Quantum Optical Model Nonintegrability Quantum Fluctuation

Quantum Optical Model Nonintegrability Quantum Fluctuation Nonintegrability and quantum fluctuations in a quantum optical model Nilakantha Mehera and S. Sivakumarb Abstract Integrability in quantum theory has been defined in more than one ways. Recently, Braak suggested that a quantum system is integrable relating the number of parameters required to specify the eigenstates and the number degrees of freedom (both discrete and continuous). It is argued that the dependence of uncertainty product of suitable operators on the atom-field interaction strength is distinctly different for the integrable and nonintegrable cases. These studies indicate that the uncertainty product is able to identify the nonintegrable systems from the integrable ones in the context of this new definition. Introduction A classical dynamical system with n degrees of freedom (DOF) is integrable, Liouvillean integrable to be precise, if there are equal number of suitable constants of motion (COM) that have vanishing Poisson bracket among themselves and with the Hamiltonian1.. Otherwise, the system is nonintegrable. While this definition is based on a sound mathematical footing, the situation in quantum dynamics is not very clear, essentially arising from the difficulty in defining or identifying DOF in quantum theory2. One possibility is define integrablity by the existence of sufficient number of observables which commute with the Hamiltonian and pair-wise commute among themselves. However, this is wrought with difficulties as it may not be possible to arrive at classical limits of some quantum systems. One such example is the case of a single two-level atom interacting with a single mode of the electromagnetic field. The former is a discrete DOF (finite dimensional Hilbert space) and the later is a continuous DOF (infinite dimensional Hilbert space). While the continuous DOF, namely, the electromagnetic field, has a proper classical limit, the two-level atom does not have a suitable classical limit. According to a new definition introduced by Braak, a system is integrable if the number of parameters required to specify the eigenstates of the Hamiltonian is equal to the sum of the number of discrete DOF and continuous DOF2. This definition does not involve the existence of constants of motion, though all such cases are covered by this definition. In this new definition of integrability, some of the nonintegrable systems based on the Liouvillean definition become integrable. A simple example of such a system is the Rabi model describing the interaction between a two-level atom and a single mode of the electromagnetic field with Hamiltonian3; (1) Here, are Pauli matrices, is the atomic transition frequency, () denote the annihilation (creation) operators of field with frequency. is the atom-field coupling constant. This Hamiltonian has only one COM, namely itself. Since there are two DOF, the field and the two-level atom, the Hamiltonian is nonintegrable in the sense of Liouville. However, exploiting the parity symmetry in , the system has been shown to be integrable2. Another interesting case is the rotating wave approximation of , yielding the well known Jaynes-Cummings model4,5: (2) This Hamiltonian has two COM, the Hamiltonian itself and the operator for the number of excitations . Existence of these two COMs renders the Hamiltonian integrable. The eigenstates are labelled by two parameters,an integer n corresponding to the number of excitations and the total energy. Also, both the Hamiltonians and exhibit level-crossings of the eigenvalues as the interaction strength g is varied, which is an indication that the models are integrable2. Level-crossing refers to the phenomenon where in the eigenvalues depend on the interaction strength g in such a way that the eigenvalues corresponding to two different eigenstates become degenerate at a specific values of g and reverse their order for other values of g. Nonintegrable Model : An interesting modification to to make it nonintegrable is to break the à ¯Ã‚ Ã… ¡2 symmetry by adding and this leads to; (3) Within the scope of the Braak’s definition of integrability, this model is integrable only when ÃŽ µ is an integral multiple of ω/2. This is also borne out by the existence of level crossings as shown in Fig.1. This criterion is sufficient for nonintegrability. We assume resonance . For the results presented here, it is assumed that =1 