The Fermi energy at 0°K is evaluated for electrons confined to cubical and spherical rigid-walled boxes of equal volume, respectively, in the Sommerfeld approximation. Due primarily to large differences in single-particle degeneracies, Fermi energies compared for equal numbers of particles in these two configurations are found to be unequal. Approximate expressions of the Fermi energy in the large particle-number limit for the spherical case reveal that it agrees in form with the Fermi energy (...) for the cubical configuration. The finite cylindrical box is also examined and it is found that for fixed number of particles and fixed volume, the Fermi energy varies as the two characteristic lengths of the box are changed. Experimental corroboration of the theory is suggested through measurement of the work function for equal-volume micrometallic samples in the different geometries. (shrink)

Spin-half fermions are considered to be limited in a spherical potential well with periodic boundary conditions. The whole system is treated like a relativistic Fermi Gas. Solving the corresponding Dirac equation, the density of states, the Fermi energy, the average energy, the density of states of nucleons and the total energy of the ground-state are obtained.

A cornerstone of physics, Maxwell‘s theory of electromagnetism, apparently contains a fatal flaw. The standard expressions for the electromagnetic field energy and the self-mass of an electron of finite extension do not obey Einstein‘s famous equation, \, but instead fulfill this relation with a factor 4/3 on the left-hand side. Furthermore, the energy and momentum of the electromagnetic field associated with the charge fail to transform as a four-vector. Many famous physicists have contributed to the debate of this (...) so-called 4/3-problem without arriving at a complete solution. It has generally been assumed that, as originally suggested by Poincaré, the problems are connected to the question of stability of the charge distribution, and that relativistic equivalence between energy and self-mass can only be restored by inclusion of stabilizing forces. Alternative solutions to the problems have also been proposed. Nearly a century ago Fermi suggested a covariant definition of the electromagnetic energy and momentum, and sixty years later Kalckar et al. argued that the 4/3 problem is caused by omission of a relativistic correction in the standard evaluation of the self-force from Coulomb self-interaction. However, the relation between these suggestions has not been clear. We show that the relativistic correction implies that the mechanical momentum of an accelerated rigid body must be defined as the sum of the momenta of its parts for fixed time in the momentary rest frame of the body. For the total momentum of particles and field to be conserved, the total energy–momentum tensor must be divergence free, and this then requires that the momentum of the associated electromagnetic field be defined in the same way, consistent with the suggestion by Fermi. This comprehensive solution of the 4/3-problem demonstrates that there is no conflict of Maxwell‘s theory with special relativity and the questions of equivalence of electromagnetic energy and self-mass and of stability of a classical charge distribution are independent. In appendices we discuss the relations of our treatment with Fermi‘s seminal paper and with a classic paper by Dirac where he evaluated the damping self-force on a point electron from transport of energy and momentum in the electromagnetic field. (shrink)

The chromodielectric soliton model is Lorentz and chirally invariant. It has been demonstrated to exhibit dynamical chiral symmetry breaking and spatial confinement in the locally uniform approximation. We here study the full nonlocal quark self-energy in a color-dielectric medium modeled by a two-parameter Fermi function. Here color confinement is manifest. The self-energy thus obtained is used to calculate quark wave functions in the medium which, in turn, are used to calculate the nucleon and pion masses in the (...) one-gluon-exchange approximation. The nucleon mass is fixed to its empirical value using scaling arguments; the pion mass (for massless current quarks) turns out to be small but nonzero, depending on the model parameters. (shrink)

L’esame di un passo della Vita di Platone laerziana (III 2) – e di luoghi paralleli riferibili alla medesima tradizione – permette di mettere a fuoco un motivo mitico cruciale nei racconti sulla concorrenza tra essere umano e divinità nel gamos con una mortale: il divieto, che il nume impone al ‘rivale’ di inferiore natura, di congiungersi con la donna che reca in grembo il suo seme, finché l’eroina non abbia portato a termine la gravidanza. Le storie di personaggi femminili (...) colpevoli di duplice gamos – prima con il dio e poi, a breve distanza di tempo, con il mortale – si focalizzano sulle conseguenze nefaste del secondo contaminante connubio: uno per tutti il caso di Coronide madre di Asclepio (e forse il motivo era associato anche ad altre note madri mitiche, come Arianna, Semele e Alcmena). In questi esempi, tuttavia, in genere non è esplicitato il tema della proibizione divina, come invece accade nel bios di Diogene Laerzio. Qui infatti Apollo, dopo aver concepito Platone con Perictione, appare ad Aristone per vietargli il rapporto sessuale con la parthenos. (shrink)

