photon emerging towards detector D${}_2$. In other words, according to standard view of QM, the photon’s state within the MZI (i.e, after BS${}_1$ and before reaching BS${}_2$) is only $|\Psi \rangle =ir|A\rangle +t|B\rangle$.

To our opinion, these considerations support once more the novel character and predictive power of WV-TSVF.

3. Short Remarks on the Logical-Axiomatic Status of “Wavefunction”

As already mentioned above, “interpretational issues” of quantum mechanics, WV and TSVF, and especially of the term “wavefunction” (or state vector; we do not differentiate between the two here) are outside the scope of this paper. Nevertheless, the experimental findings and their theoretical interpretation presented below, appear to shed some more light on such issues. Especially, the discussions of the experimental INS results seem to provide further evidence that the concept of wavefunction represents a fundamental property of single quantum systems.

The interested reader may find in Appendix B a few examples of selected additional theoretical and also technological advances, which, in our view, do support this interpretational view. For instance, it will be argued that the wavefunction represents a physical entity of a single system, being as real as its energy or momentum—and not of an ensemble of systems.

4. Experimental Context and Conventional Theory

This section and the associated Appendix A are essential in order to demonstrate in full detail why the INS observations considered below are “anomalous” in view of, i.e., they have no interpretation within, standard INS theory.

Moreover, here is also explained in full detail what exactly means the term effective mass ($M_{eff}$) in the here considered specific INS experimental context, i.e.,  neutron-proton scattering. This detail seems necessary, in order to prevent possible confusion with the many different definitions of “effective mass” appearing in the frames of other physical fields and/or contexts, (e.g., theory of electrons and holes in band structures), and where a small, or even a negative, value of it represents no anomaly at all. Such well known effects can be found in textbooks of solid state physics, see e.g., [15].

4.1. INS Measurement—Outline of Some Concepts

The method applied in most of the following experiments is INS, and the associated spectrometers are time-of-flight (TOF) instruments. In Appendix A are presented all essential technical details of an INS TOF-spectrometer, and a concise description of “what is directly measured” in a real INS TOF-experiment—a key point for the understanding of the experimental observations.

In this section we present some theoretical results of conventional neutron scattering theory [9,16,17], which are needed for the interpretation of the INS experiments under consideration.

4.2. Momentum and Energy Conservation in Non-Relativistic Two-Body Collisions—Conventional Theory

Let us consider a neutron colliding with an atom (better: nucleus) of mass M, using the notations defined in Appendix A. The collision is impulsive, governed by the ultashort-range (ca. femto-meter) strong force as captured by the Fermi pseudo-potential [9].

The atomic initial momentum $\mathbf{P}$ causes the energy transfer $E=E_0-E_1\equiv \hslash \omega$ being transferred from the neutron to the atom ($E_0$: neutron’s initial energy; $E_1$: neutron’s final energy; see Figure A1 of Appendix A):

$$E=E_0-E_1\equiv \hslash \omega =\frac{{(\hslash \mathbf{K}+\mathbf{P})}^2}{2M}-\frac{{\mathbf{P}}^2}{2M}=\frac{{(\hslash \mathbf{K})}^2}{2M}+\frac{\hslash \mathbf{K}·\mathbf{P}}{M}$$

(8)

which represents the conventional energy conservation in a collision of two mass points. $\hslash \mathbf{K}$ is the momentum transfer from the neutron to the atom. In the following we use the common notation |$\mathbf{X}$| = X, for any vector $\mathbf{X}$.

The first term in the right-hand side (rhs) of the last equation is called recoil energy,

$$E_{rec}=\hslash {\omega }_{rec}=\frac{{(\hslash K)}^2}{2M}$$

(9)

which is the gained kinetic energy of a struck atom being initially at rest. Since $\langle P\rangle =0$ before collision (i.e the atom is initially at rest), one obtains for the involved mean values:

$$\langle E\rangle =\frac{{\hslash }^2{\langle K\rangle }^2}{2M}\equiv E_{rec}$$

(10)

(Note that, for simplicity, both last equations are often used to define the same quantity “recoil energy”.) This equation refers to the center of a measured peak, which usually has a finite width. Thus, INS from a gaseous sample of atoms leads to an intensity peak centered at energy transfer $E_{rec}$, and having a finite width caused by the additional term $\hslash \mathbf{K}·\mathbf{P}/M$, which represents the well known Doppler broadening (see details in the Appendix A).

For illustration of the above equations, Figure 2 shows experimental data of incoherent neutron scattering from liquid helium, ${}^4$He; for details see [18].

A theoretical result illustrating the same effect is given by the calculation of $S(K,E)$ of a quantum harmonic oscillator (modeling INS from H of the palladium-hydrogen system) by Kofu and Yamamuro [20]; see Figure 6 therein.

The above simple formulas capture the main features of the so-called Impulse Approximation (IA) [11,16]. The IA describes ultrafast two-body collisions, since the Fermi pseudo-potential of the neutron-nucleus collision is very short (in the ${10}^{-15}$ m range) [9,16].

4.3. Experimental Determination of the Mass of the Scattering Nucleus

Small deviations from the IA, often called final-state effects (FSE), may be observed in experiments, which however are well understood (see, e.g., [11,16]) as follows.

When a particle with mass M is interacting with its environment, the impinging neutron effectively scatters from a particle having effectively a higher mass than M. This is because the adjacent particles are exerting attractive forces on the scattering particle, the free motion of the latter can only be hindered by these forces. (Repulsive forces would spontaneously separate the considered atom from its environment.) Obviously, within conventional theory, this can only lead to and increase—and never a decrease—of its effective mass $M_{eff}$

$$M_{eff}\ge M\equiv M_{free}.$$

(11)

cf. [11,12,16] for more details.

This inequality also follows more formally from the aforementioned energy conservation relation, if one includes an additional term $E_{int}>0$ caused by the atom-environment interactions:

$$\overline{E}=\frac{{\hslash }^2{\overline{K}}^2}{2M}+E_{int},$$

(12)

$\overline{E}$ and $\overline{K}$ refer to the center of a measured recoil peak. Thus there is a reduced amount of energy, $\overline{E}-E_{int}$, available to the atom as its kinetic energy after the collision. $\overline{E}$ and $\overline{K}$ are determined from the neutron-atom collisional kinematics, in contrast to $E_{int}$, which is a quantity of the sample (i.e., of the scatterer).

