Hbar Research Articles

CODATA recommended values of the fundamental physical constants: 2018.

We report the 2018 self-consistent values of constants and conversion factors of physics and chemistry recommended by the Committee on Data of the International Science Council (CODATA). The recommended values can also be found at physics.nist.gov/constants. The values are based on a least-squares adjustment that takes into account all theoretical and experimental data available through 31 December 2018. A discussion of the major improvements as well as inconsistencies within the data is given. The former include a decrease in the uncertainty of the dimensionless fine-structure constant and a nearly two orders of magnitude improvement of particle masses expressed in units of kg due to the transition to the revised International System of Units (SI) with an exact value for the Planck constant. Further, because the elementary charge, Boltzmann constant, and Avogadro constant also have exact values in the revised SI, many other constants are either exact or have significantly reduced uncertainties. Inconsistencies remain for the gravitational constant and the muon magnetic-moment anomaly. The proton charge radius puzzle has been partially resolved by improved measurements of hydrogen energy levels.

First-Principles Calculation of the Optical Rotatory Power of Periodic Systems: Application on α-Quartz, Tartaric Acid Crystal, and Chiral (n,m)-Carbon Nanotubes.

The self-consistent coupled-perturbed (SC-CP) method in the CRYSTAL program has been adapted to obtain electromagnetic optical rotation properties of chiral periodic systems based on the calculation of the magnetic moment induced by the electric field. Toward that end, an expression for the magnetic transition moment is developed, which involves an appropriate electronic angular momentum operator. This operator is forced to be hermitian so that the chiroptical properties are real. In our formulation, the trace of the optical rotatory power matrix is gauge-origin-invariant as long as the electric dipole transition matrix elements are obtained using the velocity (rather than position) operator. On the other hand, the component along the optic axis is invariant in general for uniaxial and biaxial crystals. Under the same conditions, these properties also do not depend on the so-called missing integers that occur in the treatment of the electric dipole moment of quasi-one-dimensional periodic systems or the analogue of missing integers for the case of higher dimensionality. Tests on a model H2O2 polymer confirm the formalism and, as desired, show that the calculated properties are independent of the size and definition of the unit cell. In addition, an empirical relation to a finite oligomer gauge-including atomic orbital (GIAO) calculation is found. Applications, with comparison to experiment, are carried for α-quartz, tartaric acid crystal, and carbon nanotubes. Future developments of this initial approach to chiroptical properties in the solid state are noted.

Correspondence between classical and quantum resonances

Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li⇆Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to ℏ=0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit ℏ→0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.

Is the Devil in h?

This note is a part of my effort to rid quantum mechanics (QM) nonlocality. Quantum nonlocality is a two faced Janus: one face is a genuine quantum mechanical nonlocality (defined by the Lüders’ projection postulate). Another face is the nonlocality of the hidden variables model that was invented by Bell. This paper is devoted the deconstruction of the latter. The main casualty of Bell’s model is that it straightforwardly contradicts Heisenberg’s uncertainty and Bohr’s complementarity principles generally. Thus, we do not criticize the derivation or interpretation of the Bell inequality (as was done by numerous authors). Our critique is directed against the model as such. The original Einstein-Podolsky-Rosen (EPR) argument assumed the Heisenberg’s principle without questioning its validity. Hence, the arguments of EPR and Bell differ crucially, and it is necessary to establish the physical ground of the aforementioned principles. This is the quantum postulate: the existence of an indivisible quantum of action given by the Planck constant. Bell’s approach with hidden variables implicitly implies rejection of the quantum postulate, since the latter is the basis of the reference principles.

Natural and magnetically induced entanglement of hyperfine-structure states in atomic hydrogen

The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron and nuclear spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, whose two-qubit concurrence and quantum coherence can be expressed in terms of the fundamental physical constants -- the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a $\tau_c$ $\approx$ 5.35 $\mu$eV threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement. As one of the central findings of this work, we show that an external magnetic field can induce and sustain an HFS entanglement, against all the odds of thermal effects, at temperatures well above the $\tau_c$ threshold, thus enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.

Realizing the Kilogram from the Planck Constant: The Kibble Balance and the Electrical Units

The international System of Units (SI) was redefined in 2018 [1]. This historical event implemented a beautiful idea—a system of units defined entirely by constants of nature. The most notable modification to the SI involved abandoning the artifact definition of the kilogram in favor of fixing the numerical value of the Planck constant. Apart from its physical and mathematical elegance, the new SI solves the problem of providing an absolute reference for the kilogram. But, how do we understand the link between the kilogram and the Planck constant? And, how can we realize the kilogram now, using the new definition? This paper attempts to give a simple answer to these complicated questions.