Summary In this paper we examine the reaction of the Leiden low-temperature laboratory of Heike Kamerlingh Onnes to new ideas in quantum theory. Especially the contributions of Albert Einstein (1906) and Peter Debye (1912) to the theory of specific heat, and the concept of zero-point energy formulated by Max Planck in 1911, gave a boost to solid state research to test these theories. In the case of specific heat measurements, Kamerlingh Onnes's laboratory faced stiff competition from Walter Nernst's Institute (...) of Physical Chemistry in Berlin. In fact, Berlin got the better of it because Leiden lacked focus. After the liquefaction of helium in 1908, Kamerlingh Onnes transformed his laboratory into an international facility for low temperature research, and for this reason it was impossible to make headway with the specific heat measurements. In the case of zero-point energy, Leiden developed a magnetic research programme to test the concept. Initially the balance of evidence seemed to be tipping in favour of zero-point energy. After 1914, however, Leiden would desert the theory in fovour, of a concept from calssical physics. A curious move that illustrates Kamerlingh Onnes's discomfort with the new quantum theory. (shrink)

The organization of the mobile atomic exhibition, Mostra Atomica, designed by the United States Information Service to travel through Italy in 1954–55, had to meet technical, scientific, artistic, and political challenges. The head of the group in charge of the exhibition was architect Peter G. Harnden whose pedigree in the intelligence and training in architecture were an ideal match for leading the unit dedicated to exhibitions. The political sensitivity of the Mostra Atomica also required the intervention of the Italian Ministry (...) of the Interior to guarantee safe mobility and secure shows. In every major town, American and British diplomats attended the local opening ceremony, while the very symbol of science diplomacy was Enrico Fermi, whose recorded message praised international cooperation. All in all, the USIS campaign promoting peaceful applications of nuclear physics was successful in reaching and involving Italian society. Visual and spatial aesthetics were particularly relevant: the geometrical design of the exposition rooms conveyed a strong sense of modernity that contrasted with the artistic heritage of Italian cities. The present article is based on archival files, newspaper reports, and photographs that document who was responsible for planning, setting up, and reporting this Cold War propaganda event. (shrink)

SummaryDuring the 1920s and 1930s, Italian physicists established strong relationships with scientists from other European countries and the United States. The career of Bruno Rossi, a leading personality in the study of cosmic rays and an Italian pioneer of this field of research, provides a prominent example of this kind of international cooperation. Physics underwent major changes during these turbulent years, and the traditional internationalism of physics assumed a more institutionalized character. Against this backdrop, Rossi's early work was crucial in (...) transforming the study of cosmic rays into a branch of modern physics. His friendly relationships with eminent scientists — notably Enrico Fermi, Walther Bothe, Werner Heisenberg, Hans Bethe, and Homi Bhabha — were instrumental both for the exchange of knowledge about experimental practices and theoretical discussions, and for attracting the attention of physicists such as Arthur Compton, Louis Leprince-Ringuet, Pierre Auger and Patrick Blackett to the problem of cosmic rays. Relying on material from different archives in Europe and the United States, this case study aims to provide a glimpse of the intersection between national and international dimensions during the 1930s, at a time when the study of cosmic rays was still very much in its infancy, strongly interlaced with nuclear physics, and full of uncertain, contradictory, and puzzling results. Nevertheless, as a source of high-energy particles it became a proving ground for testing the validity of the laws of quantum electrodynamics, and made a fundamental contribution to the origins of particle physics. (shrink)

Now the visionaries of the high energy physics community are ready to take the next step, the Superconducting Super Collider or SSC. By now you must have heard or read something about this machine. It is being pushed by President Reagan and the Department of Energy. It will have a total cost in 1987 dollars of $4.4 billion and will be built at a location now in the process of selection. It will be a big machine, the largest (...) proton synchrotron ever planned, "the greatest public works project in the history of mankind", as one congressman put it. The tunnel in which it will be constructed does not circle the Earth. It is more modest, with a circumference of "only" 52 miles. Yet the SSC will produce proton-proton collision energies 40 times higher than those of the FermiLab Tevatron or Fermi's dream machine. (shrink)