Moreover, having in mind the impulsive character of the neutron-proton and the small range (about 10${}^{-15}$ m) of the strong nuclear force, one can try to fulfill this equation by using the H-mass as a fitting parameter ($M_{eff}$), as follows:

$$\overline{E}=\frac{{{\hslash }^2\overline{K}}^2}{2M_{eff}}.$$

(13)

Here, energy and momentum transfers are assumed to be equal to the corresponding measured quantities; see also Appendix A. Obviously, the last two equations imply necessarily

$$M_{eff}>M$$

(14)

Again, these considerations are fully in line with conventional theory of INS. It should be emphasized that the quantity $M_{eff}$ introduced here, which refers to the non-relativistic impulsive collision of two mass points (here; neutron and proton), is fully unrelated with other definitions of the term “effective mass” used in other fields, e.g.,  in traditional solid state physics [15].

The above conclusion can be reformulated to be applicable to a visual inspection of a given experimental intensity map $S(K,E)$ of an incoherent scattering experiment. Namely, the recoil parabola of a free atom with mass M divides the momentum-energy transfer plane $K-E$ into two parts:

• Normal Part. The part to the right (and thus also below) of the recoil parabola, is the conventional scattering regime, which corresponds to the discussed case $M_{eff}>M$ and thus has a conventional interpretation. Of course, the recoil parabola itself belongs to this part, too.
• Anomalous Part. The part to the left (and thus also above) of the recoil parabola is the non-conventional, or “anomalous”, scattering regime, since it corresponds to the case of $M_{eff}<M$ which has no interpretation within standard (or conventional) INS theory.

For convenience, let us summarize here the indispensable assumptions of conventional INS theory, which conventional expectations are based on:

$\left(A1\right)$

The neutron-nucleus collision is impulsive, due to short Fermi pseudo-potential describing the strong nuclear a force.

$\left(A2\right)$

The collision is a non-relativistic two-body process between two mass points (i.e., neutron and the H-nucleus).

$\left(A3\right)$

Energy and momentum transfers hold for this two-body process.

$\left(A4\right)$

Prior to collision the scatterer is in thermal equilibrium (and consequently, $E_{int}$ cannot be negative).

$\left(A5\right)$

The first-order quantum perturbation theory, and specifically Fermi’s Golden Rule, which the standard INS theory is based on [9], are sufficient approximations.

In other words, it holds:

• All experimental results satisfying conventional theory must lie in the normal part of the $K-E$ plane, in which holds $M_{eff}\ge M$. Violation of relation 11, i.e., $M_{eff}<M$, is tantamount to violation of at least one of conditions $A1$–$A5$.

For further experimental results and associated details of FSE and effective mass, the interested reader may consult Refs. [11,13,16] and references therein.

4.4. INS from Bulk Ice—Validity of Conventional Theory

For further illustration of the concepts and relations of the preceding subsection, let us consider here one more experimental result: INS from H of H${}_2$O molecules of ice. INS from H-containing samples is usually done at very low temperatures, because this reduces the “background” (due to phonons, librations etc..) and Debye-Waller-factor effects [9] on the measured $S(K,E)$. The latter is the quantity of primary interest for quantifying material properties and testing theoretical predictions [9].

Figure 3 shows the experimentally observed $S(K,E)$ of micro-crystalline ice Ih at $T=20$ Kelvin and 1 bar pressure [21], using the TOF spectrometer MARI [22] of ISIS neutron spallation source, UK. Neutron’s initial energy was $E_0=750$ meV. The most significant scattering contributions are caused by H, which has a much larger incoherent scattering cross-section than other atoms [9]. The black parabola line represents the conventional recoil line of a free H, i.e., with effective mass $\approx 1$ a.m.u., being initially at rest

$$E_{rec}^{free}=\frac{{\hslash }^2{K}^2}{2M_H}$$

(15)

($M_H$: atomic mass of H.) Recall that H and neutron (n) have almost equal masses, $m_n\approx M_H$. Clearly, this is equivalent to the preceding Equation (10).

The intensity peak around $E=420$ meV and $K=14$ Å${}^{-1}$ is due to the molecular stretching OH-vibrational modes of H${}_2$O. Note that the center of intensity peak lies on the free H-atom recoil line, as conventionally expected. In view of Equation (15), this experimental result shows that the observed intensity peak is well described with the common theoretical assumption that the collision is impulsive and essentially a two-body process. I.e., a “dressing” of the vibrationg H with environmental and/or intramolecular degrees of freedom is not relevant, and we have $M_H^{eff}=M_H^{free}\approx 1.0$ a.m.u.—as conventionally expected. Thus, as stated in the preceding subsection, the stretching vibrational peak lies in the “normal part” of the $K-E$ plane.

In this context, Kearley et al. [23] gave the following additional justification to the above physical picture: The very short (roughly in the pico- to femto-second) neutron-H scattering time, taken together with the significant spectral separation of the stretching modes from the other modes, effectuate a decoupling of the “fast” H-stretching motion from the other (slower) ones.

We observe a slightly different behavior of the bending vibrational mode, which gives the peak at energy transfer $E\approx 200$ meV. Obviously, this peak is centered to the “right” of the H-recoil parabola, i.e., it is still in the “normal part” of the $K-E$ plane.

Applying the aforementioned argument by Kearley et al. [23] we can also provide the following conventional explanation: The bending motion is hindered by adjacent H-bonded molecules, and thus the scattering atom is “dressed” by environmental degrees of freedom, which causes an increased effective H-mass. Indeed, a rough estimate derived from the peak-center position of the $S(K,E$) graph, Figure 3, yields for the effective mass of the struck H in the bending mode: $M_H^{eff}\approx 1.3$ a.m.u.

Summarizing, the above INS observations about the HO-vibrations of ice have a straightforward conventional-theoretical interpretation. Surprisingly, in the next section we will see that this is in blatant contrast to the case of H${}_2$O molecules being in certain nano-scale confinements, where the stretching vibrational mode of H${}_2$O exhibits a significantly large anomaly, as originally noticed in [21].