and ω=1. In Fig. 1, the absence or presence of level-crossing indicates respectively the nonintegrability or integrability of the Hamiltonian . Fig.1.Energy level (En) as a function of g for different à °Ã‚ Ã…“â‚ ¬. Level crossing occurs if à °Ã‚ Ã…“â‚ ¬=0 and 0.5 indicating integrability. No level crossing if à °Ã‚ Ã…“â‚ ¬=0.3, indicating nonintegrability. Inset shows larger view of level crossing. A pertinent question in this context is to know those features that distinguish a nonintegrable atom-field system from an integrable one. One answer to this query appears to be that uncertainty product of a pair of suitably defined operators show markedly different characteristics. Since the system is nonintegrable, it is formidable to construct an analytical solution. Therefore, extensive numerical experimentations have been carried out and the results are presented here which support the claim stated above. Nonintegrability being a feature of the Hamiltonian, it is natural to expect that the eigenstates carry signatures revealing this feature. To explore this, we define two self-adjoint operators of the two-level atom, , , where () is the atomic raising (lowering) operator. The commutation relation implies that the value of the product of uncertainties lies between 0 and 1/2. The uncertainty relation of above operators is . whereis expectation value in any eigenstate. In Fig. 2, the uncertainty product is plotted as a function of the atom-field interaction strength for different values of à °Ã‚ Ã…“â‚ ¬: à °Ã‚ Ã…“â‚ ¬= 0, 0.5 and 1.0 corresponding to the integrable case and a few other values of à °Ã‚ Ã…“â‚ ¬ corresponding to nonintegrable case. It is seen that as the parameter g is increases, the uncertainty product attains its maximum allowed value of  ½ for the integrable cases. On the other hand, for the nonintegrable cases the uncertainty product falls below the limit of  ½. In order to establish that the uncertainty product is very sensitive to the nature of the the integrable and nonintegrable cases, the plots corresponding to values of à °Ã‚ Ã…“â‚ ¬ very close to integrable cases have been chosen. Fig.2.Uncertainty product () as a function of the atom-field coupling constant g. Different plots correspond to different values of à °Ã‚ Ã…“â‚ ¬: integrable cases: à °Ã‚ Ã…“â‚ ¬=0, 0.5 and 1.0, nonintegrable cases: à °Ã‚ Ã…“â‚ ¬=-0.01,0.01, 0.49,0.51,0.2 and 0.4. In any plot, the uncertainty is plotted for the eingenstates corresponding to the first fifty eigenvalues. For instance, in the second row in Fig. 2, the sudden change in the nature of uncertainty product as à °Ã‚ Ã…“â‚ ¬ assumes values 0.49 (nonintegrable), 0.5 (integrable) and 0.51 (nonintegrable) respectively are shown. In order to bring out the features more clearly, the probability distribution of the uncertainty products in different eigenstates are shown in Fig 3 corresponding to the respective figures in Fig. 2. The sharply peaked probability distribution indicates integrability. Fig.3. Probability distribution of the uncertainty product for all the eigenstates for a particular value of g, chosen to be 1.2 here. Any higher value of g yields the same results. Summary Identification of nonintegrability in an interacting atom-field system is possible by the concentration of uncertainty product near a particular value as the atom-field interaction strength is increased. This feature seems to be related closely to the nonintegrability, also supported the absence of level crossings. This feature has been found to be able to identify nonintegrability in many other models that have been studied. In essence, suitable uncertainty product is able to identify nonintegrability, which is often difficult to establish analytically or numerically. Nevertheless, our analyses raise some important questions for which answers are to be found: Is it possible to arrive at the existence of this feature using only the definition of nonintegrability used here? Given a Hamiltonian, how to identify the correct observables whose uncertainty product will concentrate as the interaction strength is increased? How to extend this idea if the number of atoms is larger? References: M.V. Berry and M. Tabor, Proc. R. Soc. A 356, 375 (1977). D.Braak, Phys.Rev.Lett. 107, 100401(2011). I. I. Rabi, Phys. Rev. 49, 324 (1936); 51, 652 (1937). E. T. Jaynes and F.W. Cummings, Proc. IEEE 51, 89 (1963). C.Gerry and P.L. Knight, Introductory Quantum Optics (Cambridge University Press, UK, 2005).

