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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 1,063 records · Page 59
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Measurement of the production and elliptic flow of (anti)nuclei in Xe-Xe collisions at √sNN = 5.44 TeV

Measurements of (anti)deuteron and (anti) 3He production in the rapidity range |𝑦|<0.5 as a function of the transverse momentum and event multiplicity in Xe-Xe collisions at a center-of-mass energy per nucleon-nucleon pair of √𝑠𝑁⁢𝑁=5.44 TeV are presented. The coalescence parameters 𝐵2 and 𝐵3 are measured as a function of the transverse momentum per nucleon. The ratios between (anti)deuteron and (anti) 3He yields and those of (anti)protons and pions are reported as a function of the mean charged-particle multiplicity density and compared with two implementations of the statistical hadronization model and with coalescence predictions. The elliptic flow of (anti)deuterons is measured for the first time in Xe-Xe collisions and shows features similar to those already observed in Pb-Pb collisions, i.e., the mass ordering at low transverse momentum and the meson-baryon grouping at intermediate transverse momentum. The production of nuclei is particularly sensitive to the chemical freeze-out temperature of the system created in the collision, which is extracted from a grand-canonical-ensemble-based thermal fit, performed for the first time including light nuclei along with light-flavor hadrons in Xe-Xe collisions. The extracted chemical freeze-out temperature 𝑇chem=(154.2±1.1) MeV in Xe-Xe collisions is similar to that observed in Pb-Pb collisions and close to the crossover temperature predicted by lattice quantum chromodynamics calculations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Semitauonic b-hadron decays: A lepton flavor universality laboratory

The study of lepton flavor universality violation (LFUV) in semitauonic b-hadron decays has become increasingly important in light of long-standing anomalies in their measured branching fractions, and the large datasets anticipated from the LHC experiments and Belle II. Here, in this review, a comprehensive survey of the experimental environments and methodologies for semitauonic LFUV measurements at the B factories and LHCb is undertaken, along with an overview of the theoretical foundations and predictions for a wide range of semileptonic decay observables. The future prospects of controlling systematic uncertainties down to the percent level, matching the precision of standard model (SM) predictions, are examined. Furthermore, new perspectives and caveats on combinations of the LFUV data are discussed and the world averages for the R(D(*)) ratios are revisited. Here it is demonstrated that different treatments for the correlations of uncertainties from D** excited states can vary the current 3σ tension with the SM within a 1σ range. Prior experimental overestimates of D**τν contributions may further exacerbate this. The precision of future measurements is also estimated; their power to exploit full differential information, and solutions to the inherent difficulties in self-consistent new physics interpretations of LFUV observables, are explored.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

FLAG Review 2021

Abstract We review lattice results related to pion, kaon, D -meson, B -meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $$f_+(0)$$ f + ( 0 ) arising in the semileptonic $$K \rightarrow \pi $$ K → π transition at zero momentum transfer, as well as the decay constant ratio $$f_K/f_\pi $$ f K / f π and its consequences for the CKM matrix elements $$V_{us}$$ V us and $$V_{ud}$$ V ud . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $$SU(2)_L\times SU(2)_R$$ S U ( 2 ) L × S U ( 2 ) R and $$SU(3)_L\times SU(3)_R$$ S U ( 3 ) L × S U ( 3 ) R Chiral Perturbation Theory. We review the determination of the $$B_K$$ B K parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $$m_c$$ m c and $$m_b$$ m b as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $$\alpha _s$$ α s . We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

How to identify different new neutrino oscillation physics scenarios at DUNE

Next generation neutrino oscillation experiments are expected to measure the remaining oscillation parameters with very good precision. They will have unprecedented capabilities to search for new physics that modify oscillations. DUNE, with its broad band beam, good particle identification, and relatively high energies will provide an excellent environment to search for new physics. If deviations from the standard three-flavor oscillation picture are seen however, it is crucial to know which new physics scenario is found so that it can be verified elsewhere and theoretically understood. We investigate several benchmark new physics scenarios by looking at existing long-baseline accelerator neutrino data from NOvA and T2K and determine at what sensitivity DUNE can differentiate among them. We consider sterile neutrinos and both vector and scalar non-standard neutrino interactions, all with new complex phases, the latter of which could conceivably provide absolute neutrino mass scale information. We find that, in many interesting cases, DUNE will have good model discrimination. We also perform a new fit to NOvA and T2K data with scalar NSI.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2018