5. The New Scattering Effect—Momentum Transfer Deficit on H

5.1. First Example: INS from Single H${}_2$ Molecules in C-Nanotubes

In some experimental explorations in the field of material sciences, a surprising INS result was recently discovered [24]. This concerns the quantum excitation spectrum of H${}_2$ adsorbed in multi-walled nanoporous carbon. The C-nanotubes had pore diameters about 8–20 Å.

These experiments were carried out at $T=23$ Kelvin with the TOF-spectrometer ARCS of SNS [19]. Note that the energy transfers in this experiment cannot excite H${}_2$-vibrations (which lie much higher), but only rotations and translations only. Thus it holds

$$E=E_{rot}+E_{trans}.$$

(16)

The experimental two-dimensional INS intensity map $S(K,E)$ of H is shown in Figure 4b [24]. The following features should be noted. The intensive peak at $E_{rot}\approx 14.7$ meV is due to the first rotational excitation $J=0\to 1$ of H${}_2$ [17]. The observed momentum transfer of this peak is $K_{rot}\approx 2.7$ Å${}^{-1}$. Thus the peak position in the $K-E$ plane satisfies the relation $E_{rot}={(\hslash K_{rot})}^2/2M_H$ with the mass $M_H$ of the free H atom,

$${\mathrm{H}}_2\mathrm{rotation}(\mathrm{J}=0\to 1):M_{eff}=M_H=1.0079\mathrm{a}.\mathrm{m}.\mathrm{u}.$$

(17)

In other words, the spectral position of this rotational excitation is fully in line with conventional theoretical expectations. And since the scattering is incoherent, one may say that each neutron scatters from a single H atom.

Moreover, the novel finding of [24] is as follows. A detailed numerical treatment of the roto-recoil data (cf. Figure 4) was presented, which derived a significantly reduced effective mass of the recoiling H${}_2$ molecule (left parabola, green):

$${\mathrm{H}}_2\mathrm{translation}\left(\mathrm{recoil}\right):M_{eff}\left({\mathrm{H}}_2\right)\approx 0.64\pm 0.07\mathrm{a}.\mathrm{m}.\mathrm{u}.$$

(18)

Additional INS experiments with differing C-nanotube samples showed no significant variation of this astonishing numerical value. Obviously, this experimental result is in blatant contradiction to the value $M\left({\mathrm{H}}_2\right)=2.01$ a.m.u. for a freely translating H${}_2$ molecule (right parabola), as conventional theory predicts.

This observation provides direct evidence for the new “anomalous” effect of $\hslash K$-transfer deficit [11,21].

According to the associated discussions above (see Section 4.3 ), every conventional H${}_2$/substrate binding (or interaction) can never decrease the effective mass of the molecule. Thus these INS results contradict conventional theoretical expectations. However, we will see below that they find a natural interpretation in the WV-TSVF theoretical frame.

5.2. Second Example—INS from Single H${}_2$ Molecules in the Metal-Organic Framework Material “HKUST-1”

Due to its considerable magnitude and its surprising character, it is important to have another, fully independent, confirmation of this effect.

Other INS experiments, with the goal to discover novel materials for potential use in the technology of storage and transport of hydrogen, investigated the adsorption of H${}_2$ on various porous metal–organic framework materials over the past few years. The material known as HKUST-1 (see [25] for its chemical stochiometry and crystalline structure) has emerged as a characteristic model system for related studies. Here it suffices to note that HKUST-1 constitutes a three-dimensionally connected network, with  a trimodal pore structure, having pores with diameters in the range 5–12 Å.

Also in his material, the experimental intensity map $S(K,E)$ shows again a broad band attributed to the translational H${}_2$-motion in the pores, which is very similarly to the experiment presented in the preceding subsection. Again, the measurements were done at low temperatures, $T<30$ Kelvin [25].

5.3. Comparison of the Two Experiments

Most importantly, the experiment [25] provides an independent confirmation of the striking result of Ref. [24] obtained at SNS, USA. The INS measurements of Ref. [25] were done with the 2-dimensional spectrometer MARI (ISIS facility, UK)—that is, by an independent European group of scientists, and  in a different laboratory.

Figure 5 permits a direct visual comparison of the data of the two experiments, which reveals a reassuring result. Namely, The two broad continuous bands due to the translational mode are virtually identically positioned in the $K-E$ plane. This is quite astonishing given the fact that the two experiments were done with two different nano-structured materials.

Therefore, we may expect that the observed effect should be present in far more materials than known until now.

6. Theoretical Interpretation within WV and TSVF Theory

In this section, we present an outline of our recently published [11,12] theoretical model in the frame of WV-TSVF theory, which explains from first principles the anomalous momentum-transfer deficit observed in the preeceeding experiments. For full details of derivations, see [11,12].

We re-consider Equation (8) capturing E-conservation in a strict two-body collision. Due to momentum conservation, one expects (n: neutron; A: atom)

$$-\hslash K_n=+\hslash K_A\equiv \hslash K>0,$$

(19)

where $+\hslash K_A$ is the momentum transfer on the atom. Following standard notation (e.g., as in [9]) we choose $K_A$ to be positive.

Having the theory of WV and TSVF in mind, one sees that the “larger” recoil term ${(\hslash K)}^2/\left(2M\right)$ may be viewed to result from a strong impulsive interaction (associated with momentum transfer $+\hslash K$ on the atom). In the IA holds $\left|K\right|\gg \left|P\right|$, and thus the “smaller” Doppler term $\hslash \mathbf{K}·\mathbf{P}/M$ corresponds to a weaker interaction, in which the atomic momentum $\widehat{P}$ couples with an appropriate dynamical variable of the neutron; the latter turns out to be $\widehat{q}$ [11].

In view of the theory of WV and TSVF, the  mentioned smaller Doppler term is expected to cause weak deviations from the conventionally expected momentum transfer $+\hslash K$. This can be formally captured by replacing $\widehat{P}$ with a small momentum difference $\widehat{P}-\hslash K{\widehat{I}}_A$ (${\widehat{I}}_A$: identity operator in the atomic sub-space). For formal reasons, here is temporarily included a positive smallness factor, $0<\lambda \ll 1$; see Discussion below. Summarizing, the proposed von Neumann-type model interaction Hamiltonian (describing the deviation from conventional theory) is [11]

$$\Delta {\widehat{H}}_{int}\left(t\right)=+\lambda \delta \left(t\right)\widehat{q}\otimes (\widehat{P}-\hslash K{\widehat{I}}_A)$$

(20)

($\delta \left(t\right)$: delta function; $\lambda$: small real constant). It should be emphasized that the plus sign in front of this expression is not arbitrary, since it is consistent with the aforementioned definitions (19) of momentum transfer; cf. [11].