Quantum mechanics was initiated with the object of allowing only observable concepts to enter into the theory. The new mechanics has, however, inherited the old difficulty with the conservation laws involved in beta decay, and this led Pauli, about 1931, to introduce the idea of the neutrino, with the object of reconciling the facts of beta decay with the conservation laws. The neutrino, as it was proposed by Pauli and as accepted today, is a particle devoid both of mass and (...) charge, but having a spin 1/2 and obeying the Fermi statistics. In 1934 Fermi founded his theory of beta disintegration based on the neutrino. The Fermi theory assumes that this hypothetical particle is emitted simultaneously with the disintegrating electron and carries off the excess energy, thereby reconciling the conservation laws with beta disintegration. It will be remembered that the continuous distribution in energy of the beta particles is such that, except at the maximum energy end point of the beta spectra, there is more energy available from the disintegration than is emitted in the form of beta particles. (shrink)

It has been shown by several authors that in operations involving information a quantity appears which is the negative of the quantity usually defined as entropy in similar situations. This quantity ℜ = − KI has been termed “negentropy” and it has been shown that the negentropy of information and the physical entropy S are mirrorlike representations of the same train of events. In physical terminology the energy is degraded by an increase in entropy due to an increased randomness (...) in the positions or velocities of components, wave functions, complexions in phase space; in informational terminology some information about the same components has been lost or the negentropy has been decreased. In equilibrium the system has for a given energy content maximum randomness. One consequence of this dual aspect was the idea to apply the methods of statistical mechanics to problems of communication and Brillouin showed that Fermi-Dirac statistics or generalized Fermi statistics are applicable for example to a transmission of signals such as telegrams. (shrink)

The presented enhanced version of Eriksen’s theorem defines an universal transform of the Foldy–Wouthuysen type and in any external static electromagnetic field reveals a discrete symmetry of Dirac’s equation, responsible for existence of a highly influential conserved quantum number—the charge index distinguishing two branches of DE spectrum. It launches the charge-index formalism obeying the charge-index conservation law. Via its unique ability to manipulate each spectrum branch independently, the CIF creates a perfect charge-symmetric architecture of Dirac’s quantum mechanics, which resolves all (...) the riddles of the standard DE theory. Besides the abstract CIF algebra, the paper discusses: the novel accurate charge-symmetric definition of the electric-current density; DE in the true-particle representation, where electrons and positrons coexist on equal footing; flawless “natural” scheme of second quantization; and new physical grounds for the Fermi–Dirac statistics. As a fundamental quantum law, the CICL originates from the kinetic-energy sign conservation and leads to a novel single-particle physics in strong-field situations. Prohibiting Klein’s tunneling in Klein’s zone via the CICL, the precise CIF algebra defines a new class of weakly singular DE solutions, strictly confined in the coordinate space and experiencing the total reflection from the potential barrier. (shrink)

We give a review of some works where it is shown that certain quantum-like features are exhibited by classical systems. Two kinds of problems are considered. The first one concerns the specific heat of crystals (the so called Fermi–Pasta–Ulam problem), where a glassy behavior is observed, and the energy distribution is found to be of Planck-like type. The second kind of problems concerns the self-interaction of a charged particle with the electromagnetic field, where an analog of the tunnel (...) effect is proven to exist, and moreover some nonlocal effects are exhibited, leading to a natural hidden variable theory which violates Bell's inequalities. (shrink)

ABSTRACT This essay focuses on the history of one of the “atomic patents.” The patent, which described a process to slow down neutrons in nuclear reactions, was the result of experimental research conducted in the 1930s by Enrico Fermi and his group at the Institute of Physics, University of Rome. The value of the patented process became clear during World War II, as it was involved in most of the military and industrial applications of atomic energy. This ignited (...) a controversy between Fermi and U.S. government representatives over royalties to be paid for use of the process during and after the war. The controversy sheds new light on the role that the management of patents played in the context of the Manhattan Project and in the postwar U.S. nuclear program, encompassing issues of power and economic influence in the relationship between scientists, the military, and public administrators. The invention covered by Patent No. 2,206,634 covers the basic process used in the research and development leading up the production of atomic energy and the production of the atomic bomb. Such invention is of continuing importance in the production of fissionable materials and atomic weapons. The Chairman: So Jones might make the greatest invention in the field and be deprived of any award for it? Captain Lavender: If he did not want to accept the award that was offered to him. (shrink)

Observational indications combined with analyses of analogue and emergent gravity in condensed matter systems support the possibility that there might be two distinct energy scales related to quantum gravity: the scale that sets the onset of quantum gravitational effects $E_{\rm B}$ (related to the Planck scale) and the much higher scale $E_{\rm L}$ signalling the breaking of Lorentz symmetry. We suggest a natural interpretation for these two scales: $E_{\rm L}$ is the energy scale below which a special relativistic (...) spacetime emerges, $E_{\rm B}$ is the scale below which this spacetime geometry becomes curved. This implies that the first ‘quantum’ gravitational effect around $E_{\rm B}$ could simply be that gravity is progressively switched off, leaving an effective Minkowski quantum field theory up to much higher energies of the order of $E_{\rm L}$ . This scenario may have important consequences for gravitational collapse, inasmuch as it opens up new possibilities for the final state of stellar collapse other than an evaporating black hole. (shrink)

