quantum-mechanics

"... was formalized by a number of authors ... as the requirement that the wave function of a multi-electron system is in the coordinates, and , of all the electrons. This principle was incorporated into statistical mechanics by Enrico Fermi ... and Dirac, ... and for this reason particles obeying the exclusion principle are generally called 's,' just as particles like photons for which the wave function is symmetric and which obey the statistics of Bose and Einstein are called 's.'"

"… what is the wave function? Is it a complete and comprehensive representation of the world? Or do we need additional physical quantities to fully capture reality, as Albert Einstein and others suspected? Or does the wave function have no direct connection with reality at all, merely characterizing our personal ignorance about what we will eventually measure in our experiments? Until s definitively answer these questions, they can’t really be said to understand quantum mechanics — thus Feynman’s lament. Which is bad, because quantum mechanics is the most fundamental theory we have, sitting squarely at the center of every serious attempt to formulate deep laws of nature. If nobody understands quantum mechanics, nobody understands the universe."

"What is the empirical content of quantum mechanics? Or how does the wave function assigned to a relate to the results of measurements on the system? The well-known connection rule is the , which has been precisely tested by experiments. It says that a (projective) measurement of an A on a system with the wave function |ψ⟩ will randomly obtain one of the of A, and the probability of obtaining an eigenvalue ai is given by |⟨ai|ψ⟩|2, where |ai⟩ is the corresponding to the eigenvalue ai."

"... Roughly speaking, in renormalizable theories no coupling constants can have the dimensions of negative powers of . But every time we add a field or a space-time derivative to an interaction, we reduce the dimensionality of the associated coupling constant. So only a few simple types of interaction can be renormalizable. ..."

"The origins of es is one of the biggest mysteries in modern physics since they are beyond the realm of the Standard Model. As massive particles, neutrinos undergo throughout their propagation. In this paper we show that when a neutrino oscillates from a flavor state α to a flavor state β, it follows three possible paths consistent with the Quantum Yang- Baxter Equations. These trajectories define the transition probabilities of the oscillations. Moreover, we define a probability matrix for flavor transitions consistent with the Quantum Yang-Baxter Equations, and estimate the values of the three neutrino mass eigenvalues within the framework of the triangular formulation."

"It has been known for some time that the Yang-Baxter equations can be solved using s. More recently it was discovered … that the YBE for the N state could be solved using special curves of (N – 1)2."

"At an early stage the Yang-Baxter equation (YBE) appeared in several different guises in the literature, and sometimes its solutions have preceded the equation. One can trace basically three streams of ideas from which YBE has emerged: the , commuting in statistical mechanics, and factorizable in field theory."

"About 40 years ago, in the study of quantum s … , in particular in the framework of the … , new algebraic structures arose, the generalizations of which were later called quantum groups … The Yang-Baxter equations became a unifying basis of all these investigations. The most important nontrivial examples of quantum groups are quantizations (or deformations) of ordinary classical s and algebras (more precisely, one considers the deformations of the algebra of functions of a Lie group and the universal enveloping of a Lie algebra). The quantization is accompanied by the introduction of an additional parameter q (the deformation parameter), which plays a role analogous to the role of in quantum mechanics. In the limit q → 1, the quantum groups and algebras go over into the classical ones."

"That the pion-nucleon (πN) coupling constant is fundamental in our understanding of the Cosmos has been adequately emphasised in numerous works. In meson-exchange models of the , a significantly weaker coupling between the pions and the nucleons would have prevented the s from combining fast with s in the ; they would have decayed before they had any chance to be enmeshed first in s, then in other light nuclei. According to the , within half an hour of the Big Bang, all existing matter had assumed the form of free electrons, protons, and helium nuclei (as well as traces of other nuclei up to 7). On the contrary, a significantly stronger coupling would have resulted in the rapid creation of bound diprotons and would have led to a helium-dominated Universe. It is hard to imagine how life could emerge in such a Universe: typical stars burn hydrogen to helium for about 90 % of their lives."