The physical motivation of the two parts of the model Hamiltonian of Equation (20) is in conceptual agreement with the following associated remark by Aharonov et al.: “Consider, for example, an ensemble of electrons hitting a nucleus in a particle collider. $\left[\dots \right]$ The main interaction is purely electromagnetic, but there is also a relativistic and spin-orbit correction in higher orders which can be manifested now in the form of a weak interaction.” ([26], p. 3).

6.1. WV of Atomic Momentum Operator $\widehat{P}$ and the Effect of “Anomalous” Momentum-Transfer Deficit

In the following, the scattering atom represents the “system” of the general formalism, and the subscript w in $X_w$ indicates that this is a WV of the observable $\widehat{X}$. The WV of the identity operator is ${\left({\widehat{I}}_A\right)}_w=1$, and thus for the WV of the atomic coupling operator $\widehat{P}-\hslash K{\widehat{I}}_A$ in the above interaction Hamiltonian:

$${(\widehat{P}-\hslash K{\widehat{I}}_A)}_w=P_w-\hslash K$$

(21)

To proceed, we first calculate the WV $P_w$ of $\widehat{P}$ for some characteristic final state in momentum space. The derivation reveals a striking deviation—more precisely, a deficit—from the conventionally expected momentum transfer to the neutron. The latter represents the “pointer” of the general theory, and the pointer momentum variable $\widehat{p}$ is conjugated to the neutron position $\widehat{q}$.

In the following we use the momentum space representation, motivated by the fact that most scattering experiments measure momenta (rather than spatial positions).

Let atom A being initially in a spatially confined state and at rest. Here we may approximate the initial atomic wavefunction $\Xi {\left(P\right)}_i$ by a Gaussian $G_A\left(P\right)$ centered at zero momentum, $\langle P\rangle =0$, i.e.

$$\Xi {\left(P\right)}_i\approx G_A\left(P\right).$$

(22)

The width of $\Xi {\left(P\right)}_i$ is determined by the quantum uncertainty. The struck atom will move in the direction of momentum transfer $\hslash K_A=\hslash K$. Therefore, in the following calculations P may represent the atomic momentum along the direction of momentum transfer.

To make the derivations and their physical meaning more intuitively transparent, let us assume:

• The final atomic state has the same width in momentum space as the initial state.

This assumption is very common in molecular (optical) spectroscopy and/or quantum-chemical calculations of molecules; it corresponds to the well known Franck–Condon picture. Namely, it captures the intuitive view that the impulsive transition is very fast and so the atomic environment remains unchanged during the neutron-atom collision. Thus the final atomic momentum wavefunction should have the same shape as the initial state and it is shifted from zero to the transferred momentum, i.e.,

$$\Xi {\left(P\right)}_f=\Xi {(P-\hslash K_A)}_i.$$

(23)

The WV of the atomic momentum operator $\widehat{P}$ is then readily calculated:

$$\begin{array}{ccc}\hfill P_w& =& \frac{\langle {\Xi }_f\left|\widehat{P}\right|{\Xi }_i\rangle }{\langle {\Xi }_f|{\Xi }_i\rangle }=\frac{\int dP\Xi {(P-\hslash K_A)}_{i}P\Xi {\left(P\right)}_i}{\int dP\Xi {(P-\hslash K_A)}_i\Xi {\left(P\right)}_i}\hfill \\ & =& +\frac{\hslash K_A}{2}=+\frac{\hslash K}{2}\hfill \end{array}$$

(24)

The evaluation of the integral in the numerator follows immediately from the following simple considerations: (a) the two functions $\Xi$ are positioned symmetrically around the middle point $\overline{P}=\hslash K_A/2$—one function is centered at 0 and the other at $\hslash K_A$; and (b) P is a linear factor. I should be noted that this result does not depend on the width of the Gaussian $\Xi$.

This is a quite surprising result because it is associated with a considerable K-transfer correction of 50%; i.e., the scattered neutron measures a momentum kick being only half of the conventionally expected value, if one takes $\lambda =1$; see also Discussion. In more detail, it holds

$${(\widehat{P}-\hslash K{\widehat{I}}_A)}_w=+\frac{\hslash K_A}{2}-\hslash K=-\frac{\hslash K}{2}$$

(25)

More explicitly, applying the well-known general result of WV and TSVF [1,3,5] we obtain for the correction to the shift of the meter’s (i.e., the neutron’s) pointer variable:

$${\langle \widehat{p}\rangle }_f-{\langle \widehat{p}\rangle }_i=-\lambda {(\widehat{P}-\hslash K{\widehat{I}}_A)}_w=+\lambda \frac{\hslash K}{2}$$

(26)

(See [11] for full derivation) Moreover, for the total momentum transfer shown by the pointer of the measuring device (here: neutrons) we have

$$\begin{array}{ccc}\hfill {\left[{\langle \widehat{p}\rangle }_f-{\langle \widehat{p}\rangle }_i\right]}_{\mathrm{total}}& =& {\left[{\langle \widehat{p}\rangle }_f-{\langle \widehat{p}\rangle }_i\right]}_{\mathrm{conventional}}+{\left[{\langle \widehat{p}\rangle }_f-{\langle \widehat{p}\rangle }_i\right]}_{\mathrm{correction}}\hfill \\ & =& -\hslash K+\lambda \frac{\hslash K_A}{2}\hfill \end{array}$$

(27)

This result represents the effect of momentum-transfer deficit: the absolute value of momentum transfer on the neutron predicted by the new theory is smaller than conventional (or standard) theory predicts:

$$|-\hslash K+\lambda \hslash K_A/2|<|-\hslash K|$$

(28)

Here holds the aforementioned definition: $\hslash K_A=\hslash K$, Equation (19).

6.2. Plane Waves Approximation Implies Conventional Momentum Transfer

In theoretical investigations of conventional neutron scattering theory, the plane wave approximation plays a central role. Therefore, it is of paramount importance to show explicitly the consequences of using this approximation in tthe frame of the present WV-TSVF analysis.