Education: Its Aims And Objectives Essay

â€Å"Education†, says Aristotle,† is the creation of a sound mind in a sound body†. It encompasses in itself the all round development of an individual. The success of spreading education to the widest possible area lies in the way it is imparted. With the ever changing technology scenario, the methods of imparting education too have been undergoing changes. But education itself is an age old process, rather as old as the human race itself. It was man’s education through Nature, our greatest teacher, that he learned how to make fire by rubbing stones or invented the wheal to make tasks easier. Education in real earnest helps us in restraining the objectionable predisposition in ourselves. The aims of education have been categorized variously by different scholars. While Herbert Spencer believed in the ‘complete-living aim’, Herbart advocated the moral aim. The complete living aim signifies that education should prepare us for life. This view had also been supported by Rousseau and Mahatma Gandhi. They believed in the complete development or perfection of nature. All round development has been considered as the first and foremost aim of education. At the same time education ensures that there is a progressive development of innate abilities. Pestalozzi is of the view â€Å"Education is natural, harmonious and progressive development of man’s innate powers. † Education enables us to control, give the right direction and the final sublimation of instincts. It creates good citizens. It helps to prepare the kids for their future life. Education inculcates certain values and principles and also prepares a human being for social life. It civilizes the man. The moral aim of Herbart states that education should ingrain moral values in children. He is of the view that education should assist us in curbing our inferior whims and supplant them with superior ideas. This moral aim has also been stressed upon by Gandhiji in the sense of formation of character. The preachers of this aim do not undermine the significance of knowledge, vocational training or muscular strength. But simultaneously they have also laid stress on their view that the undisclosed aim of education is to assist development of moral habits. Then there is the social aim which means that education should produce effective individuals in the sense that they realize their responsibilities towards the society. And we all know that man is a social being. The interactive ability is a must as it is through interaction that we come to know of our responsibilities. Edmund Burke asks and he himself answers: â€Å"What is education? A parcel of books? Not at all, but an intercourse with the world, with men and with affairs. † Only bookish knowledge takes a child nowhere. It should be further perfected by practical usage with experience. â€Å"Reading maketh a full man; conference a ready man; and writing an exact man† is a pithy and precise statement in which the essayist Francis Bacon sums up the advantages of studies. Even Wordsworth in his poem ‘The Tables Turned’ advocated against bookish knowledge. Books! ’tis a dull and endless strife: Come, hear the woodland linnet, How sweet his music! on my life, There’s more of wisdom in it. Wordsworth was a die-hard naturalist. He wanted man to consider Nature his teacher. Naturalists believe that instincts of the child should be taken as the basis of education. The child should have freedom. Rabindra Nath Tagore was of the opinion that child should be left free in order to gather experience and to understand his own mistakes and shortcomings. The twentieth century saw the emergence of the concept of Pragmatism. Charles Pierce was the first man to introduce the concept of pragmatism in his philosophy. Later on it was popularized by John Dewey, William James, Kilpatrick and Schiller. They believed that the external world is real and the reality is being constantly created and is always changing. Knowledge and truth is one and the same thing according to them. Whatever the approach towards education, one thing we all agree: Education is for the betterment of the individual and in the long run for the society. Education helps us prepare ourselves for the life ahead. Darwin gave the theory of the ‘survival of the fittest’; we can say in a way that education prepares the individual for the struggle of life for his own survival. Knowledge combined with proper guidance can spell success. A dose of proper guidance should be commenced right from the base itself, that is, in school days. Here comes the role of the teacher in moulding a child’s mind. Educating a child, especially in the beginning years of schooling, is a very tricky job. That is probably because the child’s mind is like the unmoulded clay at that time. Therefore to get the best results and prepare well-informed and erudite adults, proper guidance is a must. For a proper system of education the teacher should encourage a student both in terms of mental encouragement and in lending a helping hand as and when needed. A student needs help for training his mind in such a way that it develops a tendency to gather knowledge from all possible sources. While