Top-philic Z' forces at the LHC

Despite extensive searches for an additional neutral massive gauge boson at the LHC, a $Z^\prime$ at the weak scale could still be present if its couplings to the first two generations of quarks are suppressed, in which case the production in hadron colliders relies on tree-level processes in association with heavy flavors or one-loop processes in association with a jet. Here, we consider the low-energy effective theory of a top-philic $Z'$ and present possible UV completions. We clarify theoretical subtleties in evaluating the production of a top-philic $Z'$ at the LHC and examine carefully the treatment of an anomalous $Z'$ current in the low-energy effective theory. Recipes for properly computing the production rate in the $Z'+j$ channel are given. We discuss constraints from colliders and low-energy probes of new physics. As an application, we apply these considerations to models that use a weak-scale $Z'$ to explain possible violations of lepton universality in $B$ meson decays, and show that the future running of a high luminosity LHC can potentially cover much of the remaining parameter space favored by this particular interpretation of the $B$ physics anomaly.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Semileptonic form factors for $$B\rightarrow D^*\ell \nu $$ at nonzero recoil from $$2+1$$-flavor lattice QCD: Fermilab Lattice and MILC Collaborations

Abstract We present the first unquenched lattice-QCD calculation of the form factors for the decay $$B\rightarrow D^*\ell \nu $$ B → D ∗ ℓ ν at nonzero recoil. Our analysis includes 15 MILC ensembles with $$N_f=2+1$$ N f = 2 + 1 flavors of asqtad sea quarks, with a strange quark mass close to its physical mass. The lattice spacings range from $$a\approx 0.15$$ a ≈ 0.15 fm down to 0.045 fm, while the ratio between the light- and the strange-quark masses ranges from 0.05 to 0.4. The valence b and c quarks are treated using the Wilson-clover action with the Fermilab interpretation, whereas the light sector employs asqtad staggered fermions. We extrapolate our results to the physical point in the continuum limit using rooted staggered heavy-light meson chiral perturbation theory. Then we apply a model-independent parametrization to extend the form factors to the full kinematic range. With this parametrization we perform a joint lattice-QCD/experiment fit using several experimental datasets to determine the CKM matrix element $$|V_{cb}|$$ | V cb | . We obtain $$\left| V_{cb}\right| = (38.40 \pm 0.68_{\text {th}} \pm 0.34_{\text {exp}} \pm 0.18_{\text {EM}})\times 10^{-3}$$ V cb = ( 38.40 ± 0 . 68 th ± 0 . 34 exp ± 0 . 18 EM ) × 10 - 3 . The first error is theoretical, the second comes from experiment and the last one includes electromagnetic and electroweak uncertainties, with an overall $$\chi ^2\text {/dof} = 126/84$$ χ 2 /dof = 126 / 84 , which illustrates the tensions between the experimental data sets, and between theory and experiment. This result is in agreement with previous exclusive determinations, but the tension with the inclusive determination remains. Finally, we integrate the differential decay rate obtained solely from lattice data to predict $$R(D^*) = 0.265 \pm 0.013$$ R ( D ∗ ) = 0.265 ± 0.013 , which confirms the current tension between theory and experiment.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Supersymmetric D3/D5 for massive defects on curved space

We construct the holographic dual for N=4 SYM on S 4 and AdS 4 coupled to massive N=2 supersymmetric quenched flavor fields on a codimension-1 defect, which is S 3 and AdS 3 , respectively. The holographic description is in terms of a D 3 /probe D 5 brane system. We set up and reduce the BPS equations for D 5 -brane embeddings with arbitrary supersymmetric deformations and partly solve them at the non-linear level. The remaining equations are solved explicitly in a small-mass expansion. We compute the contribution of the defect fields to the partition function on S 4 and compare to a field theory computation using supersymmetric localization, for which we set up the matrix model. Both computations agree, lending strong support to holographic probe brane constructions using D 3 /D 5 configurations in general.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Beam energy dependence of net-hyperon yield and its implication on baryon transport mechanism