So let us here make the assumption:

• The final state of the struck atom should be a plane wave, i.e., it has vanishing width in momentum space—as e.g., in the IA.

Interestingly, now the well known result of conventional theory (i.e., the conventional momentum transfer $\hslash K$) follows straightforwardly within our formalism—the preceding correction due to WV and TSVF just disappears. Now the atomic final-state wavefunction of A is a delta function ${\delta }_A$ centered at the assumed transferred momentum $\hslash K_A$,

$$\Xi {\left(P\right)}_f={\delta }_A(P-\hslash K_A)$$

(29)

For the WV of $\widehat{P}$ one then obtains:

$$\begin{array}{ccc}\hfill P_w& =& \frac{\langle {\Xi }_f\left|\widehat{P}\right|{\Xi }_i\rangle }{\langle {\Xi }_f|{\Xi }_i\rangle }=\frac{\int dP{\delta }_A(P-\hslash K_A)P\Xi {\left(P\right)}_i}{\int dP{\delta }_A(P-\hslash K_A)\Xi {\left(P\right)}_i}=\frac{\hslash K_A\Xi {(\hslash K_A)}_i}{\Xi {(\hslash K_A)}_i}\hfill \\ & =& +\hslash K_A\equiv +\hslash K\hfill \end{array}$$

(30)

Hence, the WV of the system coupling operator, $(\widehat{P}-\hslash K{\widehat{I}}_A)$, becomes equal to zero, that is,

$${(\widehat{P}-\hslash K{\widehat{I}}_A)}_w=P_w-\hslash K=0$$

(31)

Parenthetically, it may be noted that, according to standard theory, the scattered wave may acquire an additional phase factor, e.g., $e^{ı\chi }$. Howeever this does not affect the preceding result because this factor cancels out in the fractions of Equations (24) and (30).

This result implies that, under the usual assumption of “plane waves”, the WV-TSVF theory yields no correction to the conventionally expected value of momentum transfer. Hence we have:

$${\left[{\langle \widehat{p}\rangle }_f-{\langle \widehat{p}\rangle }_i\right]}_{\mathrm{total}}={\left[{\langle \widehat{p}\rangle }_f-{\langle \widehat{p}\rangle }_i\right]}_{\mathrm{conventional}}=-\hslash K$$

(32)

In other words, the pointer of the measuring device—i.e., the experimentally determined momentum loss of a neutron—will now exhibit the conventionally expected value $-\hslash K$. Importantly, the result of Equation (31) is consistent with conventional theory of the IA, and also of general incoherent neutron scattering theory—the standard expectation of conventional theory is here reproduced. This is a very satisfactory result.

Comparison of the aforementioned two derivations shows that the magnitude of the momentum transfer deficit under consideration depends on the “deformation” of the shape of the final atomic state. e.g., if the final state is, say, “almost” a delta function (plane wave), then the anomalous momentum transfer deficit will be “very small”—and perhaps remain undetectable. Recall that the plane-wave assumption is standard in NCS and/or DINS data analysis; cf. [16].

The general case of an arbitrary shape of the final-state wavefunction was derived and discussed in Ref. [11].

6.3. On Pre- and Post-Selection of System’s States

In the physical set-up of a MZI and the gedanken experiment of Section 2, one recognizes easily how the theoretical concepts of pre- and post-selected states of WV and TSVF theory appear. Namely, it is clear that these states are simply chosen by the experimentalist.

In contrast, this is not the case in the theoretical model presented in this section—the experimentalist has not the capability to choose the atomic (or molecular) states of H in a material probe, like those described above, at his/her will.

First, is intuitively easy to understand that the initial (before collision) H-state corresponds to the concept of a pre-selected state in the sense of WM-TSVF. But what about the post-selected states, which, as we saw in the above model calculations, can greatly affect the magnitude of the WV of the H-momentum operator?

Looking at the INS experiments on H${}_2$ in C-nanotubes (Section 5.1), one may realize the following details.

After neutron-H collision, the H${}_2$ molecule can translate only along the C-nanotube axis, but not perpendicular to it. (due to the nanoscale diameter of the C-nanotube). In other words,

(1)

the C-nanotube’s geometry, taken together with

(2)

the orientation of the sample with respect to the neutron-beam, and

(3)

the associated chosen direction of the momentum transfer ℏK,

correspond to, or effectuate, the choice of a specific post-selection of the final state of the roto-translating H${}_2$. And, as we saw in Figure 4, this specifically selected state may led to the dramatic effect of momentum transfer deficit (and the corresponding non-conventional result $M_{eff}<M_H$).

Furthermore, the INS results from H${}_2$ in C-nanotubes presented above might appear contradictory, because:

$\left(i\right)$

The $J=0\to 1$ rotational excitation of H${}_2$ shows a conventional $M_{eff}\approx 1$ a.m.u.

$\left(ii\right)$

In contrast, the $M_{eff}$ of the observed translational roto-recoil response of the whole H${}_2$ is not 2 a.m.u. (as expected by conventional theory because the whole H${}_2$ undergoes a translational motion), but only $M_{eff}\approx 0.64$ a.m.u., see Equation (18).

However, This “conflict” just disappears in the light of the WV-TSVF theoretical frame, because it simply implies that the quantum environments $\mathcal{E}$ of H in the two cases $\left(i\right)$ and $\left(ii\right)$ are different. In illustrative terms, the locally rotating H${}_2$ is much less influenced by its environmental interactions than the translating H${}_2$ that necessarily interacts with a greater part of the nanotube during its translational motion initiated by the collision. In short: Different environments effectuate different post-selections of the H${}_2$ final state.

These considerations indicate a potential applicability of these ideas in the field of material sciences. Namely, the discussed post-selected H-states are effectuated by the H-environment, i.e., the nano-structure of the host material, besides its physico-chemical properties. Further studies along these lines of thought may be of technological/industrial significance, too.

7. Expectations and/or Predictions of New Effects in Various Materials

Motivated by the striking experimental results of Section 5, we propose and/or qualitatively predict further concrete experimental topics which are expected to show similar “anomalous” momentum-transfer deficit effects. Two of them, i.e., the ultra-fast proton conductor H${}_3$OSbTeO${}_6$ and the metallic hydride system PdH${}_x$ are even of technological (industrial) relevance.