Entropic uncertainty principle, partition function and holographic principle derived from Liouville’s Theorem

An entropic version of Liouville’s Theorem is defined in terms of the conjugate variables (“hyperbolic position” and “entropic momentum”) of an entropic Hamiltonian. It is used to derive the Holographic Principle as applied to holomorphic structures that represent maximum entropy configurations. The Bekenstein-Hawking expression for black hole entropy is a consequence. Based on the entropic commutator derived from Liouville’s Theorem and the same entropic conjugate variables, an entropic Uncertainty Principle (in units of Boltzmann’s constant) isomorphic to the kinematic Uncertainty Principle (in units of Planck’s constant) is also derived. These formal developments underpin the previous treatment of Quantitative Geometrical Thermodynamics (QGT) which has established (entirely on geometric entropy grounds) the stability of the double-helix, the double logarithmic spiral, and the sphere. Since in the QGT formalism the Boltzmann and Planck constants are quanta of quantities orthogonal to each other in Minkowski spacetime, a solution of the Schrödinger Equation is demonstrated isomorphic to a probability term of an entropic Partition Function, where both are defined by path integrals obeying the stationary principle: this isomorphism represents an important symmetry of the formalism. The geometry of a holomorphic structure must also exhibit at least C2 symmetry.

WKB electron wave functions in a tightly focused laser beam

Available laser technology is opening the possibility of testing QED experimentally in the so-called strong-field regime. This calls for developing theoretical tools to investigate strong-field QED processes in electromagnetic fields of complex spacetime structure. Here, we propose a scheme to compute electron wave functions in tightly focused laser beams by taking into account exactly the complex spacetime structure of the fields. The scheme is solely based on the validity of the Wentzel-Kramers-Brillouin (WKB) approximation and the resulting wave functions, unlike previously proposed ones [Phys. Rev. Lett. \textbf{113}, 040402 (2014)], do not rely on approximations on the classical electron trajectory. Moreover, a consistent procedure is indicated to take into account higher-order quantum effects within the WKB approach depending on higher-and-higher powers of the Planck constant. In the case of a plane-wave background field the found wave functions exactly reduce to the Volkov states, which are then written in a new and fully quasiclassical form. Finally, by using the leading-order WKB wave functions to compute the probabilities of nonlinear Compton scattering and nonlinear Breit-Wheeler pair production, it is explicitly shown that, if additionally the energies of the charges are sufficiently large that the latter are not significantly deflected by the field, the corresponding Baier's formulas are exactly reproduced for an otherwise arbitrary classical electron/positron trajectory.

Losing the trace to find dynamical Newton or Planck constants

We show that promoting the trace part of the Einstein equations toa trivial identity results in the Newton constant being an integrationconstant. Thus, in this formulation the Newton constant is a globaldynamical degree of freedom which is also a subject to quantizationand quantum fluctuations. This is similar to what happens to the cosmologicalconstant in the unimodular gravity where the trace part of the Einsteinequations is lost in a different way. We introduce a constrained variationalformulation of these modified Einstein equations. Then, drawing onanalogies with the Henneaux-Teitelboim action for unimodular gravity,we construct different general-covariant actions resulting in thesedynamics. The inverse of dynamical Newton constant is canonicallyconjugated to the Ricci scalar integrated over spacetime. Surprisingly,instead of the dynamical Newton constant one can formulate an equivalenttheory with a dynamical Planck constant. Finally, we show that anaxion-like field can play a role of the gravitational Newton constantor even of the quantum Planck constant.

Procedimiento didáctico para el estudio de la fórmula de Planck en carreras de ingeniería

En este artículo se analiza la ley de distribución de energía en el espectro de la radiación del cuerpo negro, y la fórmula de Planck que la sintetiza, un descubrimiento revolucionario, que marca el nacimiento de la teoría cuántica. Se propone un procedimiento didáctico para el estudio de la fórmula de Planck en carreras de ingeniería, siguiendo la secuencia del ciclo de la creación científica: Hechos → Modelo-hipótesis → Consecuencias → Verificación experimental. Se ejemplifican los pasos metodológicos y los elementos esenciales de la idea básica de la cuantización de la energía, con notas históricas que facilitan la comprensión del fenómeno estudiado y el papel que jugaron los científicos en su determinación. Se consideran hechos iniciales: Las leyes de Kirchhoff, Stefan-Boltzmann y Wien, además de la fórmula de Rayleigh-Jeans y se desarrollan las ideas fundamentales que permitieron a Planck encontrar la solución teórica, que culmina en 1900 con la propuesta del cuanto de acción, expresado por la constante h que lleva su nombre. De la fórmula de Planck, se deducen las leyes antecedentes y se obtienen otras relaciones de importancia para la física y la química. Para cerrar el ciclo de la creación científica, se realiza la verificación experimental, con la medición de la radiación de fondo cósmico, que se comporta como la radiación de un cuerpo negro a una temperatura de 2,73 K. Se concluye con ejemplos de aplicaciones de la radiación térmica, que involucran algunos conceptos básicos de la elaboración teórica de Planck: Energía solar, recubrimientos térmicos y pirometría.