In the constituent quark model, each quark inside a baryon carries 1/3 unit of the baryon number. An alternative picture exists where the center of a Y-shaped topology of gluon fields, called the baryon junction, carries a unit baryon number. Studying baryon transport over a large rapidity gap (δy) in nuclear collisions provides a possible tool to distinguish these two pictures. A recent analysis of global data on net-proton yield at mid-rapidity in Au+Au collisions showed an exponential dependence on δy and the exponential slope does not vary with event centrality, favoring the baryon junction picture. Since junctions are flavor blind, hyperons – baryons containing valence strange quarks – are expected to exhibit a similar behavior as the proton. This study aims to test this prediction by analyzing hyperon yields in Au+Au collisions at various energies. We observe that net-hyperon yields, after correcting for the strangeness production suppression, adhere to the expected exponential form. The extracted slope parameters for net-Λ, net-$Ξ$ and net-Ω are consistent with each other and with those of net-proton within uncertainties, and exhibit no centrality dependence. Various implementations of the PYTHIA event generator, primarily based on valence quarks for baryon transport, are unable to simultaneously describe the slope parameters for all baryons.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Evidence for $CP$ violation and measurement of $CP$-violating parameters in B$^0_\mathrm{s}$ $\to$ J/$\psi\,\phi$(1020) decays in pp collisions at $\sqrt{s} =$ 13 TeV

A pioneering machine-learning-based flavor-tagging algorithm combining same-side and opposite-side tagging is used to obtain the equivalent of 27$\,$000 tagged B$^0_\mathrm{s}$$\to$ J/$\psi\, \phi$(1020) decays from pp collisions at $\sqrt{s} =$ 13 TeV, collected by the CMS experiment and corresponding to an integrated luminosity of 96.5 fb$^{-1}$. A time- and flavor-dependent angular analysis of the $\mu^+\mu^-$K$^+$K$^-$ final state is used to measure parameters of the $\mathrm{B}^0_\mathrm{s}$-$\overline{\mathrm{B}}^0_\mathrm{s}$ system. The weak phase is measured to be $\phi_\mathrm{s}$ = $-$73 $\pm$ 23 (stat) $\pm$ 7 (syst) mrad, which, combined with a $\sqrt{s}$ = 8 TeV CMS result, gives $\phi_\mathrm{s}$ = $-$74 $\pm$ 23 mrad. This value differs from zero by 3.2 standard deviations, providing evidence for $CP$ violation in B$^0_\mathrm{s}$$\to$ J/$\psi\,\phi$(1020) decays. All measured physics parameters are found to agree with standard model predictions where available.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

FLAG Review 2019: Flavour Lattice Averaging Group (FLAG)

Abstract We review lattice results related to pion, kaon, D -meson, B -meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $$f_+(0)$$ f + ( 0 ) arising in the semileptonic $$K \rightarrow \pi $$ K → π transition at zero momentum transfer, as well as the decay constant ratio $$f_K/f_\pi $$ f K / f π and its consequences for the CKM matrix elements $$V_{us}$$ V us and $$V_{ud}$$ V ud . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $$SU(2)_L\times SU(2)_R$$ S U ( 2 ) L × S U ( 2 ) R and $$SU(3)_L\times SU(3)_R$$ S U ( 3 ) L × S U ( 3 ) R Chiral Perturbation Theory. We review the determination of the $$B_K$$ B K parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $$m_c$$ m c and $$m_b$$ m b as well as those for D - and B -meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $$\alpha _s$$ α s . Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2016

Search for new physics with the M T2 variable in all-jets final states produced in pp collisions at $ \sqrt{s}=13 $ TeV

A search for new physics is performed using events that contain one or more jets, no isolated leptons, and a large transverse momentum imbalance, as measured through the M T2 variable, which is an extension of the transverse mass in events with two invisible particles. The results are based on a sample of proton-proton collisions collected at a center-of-mass energy of 13 TeV with the CMS detector at the LHC, and that corresponds to an integrated luminosity of 2.3 fb –1 . The observed event yields in the data are consistent with predictions for the standard model backgrounds. The results are interpreted using simplified models of supersymmetry and are expressed in terms of limits on the masses of potential new colored particles. Assuming that the lightest neutralino is stable and has a mass less than about 500 GeV, gluino masses up to 1550-1750 GeV are excluded at 95% confidence level, depending on the gluino decay mechanism. For the scenario of direct production of squark-antisquark pairs, top squarks with masses up to 800 GeV are excluded, assuming a 100% branching fraction for the decay to a top quark and neutralino. Moreover, bottom squark masses are excluded up to 880 GeV, and masses of light-flavor squarks are excluded up to 600-1260 GeV, depending on the degree of degeneracy of the squark masses.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2015