7.1. INS Investigation of Momentum Transfer Deficit on H of the Fast Proton Conductor H${}_3$OSbTeO${}_6$

In this subsection, a qualitative prediction concerning a possible technological application is considered.

Solids with high proton mobility are of considerable technical (and industrial) interest as electrolyte materials in fuel cells working in the temperature range below 600 K. Nowadays, the commonly used so-called PEM cells, working below 373 K, are equipped with specific polymers, which are very sensible to various disturbances or inefficiencies, e.g.,  their intolerance to impurities, water and heat management, additionally to their high costs.

Therefore, the search for alternative fast proton conductors based on inorganic solids has been under intense investigation. In 1985, Turrillas et al. [27] reported an antimonic acid-based compound with an outstanding proton conductivity of ∼${10}^{-2}{\Omega }^{-1}{\mathrm{cm}}^{-1}$, which opened up a new road of materials research. The most promising composition of this class of materials is H${}_3$OSbTeO${}_6$, a potential candidate for advanced fuel cells technology [27,28].

This material consists of a three-dimensional interconnected (Sb,Te)O${}_6$ framework, built from randomly distributed corner-shared SbO${}_6$ and TeO${}_6$ octahedra, forming large cages interconnected to a three-dimensional network, in which mobile H${}_3$O${}^{+}$ ions are contained; see Figure 6. The observed outstanding proton conductivity is due to the dynamics of these ions.

Extensive experimental investigations with various techniques [28], accompanied by quantum chemical calculations, have identified three main types of motion of protonic species: (1) stochastic rotations of the H${}_3$O group around its 3-fold axis, (2) jumps between four equivalent positions within the octraedral cages, and  (3) long-range translational motion (diffusion) over several cages. Specific details of the complex kinetic mechanisms were given in Ref. [28]. Note that INS experiments similar to those presented in the preceding section have not been reported thus far.

Making contact with the findings of the preceding section, in the following our interest lies with the translational motion of the H${}_3$O group (or H${}_3$O${}^{+}$-ion). Motivated by our theoretical WV-investigations on the quantum momentum transfer deficit and associated enhanced mobility of H${}_2$, we are led to the following prediction:

• The 2-dim. $S(K,E)$ of this material should reveal an anomalous momentum-transfer deficit on H${}_3$O${}^{+}$, which may be qualitatively similar to the effect of Figure 5 shown above. The roto-translational continuum of the H${}_3$O${}^{+}$, if measurable, would be again deeply in the quantum domain of the $K-E$, i.e.,  it should be positioned “on the left” of the H${}_3$O${}^{+}$-recoil line. Furthermore, observation of a (howsoever weak) translational continuum of single protons (H${}^{+}$) caused by neutron collisions would provide neew insights into the conductivity mechanism.

These suggested INS experiments should be carried out at incident neutron energies (1) similar to the preceding experiments, and additionally also (2) at $E_0$ about 1000 meV, so that also the stretching HO-vibration can be excited and observed in a 2-dim. spectrometer (like ARCS). In the latter case,

• the stretching vibrational HO-peak is expected to lie again in the quantum regime of the $K-E$ plane similarly to the quantum effect observed on the HO-stretching vibrations of H${}_2$O molecules in some nanoscopic channels of inorganic materials [29],

as earlier discussed and analyzed in Refs. [12,13,14,21]. For the convenience of the reader, here we display again the mentioned effect, as shown in Figure 7.

The following specific remark may be of interest. Figure 8 shows experimentally determined widths ${\sigma }_H$ of the momentum distributions of H in H${}_3$OSbTeO${}_6$ and of solid polyethylene at room temperature [30] recorded with the electron-Volt TOF-spectrometer eVS/Vesuvio of ISIS, Oxfordshire, UK. The scattering method is termed neutron Compton cattering (NCS), also known as deep-inelastic neutron scattering (DINS). This technique measures the momentum-peak shapes of the recoiling atoms, especially of H, and thus it determines the H-momentum distributions; see e.g., the concise review article by Watson [16].

The measured widths ${\sigma }_H$ of the H-momentum distributions in the two materials appear to be approximately equal, which shows that the scattering H species in the two materials are covalently bonded with comparable bonding energies. Any indication of quasi-free protons (H${}^{+}$) “jumping” between water molecules, and/or translating in the channels of the proton conductor H${}_3$OSbTeO${}_6$ is absent in the data. The characteristic scattering time of NCS is shorter than that of INS, due to the higher incident energies of NCS being the the range 1–100 eV. It is expected that the proposed new INS experiment on H${}_3$OSbTeO${}_6$ should show an “anomalous” vibro-translational motion of H${}_3$O${}^{+}$, in an analogous way as the INS experiments in the preceding section show the striking roto-translational motion of the H${}_2$ molecule.

If the aforementioned prediction would be confirmed by the INS experiment, this would demonstrate that the outstanding high conductivity of H${}_3$OSbTeO${}_6$ is intrinsically connected with the quantum effect of momentum transfer deficit [11] on the H protons of H${}_3$O${}^{+}$. This knowledge would facilitate further investigations for the optimization of fast proton conductors.

7.2. INS from H of Metal-Hydrogen Systems

Structural and dynamical effects of H in various metals has attracted considerable interest in fundamental and applied research areas for many decades. For instance, Palladium (Pd) and Niobium (Nb) are interesting, because they can absorb large quantities of hydrogen, and H-atoms are highly mobile in some lattices, even at room temperature. The physical nature of this high mobility is not fully understood thus far [31]. Metallic hydrides is also of certain technological and/or industrial relevance, since they have been suggested to represent possible hydrogen storage materials at room temperature.

For example, INS has been used to study the dynamics of hydrogen in Pd (even at room temperature), and such studies provided many insights into the microscopic dynamics of H atoms in this system; see e.g., Ref. [20] and the extensive literature cited therein. Of particular interest for our purposes is the fact that, in this metallic hydride, INS takes place from atomic H—which extends the aforementioned class of experiments on INS from molecular hydrogen. Of course, there are no rotations present in in the PdH${}_x$ system, and so any spectral contribution of the translational modes of H will be coupled with the vibrational modes of H against Pd.