Pancharatnam metals with integer and fractional quantum Hall effects

ABSTRACT Weakly interacting Fermi liquid can exhibit integer or fractional quantum Hall effect at sufficiently low temperatures, and in the presence of variable applied magnetic and/or electric fields. The said effects have been captured only incompletely by means of the Klitzing conductance formula, multiplied by the dimensionless filling factor (integer or fractional), ν because scattering rate cannot be absent for any conductor that is neither a perfect conductor at zero Kelvin (in the absence of phonons, scattering rate, Cooper pairing and Meissner effect) nor a superconductor. Here, we shall derive the proper microscopic resistance formula based on the notion of Pancharatnam phase retardation that captures both integer and fractional quantum Hall effects. We then compute the Planck constant from the Pancharatnam resistance formula, as well as provide a formal proof for this new formula. We shall unambiguously elaborate why the Pancharatnam resistance formula gives the complete physics of electron transport phenomena responsible for the quantum Hall effects, which include the changes to longitudinal and Hall resistance, as well as the temperature effect.

Global Well-Posedness, Blow-Up and Stability of Standing Waves for Supercritical NLS with Rotation

We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation $$\begin{aligned} iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega }u,\quad (x,t)\in {\mathbb {R}}^{N}\times {\mathbb {R}}, \end{aligned}$$where \(N=2\) or 3 and V(x) is an anisotropic harmonic potential. Here \(L_{\Omega }\) is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are \(L^{2}\)-supercritical.

A program for simplifying summation of Wigner 3[formula omitted]-symbols

The program presented in this paper aims to simplify formulas involving products of a large amount of Wigner 3j-symbols summed over various magnetic quantum numbers. The algorithm used in our program is based on the graphical techniques originally developed by Yutsis, Levinson, and Vanagas, and later revised by some others. The output of our program is expressed as a weighted sum of products of 3j-, 6j- and 9j-symbols, with appropriate phase factors embedded. Compared with the previous literature, the present program is more flexible in a sense that it can deal with not only closed diagrams, such as the angular momentum recoupling coefficients, but also open diagrams. Our program is particularly useful to high precision atomic and molecular structure calculations using correlated electron–electron coordinates. Program summaryProgram Title: Reduce3jCPC Library link to program files:https://doi.org/10.17632/9zmx4b5sdd.1Licensing provisions: BSD 3-clauseProgramming language: Python 3.6.5Nature of problem: In atomic and molecular structure calculations, one often encounters expressions containing products of a large amount of Wigner 3j-symbols summed over various magnetic quantum numbers. It is desirable to simplify these expressions to make the final results compact, which is the goal of this paper.Solution method: The program uses the graphical techniques of quantum mechanical angular momentum theory, originally developed by Yutsis, Levinson, and Vanagas, and later modified by Brink and Satchler. The program is implemented in Python language, with its package “Networkx” extensively used. A friendly graphical input and output interface is developed. The input can be either a recoupling coefficient or an arbitrary expression containing products of the 3j-symbols summed over various magnetic quantum numbers. A strategically designed optimization method is also developed to help achieve our goal. Our program has significantly expanded the scope of usefulness of the existing programs so that it can now meet most of the demands from high precision atomic and molecular structure calculations.Additional comments including restrictions and unusual features: Reduce3j can effectively handle any open or closed graph, as long as the graph has more than one free line, and all the magnetic angular momentum quantum numbers from internal lines are summed over.

Ehrenfest Time at the Transition from Integrable Motion to Chaotic MotionSupported by the National Key R&D Program of China (Grants Nos. 2017YFA0303302 and 2018YFA0305602), the National Natural Science Foundation of China (Grant No. 11921005), and the Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01).

Ehrenfest time depends differently on the Planck constant in integrable motion and chaotic motion. We study how its dependence on the Planck constant changes when there is a continuous transition from regular motion to chaotic motion. We find that the dependence is a weighted compromise between its two distinct dependences in regular and chaotic motions. The study is carried out with the system of periodically driven anharmonic oscillator. As the system is quite typical, the result may apply generally.