on the other hand too much help if lent to him will make him dependant and used to spoon feeding. Self-study is the most sought after quality in a student. It helps them at the later stages. But because the ‘child is the father of man’ (Wordsworth) all the qualities have to be inculcated right in childhood. And teacher along with parents plays a very significant role. The aims of education should be kept in mind, although a thorough study of these aims may not be imperative. A teacher should make a child ready to face the society, inculcate moral habits in him and thus, assist him in his all round development. Education should not be considered synonymous with all that we learn. It does not signify the things we mug up before appearing for an examination. Education is what remains behind, when we fail to remember the mugged up portion. After we have left school, we realize that although we have forgotten quite a few things we learnt but still retain a very large part of it. The latter part is education. Education formally begins in school but actually it begins the day we are born and the process goes on for the whole of our life. This is where the aims of education come in. Education is not only the formal part we gain in schools, colleges or universities. It also includes the lessons life teaches us in various forms. For instance, when a child gets his finger pricked by a needle accidentally he learns that a needle is sharp and can hurt a person, so he will learn to avoid hurting himself in the future. This is only one example from thousands of other instances. We can even learn a lesson of a lifetime from a beggar. The birds inspire us to rise high. An ant motivates us for hard work. We learn some things just by doing them on our own, they are never taught in a school. A child’s first teacher is his mother, then his home and then come the formal agencies of education. Nature too is a great teacher. English poetry too gives us quite a few guidelines for leading a better life. It was not for nothing that Wordsworth went on to remark: â€Å"One impulse from the vernal wood May teach you more of a man Of moral evil and of good Than all the sages can. † As long as there is life, we require education; we need ways to modify our views about life, to face it, to live it in a better way. And education teaches us all this. Even when you read a comic strip, it educates you in some way. They improve our language and make us realize that life isn’t so bad after all that it can’t get worse, as states Bill Watterson in ‘Calvin and Hobbes’. The witty humour of ‘Dennis the Menace’ enriches us no end. â€Å"The aim of education,† says Walter Grophices, â€Å"is not the specialist but the man of vision who can humanize our life by integrating emotional demands with our new knowledge. † In another way too, the insects and animals also teach you a lot. The easiest example is that of an ant. It inspires you to work hard. Therefore we can say there are innumerable modes of education, all that one needs is to have a discerning eye. Education enriches a person in terms of accepting a defeat. A student should first of all be taught so that he is encouraged to study. Side by side he should be readied to face a failure. As Charles F. Kettering rightly says,† The chief job of the education is to teach people how to fail intelligently. † This will help the child coping with the other adversities of life. Education, thus, makes a person an improved version of himself and the world a much better place to live in.

Manufacturing alumina

The production of aluminum begins with the mining and beneficiation of bauxite. At the mine (usually of the surface type), bauxite ore is removed to a crusher. The crushed ore is then screened and stockpiled, ready for delivery to an alumina plant. At the alumina plant, the bauxite ore is further crushed or ground to the correct particle size for efficient extraction of the alumina through digestion by hot sodium hydroxide liquor. After removal of â€Å"red mud† (the insoluble part of the bauxite) and fine solids from the process liquor, aluminum trihydrate crystals are precipitated and calcined in rotary kilns or fluidized bed calciners to produce alumina (Al2O3). (Bounicore & Wayne 1992) Some alumina processes include a liquor purification step. Primary aluminum is produced by the electrolytic reduction of the alumina. The alumina is dissolved in a molten bath of fluoride compounds (the electrolyte), and an electric current is passed through the bath, causing the alumina to dissociate to form liquid aluminum and oxygen. The oxygen reacts with carbon in the electrode to produce carbon dioxide and carbon monoxide. Molten aluminum collects in the bottom of the individual cells or pots and is removed under vacuum into tapping crucibles. . Depending on the desired application, additional refining may be necessary. For demagging (removal of magnesium from the melt), hazardous substances such as chlorine and hexachloroethane are often used, which may produce dioxins and dibenzofurans. (Bounicore & Wayne 1992) Industrial forms of aluminum include commercially pure metal and alloys with other metals such as chromium, copper, iron, magnesium, manganese, nickel, titanium