"The fundamental scientific purpose of the LHC is to explore the inner structure of matter and the forces that govern its behavior, and thereby understand better the present content of the Universe and its evolution since the Big Bang, and possibly into the future. The unparalleled high energy of the LHC, which is designed to be 7 TeV per proton in each colliding beam, and its enormous collision rate, which is planned to attain about a billion collisions per second, will enable the LHC to examine rare processes occurring at very small distances inside matter. It will be a microscope able to explore the inner structure of matter on scales an order of magnitude smaller than any previous collider. The energies involved in the proton-proton collisions will be similar to those in particle collisions in the first trillionth of a second of the history of the Universe. By studying these processes in the laboratory, the LHC experiments will, in a sense, be looking further back into time than is possible with any telescope."

"With the discovery of the Higgs boson, the next burning question at the LHC is why its mass is so low. Nobody knows the answer to that question, but it is definitely the next hot topic for LHC physicists ..."

"On July 4, scientists working with data from ongoing experiments at the Large Hadron Collider (LHC) announced the discovery of a new particle "consistent with" the Higgs boson — a subatomic particle also colloquially referred to as the "God particle." After years of design and construction, the LHC first sent protons around its 27 kilometer (17 mile) underground tunnel in 2008. Four years later, the LHC's role in the discovery of the Higgs boson provides a final missing piece for the Standard Model of Particle Physics — a piece that may explain how otherwise massless subatomic particles can acquire mass. Gathered here are images from the construction of the massive $4-billion-dollar machine that allowed us peer so closely into the subatomic world."

"… The relied not on the detection of photon pairs with a certain energy but on the detection of more of those pairs than expected. That reliance on probabilities is why the L.H.C. and other major collider experiments often have independent teams, working with separate detectors, analyzing the same types of collisions—to avoid biasing each other. It is also the reason for the ."

"A new conceptual foundation for Tμν on locally flat —to obtain the so-called Casimir effect—is presented. The Casimir ground state is viewed locally as a (nonvacuum) state on Minkowski space-time and the expectation value of the normal-ordered is taken. The same ideas allow us to treat, for the first time, self-interacting fields for arbitrary mass in —using traditional flat-space-time renormalization theory. First-order results for zero-mass λφ4 theory agree with those recently announced by . We point out the crucial role played by the simple renormalization condition that the vacuum expectation value of Tμν must vanish in Minkowski space-time, and in a critical discussion of other approaches, we clarify the question of renormalization ambiguities for Tμν in curved space-times."

"Vacuum expectation values of products of neutral operators are discussed. The properties of these distributions arising from , the absence of states and the of the scalar product are determined. The vacuum expectation values are shown to be s of s. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved."

"I remember a lunch in which Schwinger began by saying to Weisskopf, “Now I will make you a world.” The “world” was written down on a few paper napkins, one of which I saved. In any event, one of the things that he said, which has stuck with me ever since, was that scalar particles were the only ones that could have nonvanishing vacuum expectation values. He then went on to say that if you couple one of these to a fermion \Psi by a of the form \Phi \overline \Psi\Psi, then this vacuum expectation value would act like a mass. This sort of coupling is how mass generation is done in principle for the fermions. All particles in this picture would acquire their masses from the vacuum."

"There seems to be a vast landscape of possible universes. ... We live in one in which life is possible, but if the universe were only slightly different, beings like us could not exist. What are we to make of this fine-tuning? Is it evident that the universe, after all, was designed by a benevolent creator? Or does science offer a different explanation?"

"The old cosmological constant problem is to understand why the is so small; the new problem is to understand why it is comparable to the present mass density. ... does not help with either; anthropic considerations offer a possibility of solving both. In theories with a that takes random initial values, the anthropic principle may apply to the cosmological constant, but probably to nothing else."

"Once one starts to admit anthropic interpretations of fine-tuning problems like the cosmological constant, it is clear that such a proposal might be made for other fine-tuning problems, such as the problem of the Higgs boson mass. Certainly, we would not be here if the Higgs boson mass, and hence also the and and and masses, were greatly bigger. If they were near the , for example, any collection of more than a few elementary particles would collapse into a Black Hole. More generally, if the elementary particle masses were scaled up by a factor N, the number of elementary particles in a star or planet would scale down like N–3, and for very modest N the stars would stop shining."