All waves have a zero tunneling time

Abstract Zero tunneling time and thereby a faster than light traversal velocity was calculated nearly a hundred years ago and has been observed recently. We report about experimental results and estimations, which confirm the zero time tunneling for elastic as well as for electromagnetic and Schrödinger waves. Zero time tunneling was first observed with microwaves 1992 (H. Aichmann and G. Nimtz, Found. Phys., vol. 44, p. 678, 2014; A. Enders and G. Nimtz, J. Phys. I, vol. 2, p. 169, 1992). In 2008, zero time was also observed for tunneling electrons (P. Eckle, A. N. Pfeiffer, C. Cirelli, et al., Science, vol. 322, p. 1525, 2008). Presumably, this effect took place with atoms quite recently (R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature, vol. 583, p. 529, 2020). The Einstein relation E 2 = (ħk)2 c 2 is not satisfied in the tunneling process, since the wave number k is imaginary (E is the total energy, ħ the Planck constant, and c the vacuum velocity of light), Zero time tunneling is described by virtual photons (A. Stahlhofen and G. Nimtz, Europhys. Lett., vol. 76, p. 189, 2006). The tunneling process itself violates the Special Theory of Relativity. Remarkably, Brillouin conjectured that wave mechanics is valid for all waves independent of their field (L. Brillouin, Wave Propagation in Periodic Structures, Chap. VIII, New York, Dover Publications, 1953).

Change of energy and magnetic moment of a quantum charged particle after a fast jump of the magnetic field in solenoids of arbitrary cross sections

We consider a quantum charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the most general linear vector potential of the form A=B(t)−y(1+α),x(1−α)∕2. Such potentials with α≠0 can be created inside infinite solenoids with non-circular cross sections. The physical properties, such as energy and magnetic moment, do not depend on the choice of gauge in the thermodynamic equilibrium state. But systems with different values of α are not equivalent for nonstationary magnetic fields, due to different structures of induced electric fields. Using the approximation of the stepwise variation of the magnetic field, we obtain explicit formulas describing the change of the energy and magnetic moment from the initial equilibrium state, created with the aid of a weak anisotropic harmonic potential. The results are quite different for different values of the gauge parameter α, as well as for the initial high temperature and low temperature states. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields. In addition, the magnetic moment becomes a strongly oscillating function of time after the jump of the field. Strong fluctuations of the magnetic moment (described in terms of the variance) are discovered in all regimes, including the initial equilibrium state. These fluctuations are sensitive to the shape of the trap confining the particle initially. Moreover, these fluctuations do not depend on the Planck constant in the high temperature case, and the magnetic moment variance is much bigger than the square of mean magnetic moment.

Quantum postulate vs. quantum nonlocality: on the role of the Planck constant in Bell\u2019s argument

We present a quantum mechanical (QM) analysis of Bell’s approach to quantum foundations based on his hidden-variable model. We claim and try to justify that the Bell model contradicts to the Heinsenberg’s uncertainty and Bohr’s complementarity principles. The aim of this note is to point to the physical seed of the aforementioned principles. This is the Bohr’s quantum postulate: the existence of indivisible quantum of action given by the Planck constant h. By contradicting these basic principles of QM, Bell’s model implies rejection of this postulate as well. Thus, this hidden-variable model contradicts not only the QM-formalism, but also the fundamental feature of the quantum world discovered by Planck.

Explaining to different audiences the new definition and experimental realizations of the kilogram

Abstract. Different options were discussed before reaching the final agreement on the new definitions of the SI units, effective from 20 May 2019, especially with regard to the kilogram, now defined in terms of the numerical value of the Planck constant (h). Replacing the artefact definition of the kilogram with a new one based on the mass of a particle, or the atomic mass constant (mu), would have been preferable for ease of understanding, among other reasons. In this paper we discuss some limitations of teaching to different audiences what a kilogram is in the redefined International System of Units (SI), including realizations of the new definition.

On the Asymptotic Dynamics of 2-D Magnetic Quantum Systems

In this work we provide results on the long time localisation in space (dynamical localisation) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form $H=H_0+W$, where $H_0$ is rotationally symmetric, has dense point spectrum, and $W$ is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.

WKB Estimate of Bilayer Graphene's Magic Twist Angles.

Graphene bilayers exhibit zero-energy flatbands at a discrete series of magic twist angles. In the absence of intrasublattice interlayer hopping, zero-energy states satisfy a Dirac equation with a non-Abelian SU(2) gauge potential that cannot be diagonalized globally. We develop a semiclassical WKB approximation scheme for this Dirac equation by introducing a dimensionless Planck's constant proportional to the twist angle, solving the linearized Dirac equation around AB and BA turning points, and connecting Airy function solutions via bulk WKB wave functions. We find zero-energy solutions at a discrete set of values of the dimensionless Planck's constant, which we obtain analytically. Our analytic flatband twist angles correspond closely to those determined numerically in previous work.