Following with the discovery of the electron by J. J. Thomson at the end of the nineteenth century a steady elucidation of the structure of the atom occurred over the next 40 years culminating in the discovery of nuclear fission in 1938–1939. The significant steps after the electron discovery were: discovery of the nuclear atom by Rutherford (Philos Mag 6th Ser 21:669–688, 1911 ), the transformation of elements by Rutherford (Philos Mag 37:578–587, 1919 ), discovery of artificial radioactivity by Joliot-Curie (...) and Joliot-Curie (Comptes Rendus Acad Sci Paris 198:254–256, 1934 ), and the discovery of the neutron by Chadwick (Nature 129:312, 1932a , Proc R Soc Ser A 136:692–708, 1932b ; Proc R Soc Lond Ser A 136:744–748, 1932c ). The neutron furnished scientists with a particle able to penetrate atomic nuclei without expenditure of large amounts of energy. From 1934 until 1938–1939 investigations of the reaction between a neutron and uranium were carried out by E. Fermi in Rome, O. Hahn, L. Meitner and F. Strassmann in Berlin and I. Curie and P. Savitch in Paris. Results were interpreted as the formation of transuranic elements. After sorting out complex radio-chemistry and radio-physics O. Hahn and F. Strassmann came to the conclusion, beyond their belief, that the uranium nucleus split into smaller fragments, that is nuclear fission. This was soon followed in 1939 by its theoretical interpretation by L. Mietner and O. Frisch. (shrink)

According to prevailing theory, relativistic degenerate stars with masses beyond the Chandrasekhar and Oppenheimer–Volkoff (OV) limits cannot achieve hydrostatic equilibrium through either electron or neutron degeneracy pressure and must collapse to form stellar black holes. In such end states, all matter and energy within the Schwarzschild horizon descend into a central singularity. Avoidance of this fate is a hoped-for outcome of the quantization of gravity, an as-yet incomplete undertaking. Recent studies, however, suggest the possibility that known quantum processes may (...) intervene to arrest complete collapse, thereby leading to equilibrium states of macroscopic size and finite density. I describe here one such process which entails pairing (or other even-numbered association) of neutrons (or constituent quarks in the event of nucleon disruption) to form a condensate of composite bosons in equilibrium with a core of degenerate fermions. This process is analogous to, but not identical with, the formation of hadron Cooper pairs that give rise to neutron superfluidity and proton superconductivity in neutron stars. Fermion condensation to composite bosons in a star otherwise destined to collapse to a black hole facilitates hydrostatic equilibrium in at least two ways: (1) removal of fermions results in a decrease in the Fermi level which stiffens the dependence of degeneracy pressure on fermion density, and (2) phase separation into a fermionic core surrounded by a self-gravitating condensate diminishes the weight which must be balanced by fermion degeneracy pressure. The outcome is neither a black hole nor a neutron star, but a novel end state, a “fermicon star,” with unusual physical properties. (shrink)

We give an account of the appearance and first developments of the statistical model of atoms proposed by Thomas and Fermi, focusing on the main results achieved by Fermi and his group in Rome. Particular attention is addressed to the unknown contribution to this subject by Majorana, anticipating some important results reached later by leading physicists.

Expert reasoning combines voluminous domain‐specific knowledge with more general factual and strategic knowledge. Whereas expert system builders have recognized the need for specificity and problem‐solving researchers the need for generality, few attempts have been made to develop expert reasoning engines combining different kinds of knowledge at different levels of generality. This paper reports on the FERMI project, a computer‐implemented expert reasoner in the natural sciences that encodes factual and strategic knowledge in separate semantic hierarchies. The principled decomposition of knowledge (...) according to type and level of specificity yields both power and cross‐doman generality, as demonstrated in FERMI's ability to apply the same principles of invariance and decomposition to solve problems in fluid statics, DC‐circuits, and centroid location. Hierarchical knowledge representation and problem‐solving principles are discussed, and illustrative problem‐solving traces are presented. (shrink)

Fermi-Dirac statistics are one of two kinds of statistics exhibited by !identical quantum particles, the other being !Bose-Einstein statistics. Such particles are called fermions and bosons respectively (the terminology is due to Dirac [1902-1984] [1]). In the light of the !spin-statistics theorem, and consistent with observation, fermions are invariably spinors (of half-integral spin), whilst bosons are invariably scalar or vector particles (of integral spin). See !spin.