Towards a supersymmetric description of the Fermi Galactic center excess

We attempt to build a model that describes the Fermi galactic gamma-ray excess (FGCE) within a UV-complete Supersymmetric framework; we find this to be highly non-trivial. At the very least a successful Supersymmetric explanation must have several important ingredients in order to fit the data and satisfy other theoretical and experimental constraints. Under the assumption that a single annihilation mediator is responsible for both the observed relic density as well as the FGCE, we show that the requirements are not easily satisfied in many TeV-scale SUSY models, but can be met with some model building effort in the general NMSSM with ~ 10 parameters beyond the MSSM. We find that the data selects a particular region of the parameter space with a mostly singlino lightest Supersymmetric particle and a relatively light CP-odd Higgs boson that acts as the mediator for dark matter annihilation. We study the predictions for various observables within this parameter space, and find that searches for this light CP-odd state at the LHC, as well as searches for the direct detection of dark matter, are likely to be quite challenging. It is possible that a signature could be observed in the flavor sector; however, indirect detection remains the best probe of this scenario.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Search for e → τ charged lepton flavor violation at the EIC with the ECCE detector

The recently approved Electron-Ion Collider (EIC) will provide a unique new opportunity for searches of charged lepton flavor violation (CLFV) and other new physics scenarios. In contrast to the e ↔ μ CLFV transition for which very stringent limits exist, there is still a relatively large discovery space for the e → τ CLFV transition, potentially to be explored by the EIC. Here, with the latest detector design of ECCE (EIC Comprehensive Chromodynamics Experiment) and projected integral luminosity of the EIC, we find the τ-leptons created in the DIS process ep → τX are expected to be identified with high efficiency. A first ECCE simulation study, restricted to the 3-prong τ-decay mode and with limited statistics for the Standard Model backgrounds, estimates that the EIC will be able to improve the current exclusion limit on e → τ CLFV by an order of magnitude. As a result, the very high vertex resolution of the ECCE detector configuration plays a critical role in τ identification.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Scattering neutrinos, spin models, and permutations

We consider a class of Heisenberg all-to-all coupled spin models inspired by neutrino interactions in a supernova with N degrees of freedom. These models are characterized by a coupling matrix that is relatively simple in the sense that there are only a few, relative to N , nontrivial eigenvalues, in distinction to the classic Heisenberg spin-glass models, leading to distinct behavior in both the high-temperature and low-temperature regimes. When the momenta of the neutrinos are uniform and random in directions, we can calculate the large- N partition function for the quantum Heisenberg model. In particular, the high-temperature partition function predicts a non-Gaussian density of states, providing interesting counterexamples showing the limits of general theorems on the density of states for quantum spin models. We can repeat the same argument for classical Heisenberg models, also known as rotor models, and we find the high-temperature expansion is completely controlled by the eigenvalues of the coupling matrix, and again predicts non-Gaussian behavior for the density of states as long as the number of eigenvalues does not scale linearly with N . Indeed, we derive the amusing fact that these partition functions are essentially the generating function for counting permutations in the high-temperature regime. Finally, for the case relevant to neutrinos in a supernova, we identify the low-temperature phase as a unique state with the direction of the momenta of the neutrino dictating its coherent state in flavor-space, a state we dub the “flavor-momentum-locked” state. Published by the American Physical Society 2025

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

PHENIX Probing QCD Matter Through Heavy Flavor and Quarkonium at RHIC

Recent results from the PHENIX experiment on heavy flavors and quarkonia production in p + Al, p + Au, d + Au, and Au + Au collision systems at psNN = 200 GeV are summarized. The results are carried out by the measurements of the nuclear modification factors and elliptic flow. The nuclear modification factors measurements give insight into the energy loss of heavy quarks in the quark-gluon plasma medium along their path lengths. The elliptic flow measurements are a good tool to investigate the coupling of heavy quarks with the medium. The measurements are presented as a function of centrality, rapidity, and transverse momentum. The interpretations of the results in light of our current theoretical models, and comparison to LHCb and ALICE measurements are presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2019

Heavy neutrinos with dynamic jet vetoes: multilepton searches at $\sqrt{s}=14$, 27, and 100 TeV