and zinc. Aluminum alloys may contain as much as fifteen percent of the alloying metals. In powder form, aluminum and its alloys are combustible in air and present a potential explosion hazard. In sheet or block forms, aluminum will not normally propagate or sustain combustion. (Metals & Alloys, 1976) Hazards and Risks Entail in Processing At the bauxite production facilities, dust is emitted to the atmosphere from dryers and materials- handling equipment, through vehicular movement, and from blasting. The dust is not hazardous; it can be a nuisance if containment systems are not in place, especially on the dryers and handling equipment. Other air emissions could include nitrogen oxides (NOx), sulfur dioxide (SO2), and other products of combustion from the bauxite dryers. (Paris Com, 1992) Ore washing and beneficiation may yield process wastewaters containing suspended solids. Runoff from precipitation may also contain suspended solids. At the alumina plant, air emissions can include bauxite dust from handling and processing; limestone dust from limestone handling, burnt lime dust from conveyors and bins, alumina dust from materials handling, red mud dust and sodium salts from red mud stacks impoundments), caustic aerosols from cooling towers, and products of combustion such as sulfur dioxide and nitrogen oxides from boilers, calciners, mobile equipment, and kilns. The calciners may also emit alumina dust and the kilns, burnt lime dust. Although alumina plants do not normally discharge effluents, heavy rainfalls can result in surface runoff that exceeds what plant can use in process. (Brady & Humiston, 1982) Hydrogen Generating Reactions Aluminum is a very reactive metal, and the greatest industrial hazards associated with aluminum are chemical reactions. Aluminum is an excellent reducing agent, and should react with water readily to liberate hydrogen. However, the protective aluminum oxide coating protects it from reaction with moisture or oxygen. If the protective coating is broken, for example, by scratching or by amalgamation (the process of coating with a film of mercury in which the metallic aluminum dissolves; the aluminum oxide coating does not adhere to the amalgamated surface), rapid reaction with moisture and/or oxygen can occur. The significance of this reaction is dependent upon the quantity of aluminum available to react. Aluminum is also oxidized by heat at a temperature dependent rate. (Ogle, Beddow, Chen, Butler, 1982) Aluminum metal is amphoteric (exhibits both acidic and basic characteristics). Therefore, aluminum will react with acids or bases; both reactions liberate hydrogen, a flammable gas. However, aluminum does not react with concentrated nitric acid because the oxidizing potential of the acid contributes to the formation of the protective aluminum oxide coating. (Martin, 1976) Thermite Reactions Aluminum readily extracts oxygen from other metal oxides to form aluminum oxide with the simultaneous release of large amounts of heat (enough heat to melt the products of the reaction). For example, the reaction of aluminum with ferric oxide to produce liquid aluminum oxide and liquid iron produces temperatures approaching 3000 °C (5400 °F). This reaction, referred to as the â€Å"thermite reaction,† has been used to weld large masses of iron and steel; when enclosed in a metal cylinder and ignited by a ribbon of magnesium has been used in incendiary bombs; and, with ammonium perchlorate added as an oxidizer, has provided the thrust for the space shuttle booster rockets. (May & Berard, 1987) Dust Explosions A dust explosion is a complex phenomenon involving simultaneous momentum, energy, and mass transport in a reactive multi-phase system. Aluminum particles, when in dust, powder, or flake forms from operations such as manufacturing powder, grinding, finishing, and processing, may be suspended as a dust cloud in air and consequently may ignite and cause serious damage. If the dust cloud is unconfined, the effect is simply one of flash fire. If, however, the ignited dust cloud is at least partially confined, the heat of combustion may result in rapidly increasing pressure and produce explosion effects such as rupturing of the confining structure. Aluminum dust is not always easily ignitable, and, therefore, the hazard of dust explosions is often ignored. Minimum explosive concentrations of aluminum dust have been reported upwards from about 40 grams per cubic meter (0.04 ounces per cubic foot) of air. (May & Berard, 1987) Effects on Health Aluminum particles deposited in the eye may cause local tissue destruction. Aluminum salts may cause eczema, conjunctivitis, dermatoses, and irritation of the upper respiratory system via hydrolysis-liberated acid. Aluminum is not generally regarded as an industrial poison, although inhalation of finely divided aluminum powder has been reported as a cause of pneumoconiosis. In most investigative cases, however, it was found that exposure was not solely to aluminum, but to a mixture of aluminum, silica, iron dusts, and other materials. Aluminum in aerosols has been referenced in studies involving Alzheimer's disease. Most exposures to aluminum occur in