In view ot the preceding experimental and theoretical considerations, a  two-dimensional INS experiment to measure the $S(K,E)$ of PdH${}_x$, e.g., in nanoparticles, would be worth studying. Depending on the neutron initial energy $E_0$ (say, a few 100 meV), various coupled vibrational-translational modes of atomic H become experimentally accessible. We again may expect a qualitatively similar anomaly as in the INS experiments on H${}_2$ in C-nanotubes [24] (see Section 5), namely:

• INS measurements should reveal an anomalous momentum-transfer deficit on the H-atoms, being qualitatively similar to the effect of H${}_2$ in nanoporous materials of Figure 5 discussed above. The vibro-translational quasi-continuum—being analogous to the roto-translational one [24]—should be again in the quantum domain of the $K-E$, i.e., “on the left” of the theoretical free-H recoil parabola.

Also in this case, the possible observation of any momentum transfer “anomalies” would shed new light onto the quantum interactions and dynamics of H with the host metals, and reveal new aspects of their quantum nature that present-day quantum-chemical calculations cannot capture.

7.3. INS from Single H${}_2$ in C${}_{60}$

It is widely believed that INS is not subject to any selection rules; see e.g., the INS textbook [17]. However, an unexpected selection rule was recently discovered, which concerns the INS spectroscopy of a H${}_2$ in a near-spherical confinement, i.e.,  a C${}_{60}$ fullerene. This new selection rule concerns the rotation-translation dynamics of the H${}_2$ molecule in C${}_{60}$, and  results in certain forbidden INS roto-translational transitions of H${}_2$ which have been also experimentally confirmed; see Refs. [32,33] and papers cited therein.

This INS topic is also of particular interest, if one recalls that the above investigated momentum-transfer deficit effect of H${}_2$ in C-nanotubes (see Section 4) does also affect the roto-translational H${}_2$-motion. Thus it seems natural to wonder, whether some anomalous momentum transfer might be present in the H${}_2$-in-C${}_{60}$ system, too. Hence, in the light of the WV-TSVF theoretical model mentioned above,

• we may expect a related “anomalous” momentum transfer deficit in H${}_2$-in-C${}_{60}$ too. Since the confinement provided by C${}_{60}$ allows only for a very small quasi-translation within the C${}_{60}$-cage, we may expect that the expected quantum momentum-transfer deficit of measurable roto-translational peaks should be much less pronounced that in the case of H${}_2$-in-C-nanotube (Section 4).

It may be noted that the quantum calculations reported in Ref. [32] can only calculate E-transfers, as all quantum chemical calculations do. But  momentum transfers are beyond the reach of present-day numerical quantum chemical methods. This effect, if observable, will indicate a new aspect of the selection rule of rotation-translation INS-dynamics.

7.4. Momentum Transfer Deficit in Scattering by Multi-Layered Structures

Recently in Ref. [12] we suggested that the momentum-transfer deficit effect could probably be observed

(1)

by neutron reflectivity from thin layers and/or coated surfaces of solid-liquid interfaces (e.g., Si/SiO${}_2$/water), and

(2)

X-ray diffraction from crystalline structures (e.g., Si and LaB${}_6$ powders). Additionally, some preliminary measurements indicating a non-conventional component in the measured signals did provide some hints of the predicted new momentum-transfer deficit effect under consideration. In both these mentioned experimental fields, the expected deviations from conventional theory would have no conventional theoretical interpretation.

For some related preliminary data and results, the interested read may consult Ref. [12].

8. Additional Remarks and Discussion

The theory of WV and the TSVF are nowadays active areas of research, providing new conceptual ideas and suggesting new experiments within a wide range of fields. In the present paper, however, interpretational issues are briefly mentioned only; see e.g., Appendix B. Instead, the focus is a new family of experiments related to momentum and/or energy transfers, thus contributing to the connection between the “theory”-community and that of experimental neutron scattering.

This novel theoretical frame reveals several counter-intuitive effects which may appear “strange” or even “wrong”, since, e.g.,—in view of the conventional neutron scattering theory [9]—they seem to violate the basic conservation laws of energy and momentum. In order to remove this and similar misunderstandings, and to facilitate further understanding, here we make several additional remarks and/or provide related explanations.

(A) We have discussed in detail why the striking experimental INS observations of Section 5 appear to be “anomalous” (i.e., they have no interpretation) in the frame of standard INS theory. In this context, one may believe that these observations can also be interpreted within “standard” formalism of QM, i.e without the need to apply WV and TSVF (as proposed in Section 6). In other terms, it is conceivable that standard INS theory could be properly extended, with the aim to interpret the INS observations using only “traditional” quantum theoretical terms.

Of course, our present work does not exclude this possibility. However, in the light of the detailed discussions of Section 4, it should be clear that any such extension of conventional INS theory has to violate and/or replace (some of) the basic INS-theoretical conditions $A1$–$A5$. Such an extension is unknown thus far.

(B) In some theoretical papers one finds the criticism that post-selection just means “throwing out some data”, and thus that weak values are not inherently quantum but rather a purely statistical feature of pre- and post-selection with disturbance; cf. [34]. As an example of other criticisms, see e.g., the early comment [35] on the original paper [2]. Furthermore, it has been argued that the conditions under which WV can be related to the system properties are rather stringent; for a recent discussion, the interested reader is referred to Ref. [36].

In the experimental context of INS and related scattering experiments discussed above, however, post-selection certainly corresponds to “performing a concrete measurement on a well positioned sample (with respect to the neutron beam direction), using a well defined detector, and analyzing the thus measured data only”; see e.g., the experiments in Section 5. The many applications and/or new predicted effects of WV-TSVF clearly support the viewpoint that post-selection is a genuine physical concept. Furthermore, and more specifically, the short analysis of the photon-mirror collision in a MZI, as presented in Section 2.1, provides considerable evidence that the TSVF part of the theory is an indispensable tool for the understanding of some new phenomena.

Recently, de Castro et al. [37] provided a broader theoretical frame of weak values and advanced formal collision theory [38], aiming to apply it in various experimental set-ups as e.g., in the case of ultracold atomic collisions.

(C) Nowadays it is recognized that a Weak Value is a unique quantum interference phenomenon of a single system with itself; see e.g., [39]. As the above INS experiments and their interpretation strongly indicate, a WV is a novel robust and well-defined property of a single pre- and post-selected quantum system. WVs may also play a direct role in the quantum dynamics of a single system. Consequently, a WV value appears to have a physical meaning beyond a conditional average of a pointer in a weak measurement procedure [40].

