#
Weak Values and Two-State Vector Formalism in Elementary Scattering and Reflectivity—A New Effect^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Motivation

#### Short Remarks on the General Theory

## 3. Elementary Neutron-Atom Scattering in View of WV and TSVF

#### 3.1. Model Hamiltonian

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

#### 3.3. Plane Waves Approximation and Conventional Momentum Transfer

## 4. Experimental Context—Incoherent Scattering

#### 4.1. Inelastic Neutron Scattering from Protons

#### 4.2. Effective Mass as Measured in the Scattering Experiment

#### 4.3. An Experiment on the New Scattering Effect—INS from Single H${}_{2}$ Molecules in C-Nanotubes

## 5. Experimental Context—Reflectivity (Coherent Scattering)

## 6. A Preliminary XRD Result—Bragg Scattering

## 7. Discussion

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

WV | Weak Value |

WM | Weak Measurement |

TSVF | Two-State Vector Formalism |

QE | Quantum Entanglement |

MZI | Mach–Zehnder Interferometer |

FSE | Final-State Effects |

IA | Impulse Approximation |

IINS | Inelastic Incoherent Neutron Scattering |

INS | Equivalent to IINS |

DINS | Deep-Inelastic Neutron Scattering |

NCS | Neutron Compton Scattering, equivalent to DINS |

TOF | Time-of-Flight |

a.m.u. | Atomic Mass Unit |

meV | Milli-Electron Volt |

XRD | X-Ray Diffraction |

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**Figure 2.**Schematic presentation of a time-of-flight neutron scattering setup. A detector, being a spatially localized system, performs a post-selection on the neutron’s final state. (Taken from [10], with permission from Quanta.)

**Figure 3.**Deep-inelastic neutron scattering (DINS) from chemisorbed H in C${}_{8}$K; data adapted from [21]. The maxima of the H peaks are about 2620 meV The vertical (red) line shows the position of the $M=1$ a.m.u. recoil peak (at 3540 meV), assuming that the conventional theory, in particular the IA, is valid. Black points and fitted line: KH, Grey points and fitted line: C${}_{8}$K${}_{0.9}$H${}_{0.13}$. The spectra were recorded with a selected final neutron energy ${E}_{f}=4280$ meV. The peak maxima correspond to ${M}_{eff}=1.2$ a.m.u. (Reproduced from Ref. [22], with permission from Institute of Physics (IOP).)

**Figure 4.**Experimental INS results from H${}_{2}$ in carbon nanotubes, with incident neutron energy ${E}_{i}=90$ meV; adapted from Figure 1 of Ref. [23]. The translation motion of the recoiling H${}_{2}$ molecules causes the observed continuum of intensity, usually called “roto-recoil” (white-blue ribbon), starting at the well visible first rotational excitation of H${}_{2}$ being centered at ${E}_{rot}\approx 14.7$ meV and ${K}_{rot}\approx 2.7$ Å${}^{-1}$ (blue ellipsoid). The $K-E$ position of the latter is in agreement with conventional theory. In contrast, a detailed fit (red parabola; full line) to the roto-recoil data reveals a strong reduction of the effective mass of recoiling H${}_{2}$, which appears to be only 0.64 a.m.u. (The red dashed line, right parabola) represents the conventional-theoretical parabola with effective mass 2 a.m.u.) For details of data analysis, see [23]. (Reproduced from Ref. [10], with permission from Quanta.)

**Figure 5.**Experimental neutron reflectivity ($R\left(Q\right)$; log-scale) from Si/water interface. (

**A**) measured and fitted reflectivity data for the Si/liquid interface of H${}_{2}$O-D${}_{2}$O mixture with D molar fraction ${x}_{D}=0.6$ at room temperature; (

**B**) the observed “anomalous” increase of the scattering length density of the liquid at the solid–liquid interface for the additional layer models investigated in [25]: Relative deviation $\Delta {\rho}_{I}$ of the observed scattering length density ${\rho}_{I}$ of the (assumed) fictitious layer at the interface (with thickness ${d}_{I}\approx 50$ Å) from the conventionally expected scattering length density ${\rho}_{th}$ of the bulk liquid, for various H${}_{2}$O-D${}_{2}$O mixtures. (Reproduced from [25], with permission from Elsevier.)

**Figure 6.**Preliminary results (left: Si, right: LaB${}_{6}$)): Measured lattice constants of Si and LaB${}_{6}$ (powders; both cubic), determined individually from the scattering angle $2\theta $ of every Bragg peak appearing in the X-ray diffraction (XRD) data. That is, the results shown in the Figures contain no fitting parameter. Results of two independent experiments for each material are shown. The typical error of the determined values of the lattice constant is given by the difference between the two sets of results in each Figure.

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**MDPI and ACS Style**

Chatzidimitriou-Dreismann, C.A.
Weak Values and Two-State Vector Formalism in Elementary Scattering and Reflectivity—A New Effect. *Universe* **2019**, *5*, 58.
https://doi.org/10.3390/universe5020058

**AMA Style**

Chatzidimitriou-Dreismann CA.
Weak Values and Two-State Vector Formalism in Elementary Scattering and Reflectivity—A New Effect. *Universe*. 2019; 5(2):58.
https://doi.org/10.3390/universe5020058

**Chicago/Turabian Style**

Chatzidimitriou-Dreismann, C. Aris.
2019. "Weak Values and Two-State Vector Formalism in Elementary Scattering and Reflectivity—A New Effect" *Universe* 5, no. 2: 58.
https://doi.org/10.3390/universe5020058