smelting and refining processes. Because aluminum may be alloyed with various metals, each metal (e.g., copper, zinc, magnesium, manganese, nickel, chromium, lead, etc.) may possibly present its own health hazards. (Buonicore & Davis, 1992) Implication Aluminum dust is strongly fibrogenic. Metallic aluminum dust may cause nodular lung fibrosis, interstitial lung fibrosis, and emphysema as indicated in animal experimentation, and effects appear to be correlated to particle size of the dust30; however, when exposure to aluminum dusts have been studied in man, most exposures have been found to be to other chemicals as well as aluminum. (Buonicore & Davis, 1992) Safety Measures: Prevention and Control The American Council of Governmental Industrial Hygienists (ACGIH) recommends the need for five separate Threshold Limit Values (TLVs) for aluminum, depending on its form (aluminum metal dust, aluminum pyro powders, aluminum welding fumes, aluminum soluble salts, and aluminum alkyls). The Occupational Safety and Health Administration (OSHA) has also established Permissible Exposure Limits (PELs) for aluminum. (May & Berard, 1987) Pollution prevention is always preferred to the use of end-of-pipe pollution control facilities. Therefore every attempt should be made to incorporate cleaner production processes and facilities to limit, at source, the quantity of pollutants generated. In the bauxite mine, where beneficiation and ore washing are practiced, tailings slurry of 7– 9% solids is produced for disposal. The preferred technology is to concentrate these tailings and dispose of them in the mined-out area. A concentration of 25–30% can be achieved through gravity settling in a tailings pond. The tailings can be further concentrated, using a thickener, to 30–50%, yielding a substantially volume reduced slurry. The alumina plant discharges red mud in slurry of 25–30% solids, and this also presents an opportunity to reduce disposal volumes. (May & Berard, 1987) Today’s technology, in the form of high-efficiency deep thickeners, and large-diameter conventional thickeners, can produce a mud of 50–60% solids concentration. The lime used in the process forms insoluble solids that leave the plant along with the red mud. Recycling the lime used as a filtering aid to digestion to displace the fresh lime that is normally added at this point can minimize these lime-based solids. Finally, effluent volume from the alumina plant can be minimized or eliminated by good design and operating practices: reducing the water added to the process, segregating condensates and recycling to the process, and using rainwater in the process. (Ogle, Beddow, Chen, Butler, 1982) References Brady, James E. and Humiston, Gerard E. (1982), General Chemistry: Principles and Structure, Third Edition, John Wiley and Sons, New York. Bounicore, Anthony J., and Wayne T. Davis, eds. (1992), Air Pollution Engineering Manual. New York: Van Nostrand Reinhold. Martin, R. (1975), â€Å"Dust-Explosion Risk with Metal Powders and Dusts,† P/M Group Annual Meeting 1975: Handling Metal Powders, Session I: Health and Safety in Powder Handling,† Powder Metallurgy, No. 2. May, David C., and Berard, David L. (1987), â€Å"Fires and Explosions Associated with Aluminum Dust from Finishing Operations,† Journal of Hazardous Materials, 17. â€Å"Metals and Alloys,† (1976), Loss Prevention Data 7-85, Factory Mutual Engineering Corporation. Paris Commission. (1992), â€Å"Industrial Sectors: Best Available Technology—Primary Aluminium Industry.† Ogle, R. A., Beddow, J. K., Chen, L. D., and Butler, P. B. (1988), â€Å"An Investigation of Aluminum Dust Explosions,† Combust. Sci. and Tech.   

Willa Cathers Issues with Realism and The Barn Burner,...

1. Willa Cather 2. Willa Cather and the Scarlet Letter by Nathaniel Hawthorne 3. The Barn Burner 4. A Rose for Emily by William Faulkner 1. Willa Cather seems to take issue with the bland and boring nature of realism above all else. She notes that realism is not in itself an artistic expression, yet so many art forms from literature to paintings—particularly from her time period—portray little more than the realism of our world. In her mind, the literalness that is realism can be successfully integrated in art, but it must be done in a specific way for it to qualify as true art. The literalism needs to fuse with the emotions and experiences of the characters and simply remain as a simple component of the artistic experience rather than the entire focus. The novelists and other artists of her time that she mentions and feels have placed too much focus on realism in their work are not real artists in her mind. Overall it seems that Cather believed that the growing prominence of realism within art forms in her time was leading to their downfall and their loss of true artistic expression. As she stated wi thin â€Å"The Novel Dà ©meublà ©,†If the novel is a form of imaginative art, it cannot be at the same time a vivid and brilliant form of journalism† (par. 7). Through this quote, she is essentially getting her aforementioned message across by comparing realism in art to mere journalism—the mere relaying of facts and realities without a shred of creativity or imagination