(D) The fact that the neutron-H collision is not immediately recognized to correspond to a weak measurement (WM)—which is believed to be a prerequisite for the applicability of WV-theory—was discussed in some detail in the original paper  [11]. Here it suffices to note the following points. Generalizing the WM formalism, Oreshkov and Brun [41] showed that WMs are universal, in the sense that any generalized measurement can be decomposed into a sequence of appropriately designed weak measurements. This interesting theoretical result may have important applications to various topics, like e.g., the INS process under consideration. In this context, a remarkable finding of Cohen and Pollak [42] should be mentioned, which says that, in principle, a WV can also be determined by strong (projective) measurements of suitably defined Hermitian  operators.

Additionally, the experimental and theoretical investigations by Dziewior et al. [43] may be mentioned, which also explicitly apply the formalism of WVs to the case of much stronger interactions than WMs. Here the authors obtain a novel expression for WVs which takes into account changes due to interactions of finite strength in the time interval between pre- and post-selection. Moreover, besides incorporating the stronger interactions, the formalism also accounts for decoherence and/or imperfections in the measurement system [43].

Summarizing, weak values are general properties of observables (operators) in association with the physical concepts of pre- and post-selection, and there is no need to associate them with WMs only.

(E) These views are also in line with the following specific observation. In our theoretical model of Section 6, the two operators $\widehat{q}$ and $\widehat{P}$ occurring in the von Neumann-type interaction Hamiltonian of Equation (20) refer to two different quantum systems. Thus, as Vaidman [44] pointed out, the concept of WV arises here due to the interference of a quantum entangled wave and therefore it has no analog in classical wave interference. This further supports the conclusion that WV is a genuinely quantum concept and not some kind of “approximation”.

(F) In contrast to our WV-TSVF model, conventional neutron scattering theory [9,16,17] treats the neutron as a classical plane wave, i.e., no dynamical variables of the neutron appear in the derivations and the main theoretical results. e.g., consider the basic formula for the partial differential cross-section, which includes the dynamic structure factor $S(K,E)$ of the scattering system. This contains degrees-of-freedom of the scatterer only—the neutron is “downgraded” to a classical object and only the scattering length $b_A$ (a c-number) describing the neutron-atom interaction appears here.

Moreover, let us also mention shortly that, in the conventional theory and experimental practice of neutron scattering, the coherence length of the neutron is believed to be fully irrelevant; cf. e.g., the above cited references. However, as one could conjecture in the light of the anomalous momentum-transfer deficit effect, this belief becomes questionable.

(G) The momentum transfer deficit, and the associated “anomalous” effective-mass reduction of the scattering particle, say H, might be seen to violate the energy and momentum conservation laws of basic physics. However, as already stressed above, this is not the case, because the neutron-H system is not a closed, but an open quantum system. Thus we may say that the participation of H’s environment to the neutron-H scattering dynamics is indispensable for the new WV-TSVF effect under consideration. In other terms, the correct conservation relations should read

$$E_{H+\mathcal{E}}=-E_n\mathrm{and}\hslash {\mathbf{K}}_{H+\mathcal{E}}=-\hslash {\mathbf{K}}_n,$$

(33)

which are valid in the case that the environment $\mathcal{E}$ of the scattering H is present; see also Appendix A. In contrast, the common view (at least for experimenters) in this context is quite different: One believes that it holds $E_H=-E_n$ and ${\mathbf{K}}_H=-\hslash {\mathbf{K}}_n$, mainly based on the belief that the n-proton scattering is extremely fast, or even “instantaneous”; see also Appendix A.

(H) Moreover, let us mention in this context the very recent publication by Aharonov et al. [45] regarding a revealing theoretical analysis of the very meaning of the basic conservation laws of quantum mechanics. The reason for doing this, in connection with the momentum-transfer deficit effect of the present paper, is the following stimulating physical insight:

“Energy … is a property of an entire wave function. However, the law of conservation of energy is often applied to processes in which a system with an extended wave function interacts with a local probe. How can the local probe “see” an extended wave function? What determines the change in energy of the local probe? These questions lead us to uncover quantum processes that seem, paradoxically, not to conserve energy” [45], p. 1

Here one may easily recognize an (intuitive, at least) analogy with the specifics of our INS process. Namely, in the case of H${}_2$ in C-nanotubes, a neutron interacts with a proton “locally” (since the Fermi-pseudo-potential has a range of some ${10}^{-15}$ m [9]), whereas the wavefunction of the roto-translating H${}_2$ is extended over a few Å, which is ca. ${10}^5$ times larger.

(I) Quantum interference and entanglement [46] have been recognized to be crucial for quantum computing and quantum information theory; cf. [47] and the very recent Refs. [48,49]. Hence it would be interesting to explore a potential applicability of the new quantum features offered by WV-TSVF to recently emerging fields of quantum computational complexity theory [50] and ICT (information and communication technology). Namely, WV-TSVF appears to make possible the exploration of a new kind of quantum information—like e.g.,  the correlation between momentum transferred on the mirror M and the photon measured with detector D${}_2$ in the example of Section 2.

(J) In view of the experimentally detected effective-mass reduction, Section 5, it seems appropriate to mention here some speculative ideas concerning a possible technological importance of the new theory. Namely, H${}^{+}$ mobility concerns the properties of fuel cells (hydrogen technology) and carbon-based materials for H-storage. Cf. the proposed experiment in Section 7.1.

(K) In view of the above discussions and experimental INS results, one may arrive at the following point of considerable interest: The quantum nature of nuclei (here: protons) may be far more important, and more widely present, that realized thus far. The strong roto-translational “anomaly” of H${}_2$ in C-nanotubes indicates that the H-substrate interactions are far more complex than the common quantum-chemical calculations can capture. (The latter consider only the electrons as quantum objects.) In other terms, it seems that the importance of the quantum nature of protons for the chemical properties of H/surface interactions (and thus also on the chemical bonding in general) should be reconsidered.

Concluding, we believe that the theoretical formalism of WV and TSVF not only sheds new light on interpretational issues of fundamental quantum concepts (like e.g., the time-inversion invariance of the basic physical laws, the meaning of quantum entanglement and correlations, etc.) but it also offers a fascinating new guide for our intuition to predict new effects, and further helps to design and perform new experiments and reveal novel quantum phenomena.