"… Remember that quark masses arise in the Standard Model because the Higgs field has a non-zero vacuum value. Roberto and I saw that one could add an additional symmetry to the theory in such a way that it is automatic that the vacuum energy is minimized for θeffective = 0. Technically this new global U(1) symmetry is not quite an exact symmetry. Like the strong CP symmetry itself, it is a pseudo-symmetry, broken only by non-perturbative or instanton (tunneling) effects. … The additional pseudo-symmetry has a consequence, as was pointed out by Weinberg … and Wilczek … namely that there is an additional pseudo-Goldstone boson, now known as the axion, associated with it. The fact that Roberto and I did not notice this obvious phenomenological consequence of our model shows that we were focused on the general solution to the strong CP problem. I, at least, was so happy to find a general solution to that that I did not stop to examine other phe- nomenological implications of the model we built to demonstrate the idea before we published it. But the axion implication is common to all such models, for it arises from the symmetry itself."

"... In my view, after more than 25 years, the preferred solution to the strong CP problem still remains the idea that the Standard Model has an additional U(1)PQ symmetry. Such a solution, necessarily, predicts the existence of a concomitant axion."

"It is pointed out that a global U(1) symmetry, that has been introduced in order to preserve the parity and time-reversal invariance of strong interactions despite the effects of instantons, would lead to a neutral pseudoscalar boson, the "axion," with mass roughly of order 100 keV to 1 MeV."

"Nuclei interact with the external environment through a number of different fields—electromagnetic, weak and hadronic. The collective excitations induced by these interactions are known as giant resonances. The best known example is the giant dipole resonance, which is stimulated when the electric field of an incident gamma ray exerts a force on the positively charged protons in a nucleus, moving them relative to the uncharged neutrons ... Other giant resonances that have been studied are the monopole, quadrupole and spin-isospin modes of oscillation. The spin-isospin mode involves charge-changing processes, in particular beta decay. The quadrupole and monopole giant resonances are most easily seen with fields that act equally on neutrons and protons, because in these modes the neutrons and protons oscillate in the same mode. The giant resonances are collective oscillations and the various modes of oscillation depend on specific aspects on the nuclear force to sustain them. In the monopole mode, the motion is radial and the frequency depends on the compressibility of the nucleus. In the dipole and spin-isospin resonances, the protons and neutrons are excited out of phase, and the proton-neutron interaction provide the restoring force."

"The spectrum of gamma-radiation emitted by a highly excited nucleus can be calculated in two ways. In the first method the transition probability for gamma emission is related to the photon absorption cross-section by detailed balance. The second method relies on the fact that an excited hot nucleus has thermal fluctuations. In particular it has a fluctuating dipole moment which produces thermal radiation. The two methods are closely related and in both cases the spectrum of the radiation emitted is dominated by the giant dipole resonance. The equivalence of the detailed balance and thermal radiation theories can be demonstrated explicitly for a coupled oscillator model of the giant resonance."

"Perhaps the best-known giant resonance in nuclei is the giant dipole resonance (GDR). The GDR is described in classical hydrodynamics as a class of nuclear motion in which the neutrons and protons within a nucleus move collectively against one another, providing a separation between the centers of mass and charge, thus creating a dipole moment."

"A powerful method to study the properties of a system is to subject it to a weak external perturbation and to examine its response. For the atomic nucleus subjected to the absorption of a photon or to the scattering of a particle (electron, proton, etc.) the response is ... a function of the energy and linear momentum transferred to the system. ... Up to about 10 MeV the nucleus responds through the excitation of relatively simple states often involving only one or a few particles. In the energy range between 10 and 30 MeV the system response exhibits broad resonances. These are the giant resonances ... Giant resonances correspond to a collective motion involving many if not all the particles in the nucleus. The occurrence of such a collective motion is a common feature of many-body quantum systems. In quantum-mechanical terms the resonance corresponds to a transition between the ground state and the collective state and its strength is described by a transition amplitude. Intuitively it is clear that the strength of the transition will depend on the basic properties of the system such as the number of particles participating in the response and the size of the system. This implies that the total transition strength should be limited by a sum rule which depends 'only' on ground-state properties. If the transition strength of an observed resonance exhausts a major part, say greater than 50%, of the corresponding sum rule we call it a giant resonance."