We present a Gedankenexperiment that leads to a violation of detailed balance if quantum mechanical transition probabilities are treated in the usual way by applying Fermi’s “golden rule”. This Gedankenexperiment introduces a collection of two-level systems that absorb and emit radiation randomly through non-reciprocal coupling to a waveguide, as realized in specific chiral quantum optical systems. The non-reciprocal coupling is modeled by a hermitean Hamiltonian and is compatible with the time-reversal invariance of unitary quantum dynamics. Surprisingly, the combination of (...) non-reciprocity with probabilistic radiation processes entails negative entropy production. Although the considered system appears to fulfill all conditions for Markovian stochastic dynamics, such a dynamics violates the Clausius inequality, a formulation of the second law of thermodynamics. Several implications concerning the interpretation of the quantum mechanical formalism are discussed. (shrink)

Buildings energy consumption is growing gradually and put away around 40% of total energy use. Predicting heating and cooling loads of a building in the initial phase of the design to find out optimal solutions amongst different designs is very important, as ell as in the operating phase after the building has been finished for efficient energy. In this study, an artificial neural network model was designed and developed for predicting heating and cooling loads of a building (...) based on a dataset for building energy performance. The main factors for input variables are: relative compactness, roof area, overall height, surface area, glazing are a, wall area, glazing area distribution of a building, orientation, and the output variables: heating and cooling loads of the building. The dataset used for training are the data published in the literature for various 768 residential buildings. The model was trained and validated, most important factors affecting heating load and cooling load are identified, and the accuracy for the validation was 99.60%. (shrink)

We consider the quantum electrodynamic problem of the velocity of transmission of an excitation from one atom to another, which was posed by Fermi in 1932 and, in an improved version, by Ferretti in 1968. The problem is solved using Heisenberg operators. The transmission velocity is shown to exceed the light velocity. We argue that this acausal result is to be explained by the inadmissibly idealized description of the signal source used by Fermi and Ferretti.

We present a reformulation of loop quantum gravity with a cosmological constant and no matter as a Fermi-liquid theory. When the topological sector is deformed and large gauge symmetry is broken, we show that the Chern–Simons state reduces to Jacobson’s degenerate sector describing 1+1 dimensional propagating fermions with nonlocal interactions. The Hamiltonian admits a dual description which we realize in the simple BCS model of superconductivity. On one hand, Cooper pairs are interpreted as wormhole correlations at the de Sitter (...) horizon; their number yields the de Sitter entropy. On the other hand, BCS is mapped into a deformed conformal field theory reproducing the structure of quantum spin networks. When area measurements are performed, Cooper-pair insertions are activated on those edges of the spin network intersecting the given area, thus providing a description of quantum measurements in terms of excitations of a Fermi sea to superconducting levels. The cosmological constant problem is naturally addressed as a nonperturbative mass-gap effect of the true Fermi-liquid vacuum. (shrink)

In this article, we put forward a new strategy for teaching the concept of energy. In the first section, we discuss how the concept is currently treated in educational programmes at primary and secondary level (taking the case of France), the learning difficulties that arise as well as the main teaching strategies presented in science education literature. In the second section, we argue that due to the complexity of the concept of energy, rethinking how it is taught should (...) involve teacher training specifically developed for the concept. In our view, Epistemology and the History of Science and Technology (EHST) is the best method for a full understanding of the concept of energy and should thus be included in the teacher training programme. In the third section, we provide a general framework for a teacher training programme on energy, based on the history and epistemology of the concept and structured on the following three questions: ‘What is the origin of the concept of energy?’, ‘What is energy?’ and ‘What purpose does the concept of energy serve?’. (shrink)

We study the conservation of energy, or lack thereof, when measurements are performed in quantum mechanics. The expectation value of the Hamiltonian of a system changes when wave functions collapse in accordance with the standard textbook treatment of quantum measurement, but one might imagine that the change in energy is compensated by the measuring apparatus or environment. We show that this is not true; the change in the energy of a state after measurement can be arbitrarily large, (...) independent of the physical measurement process. In Everettian quantum theory, while the expectation value of the Hamiltonian is conserved for the wave function of the universe, it is not constant within individual worlds. It should therefore be possible to experimentally measure violations of conservation of energy, and we suggest an experimental protocol for doing so. (shrink)