Heavy neutrinos (N) remain one of most promising explanations for the ori-gin of neutrinos’ tiny masses and large mixing angles. In light of broad advances in un-derstanding and modeling of hadron collisions at large momentum transfers, we revisit the long-standard search strategy for heavy N decaying to multiple charged leptons (ℓ), pp → NℓX → 3ℓνX. For electroweak and TeV-scale N, we propose a qualitatively new collider analysis premised on a dynamic jet veto and discriminating, on an event-by-event basis, according to the relative amount of hadronic and leptonic activity. We report that the sensitivity to heavy neutrinos at the CERN Large Hadron Collider (LHC) can be improved by roughly an order of magnitude at both $\mathcal{L}$ = 300 fb-1 and 3 ab-1. At s√=14 TeV with $\mathcal{L}$ = 3 ab -1 , we find active-sterile mixing as small as |V ℓN | 2 = 10 -2 (10 -3 ) [5 × 10 -4 ] can be probed at 95% CL for heavy Dirac neutrinos masses $m$ N ≲ 1200 (300) [200] GeV, well beyond the present |V ℓN | 2 ≲ 10 -3 - 10 -1 constraints for such heavy states set by indirect searches and precision measurements. The improvement holds also for Majorana N, and is largely independent of whether charged lepton flavor is conserved or violated. The analysis, built almost entirely from inclusive, transverse observables, is designed to be robust across increasing collider energies, and hence serves as a basis for searches at future colliders: with $\mathcal{L}$ = 15ab -1 at $\sqrt{s}$ = 27 TeV,onecanprobemixingbelow |V ℓN | 2 = 10 -2 (10 -3 )[2 × 10 -4 ] for m N ≲ 3500 (700) [200] GeV. At a hypothetical 100 TeV pp collider with $\mathcal{L}$ = 30 ab -1 , one can probe mixing down to 9 × 10 -5 for m N ≲ 200 GeV, below 10 -3 for m N ≲ 4 TeV, and below 10 -2 for m N ≲ 15 TeV. We anticipate these results can be further improved with detector-specific tuning and application of multi-variant / machines learning techniques. To facilitate such investigations, we make publicly available Monte Carlo libraries needed for the precision computations/simulations used in our study.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Chiral topologically ordered insulating phases in arrays of interacting integer quantum Hall islands

Here, we study networks of Coulomb-blockaded integer quantum Hall islands with even fillings ν = 2 k ( k being an integer), including cases with 2 k layers each of ν = 1 fillings. Allowing only spin-current interactions between the islands (i.e., without any charge transfer), we obtain solvable models leading to a rich set of insulating S U ( 2 ) k topologically ordered phases. The case with k = 1 is dual to the Kalmeyer-Laughlin phase, k = 2 to Kitaev's chiral spin liquid and the Moore-Read state, and k = 3 contains a Fibonacci anyon that may be utilized for universal topological quantum computation. Additionally, we show how the S U ( 2 ) k topological phases may be obtained also in an array of islands with ν = 2 k integer quantum Hall states and critical spin chains in a checkerboard pattern. The array and checkerboard constructions gap out the charge mode and additional “flavor” modes by virtue of their geometry. Furthermore, we find that a fine tuning of the system parameter is not needed in the checkerboard configuration and the ν = 2 case. We also discuss their bulk excitations, and show that their thermal Hall conductance is universal, reflecting the central charge c = 3 k / ( k + 2 ) of the chiral edge modes.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

P -wave π π scattering and the ρ resonance from lattice QCD

Here we calculate the parameters describing elastic I = 1, P-wave ππ scattering using lattice QCD with 2 +1 flavors of clover fermions. Our calculation is performed with a pion mass of m π ≈ 320 MeV and a lattice size of L ≈ 3.6 fm. We construct the two-point correlation matrices with both quark-antiquark and two-hadron interpolating fields using a combination of smeared forward, sequential and stochastic propagators. The spectra in all relevant irreducible representations for total momenta |$\vec{P}$| ≤ $\sqrt{3}\frac{2π}{L}$ are extracted with two alternative methods: a variational analysis as well as multiexponential matrix fits. We perform an analysis using Lüscher’s formalism for the energies below the inelastic thresholds, and investigate several phase shift models, including possible nonresonant contributions. We find that our data are well described by the minimal Breit-Wigner form, with no statistically significant nonresonant component. In determining the ρ resonance mass and coupling we compare two different approaches: fitting the individually extracted phase shifts versus fitting the t-matrix model directly to the energy spectrum. We find that both methods give consistent results, and at a pion mass of am π = 0.18295 (36) stat obtain gρππ = 5.69(13) stat (16) sys , am ρ = 0.4609(16) stat (14) sys , and am ρ /am N = 0.7476(38) stat (23) sys , where the first uncertainty is statistical and the second is the systematic uncertainty due to the choice of fit ranges.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