"Maurice Goldhaber has emphasized that the situation with respect to possible nuclear resonances in (γ,n) or (γ,fission) reactions was quite unclear at the time of George C. Baldwin and G. Stanley Klaiber’s papers on these reactions. ... This was because the rapid rise of their yield to a prominent peak with increasing energy, followed by a slower fall off was then thought to have been due to the competition between the rapidly rising density of nuclear states and the eventual domination of other reaction channels at higher energies. Goldhaber realized, however, that there could be an analogy between a possible collective nuclear resonance and the restrahl resonance (essentially the transverse optical phonon mode) in polar crystals. Goldhaber sought out Teller because of his paper with Russell Lyddane and Robert Sachs, ... relating the restrahl frequency to the asymptotic behavior of the crystal’s dielectric function. Goldhaber and Teller, in their paper together, went on to predict universal, giant photo-nuclear resonances. ..."

"In their theory of superconductivity, Bardeen, Cooper, and Schrieffer made use of a reduced Hamiltonian which included only scattering of pairs of particles of opposite momentum and spin. It is shown that the solution they obtained by a variational method is correct to O(\tfrac{1}{n}) for a large system. The single particle Green's function is derived and used to calculate the interaction energy."

"Some time in 1960 or early 1961, I learned of an idea which had originated earlier in solid-state physics and had been brought into particle physics by Heisenberg, Nambu, and Goldstone, who had worked in both areas. It was the idea of "broken symmetry," that the Hamiltonian and commutation relations of a quantum theory could possess an exact symmetry, and the physical states might nevertheless not provide neat representations of the symmetry. In particular, a symmetry of the Hamiltonian might turn out to be not a symmetry of the vacuum."

"This work is a part of an effort to analyze the physical limitations of computers due to the laws of physics. For example, Bennett ... has made a careful study of the free energy dissipation that must accompany computation. He found it to be virtually zero. He suggested to me the question of the limitations due to quantum mechanics and the uncertainty principle. I have found that, aside from the obvious limitation to size if the working parts are to be made of atoms, there is no fundamental limit from these sources either. We are here considering ideal machines; the effects of small imperfections will be considered later. This study is one of principle; our aim is to exhibit some Hamiltonian for a system which could serve as a computer."

"Gauge symmetry: Whenever the Hamiltonian is such as to conserve the total number of particles of a particular sort—or, more generally, where there is a conserved "charge"-like quantity, such as lepton or baryon number, or electric charge itself—we shall find that the Hamiltonian will exhibit a gauge invariance property."

"The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies."

"One of the simplest chemical exchange reactions involves a system of three hydrogen atoms: H+H2→H2+H. Surely, chemists have felt, one should be able to calculate the cross sections for this reaction from first principles. But the computations have not been easy. Only in the last six years or so have theorists, aided by efficient methodologies and access to supercomputers, been able to predict the cross sections in sufficient detail for comparison with experiments, which themselves have evolved in precision. The agreement has been good—well, almost. Small discrepancies, especially at higher total energies, stubbornly refused to yield to adjustments in either the calculations or the experiments. Now Yi‐Shuen Mark Wu and Aron Kuppermann of Caltech have erased these pesky discrepancies by including a topological effect known as the geometric phase. Michael Berry (University of Bristol) has called attention to the presence of this phase, which now bears his name, in a wide variety of physical systems."

"Examples of geometric phases abound in many areas of physics. Many familiar problems that we do not ordinary associate with geometric phases may be phrased in terms of them. Often, the result is a clearer understanding of the structure of the problem, and of its solution."

"One of the reasons for being interested in the geometric phase is that it connects a number of different areas."

"The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles."

"Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics."

"While at the time when I carried out this calculation on the basis of the radiation law an exact test of the value... was... impossible... it was not long before E. Rutherford and H. Geiger succeeded, by... a direct count of the α-particles, in determining the value of the electrical as 4.65 10-10, the agreement... with my value 4.69 10-10... a decisive confirmation of my theory. ...[M]ethods ...by E. Eegener, R. A. Millikan, and others... have led to a but slightly higher value."

"Nothing can better illustrate the rapid progress of experimental physics within the last twenty years than the fact that... [by] a host of methods ...the mass of a single molecule can be measured with almost the same accuracy as that of a planet."

"[T]he quantum of action, which in every one of the many and most diverse processes has always the same value, namely 6.52 10-27 erg. sec., deserves to be... incorporated into the system of the universal s."

"To work out these probability considerations the knowledge of two universal constants is required, each of... independent meaning, so... evaluation of these... from the radiation law could serve as an a posteriori test whether the... process is merely a mathematical artifice or has a true physical meaning."

"The first constant... is connected with the definition of temperature. If temperature were defined as the mean of a molecule in a , which is a minute energy indeed, this constant would have the value ⅔. But in the conventional scale of temperature the constant ...[instead] assumes an extremely small value... intimately connected with the energy of a single molecule... [I]ts accurate determination would lead to the calculation of the mass of a molecule and... associated magnitudes. This constant is frequently termed Boltzmann's constant, although to the best of my knowledge Boltzmann... never introduced it (...he, as appears from... his statements, never believed it would be possible to determine this constant accurately)..."

"[A]t just the same time as the idea of general relativity arose... nature revealed, precisely... where ...least ...expected, an absolute and strictly unalterable unit, by means of which the amount of action contained in a space-time element can be expressed by a perfectly definite number, and thus is deprived of its former relative character."

"While originally it was a question of fitting in with as little strain as possible a new and strange element into an existing system... generally regarded as settled, the intruder... having won an assured position, now has assumed the offensive; and... is about to blow up the old system... The only question... is, at what point and to what extent this will happen."

"If the latter were the case, the would have to play a fundamental role in physics, heralding the advent of a new state of things, destined, perhaps, to transform completely our physical concepts which since the introduction of the infinitesimal calculus by Leibniz and Newton have been founded upon the assumption of the continuity of all causal chains of events."

"The inverse of the process of producing light quanta by the impact of electrons is the emission of electrons on exposure to light-rays, or s, and here... energy quanta following from the action quantum and the vibration period play a characteristic role, as was early recognized from the striking fact that the velocity of the emitted electrons depends not upon the intensity but only on the colour of the impinging light. [T]the relations to the light quantum, pointed out by Einstein, have proved successful in every direction... shown especially by R. A. Millikan, by measurements of the velocities of emission of electrons, while the importance of the light quantum in inducing photo-chemical reactions was disclosed by E. Warburg."

"Its first brilliant success was the derivation of Balmer's formula for the spectrum series of and , together with the reduction of the universal constant of Rydberg to known magnitudes; and even the small differences of the Rydberg constant for these two gases appeared as a necessary consequence of the slight wobbling of the massive atomic nucleus (accompanying the motion of electrons around it). As a sequel came the investigation of other series in the visual and especially the X-ray spectrum aided by Ritz's resourceful combination principle, which only now was recognized in its fundamental significance."

"As in the case of energy, it is now possible to define an absolute value of entropy, and thus of physical probability, by fixing the additive constant so that together with the energy (or better still, the temperature) the entropy also should vanish."

"Such... led to a comparatively simple method of calculating the physical probability of a given distribution of energy in a system of resonators, which yielded precisely the same expression for entropy as that corresponding to the radiation law; and it gave me particular satisfaction, in compensation for the many disappointments... to learn from Ludwig Boltzmann of his interest and... acquiescence in my... reasoning."

"Much less simple... was the interpretation of the second universal constant of the radiation law... the product of energy and time (...a first calculation to 6.55 10-27 erg. sec.) I called the elementary quantum of action."