Eigen Value Equation: Dirac Particles and Dirac Oscillators

Eigen Value Equation: Dirac Particles and Dirac Oscillators The thermodynamic entities with the GUP for Dirac particles  and Dirac oscillators Xin-feng Diao,  Chao-yun Long,  Guang-yu sun,  Yuan-sheng wang,  Hongling liu Abstract In this paper we studied the Eigen value equation for Dirac particles and Dirac Oscillators, considering the spin and Generalized Uncertainty Principle. Then we calculated the thermodynamic entities for them with the Generalized Uncertainty Principle corrected. We find that an electron of mass m and spin 1/2 in relativistic quantum mechanics confined in a box with the length L which the eigenvalues were related to the length of box and the correction terms of the Helmholtz free energy tend to increase the thermodynamic quantities. PACS number: 03.65.-w, 11.10.Nx Key words: GUP; Dirac particles ; Dirac Oscillators; thermodynamic entities. Introduction Various candidates of quantum gravity such as string theory and black hole physics concern the existence of a minimum measurable length. When energies of particles are much smaller than the scale of Planck mass [1, 2], it gives rise to the so-called Generalized Uncertainty Principle that results in a minimum observable length [3] (1) where is the GUP parameter and is a positive constant which depends on the expectation value of the momentum operator. On the other hand, Braun Majumder has discussed the harmonic oscillators following Maxwell-Boltzmann statistics by solving the Schrà ¶dinger equation[4]. However for the high energy particles we should consider the relativistic effect, so that it is important to study the effect of the Generalized Uncertainty Principle for Dirac particles and Dirac oscillators. Further more, the canonical partition function and other thermodynamic quantities for the relativistic particles following Maxwell-Boltzmann statistics should be involved. We discused the aspects in this paper. The Generalized Uncertainty Principle can be obtained from the deformed commutation relation, (2) where . The limits andcorrespond to the ordinary quantum mechanics and extreme quantum gravity, respectively. +Email: [emailprotected]. Now let us consider an electron of mass m and spin 1/2 in relativistic quantum mechanics, confined in a box of length L. The boundaries of the box are located at and. The wave function of the particle satisfies the following GUP corrected Dirac equation inside the box, where potential energe while and outside. The Dirac equation can be given as à ¯Ã‚ ¼Ã‹â€ 3à ¯Ã‚ ¼Ã¢â‚¬ ° where and are the Dirac matrices with the following representation , à ¯Ã‚ ¼Ã‹â€ 4à ¯Ã‚ ¼Ã¢â‚¬ ° Using the Jacobi identity [5]. And we can solve the equation with the method in the paper [6]. By defining , à ¯Ã‚ ¼Ã‹â€ 5à ¯Ã‚ ¼Ã¢â‚¬ ° considering the boundary conditions, we can get the energy Eigen value à ¯Ã‚ ¼Ã‹â€ 6à ¯Ã‚ ¼Ã¢â‚¬ ° Wrong calculation And we utilized the GUP corrected energy spectrum to calculate the canonical partition function and other thermodynamic quantities for the relativistic particles following Maxwell-Boltzmann statistics. So, we first calculated the GUP corrected partition function and it can be expressed as à ¯Ã‚ ¼Ã‹â€ 7à ¯Ã‚ ¼Ã¢â‚¬ ° So the partition function is wrong too with the. For the case of indistinguishable particles we use the relation of Helmholtz free energy with partition function . For the Helmholtz free energy and it comes out to be à ¯Ã‚ ¼Ã‹â€ 8à ¯Ã‚ ¼Ã¢â‚¬ ° where N is total number of Dirac particles. We found that the correction terms tend to increase the thermodynamic quantities. Then, we consider the Dirac oscillator and get the exact solution under a harmonic term. Firstly, Dirac equation is written as [7] (9) where U0 and V0 denote scalar and vector interactions, respectively, and the matrices are à ¯Ã‚ ¼Ã…’ (10) The spin wave function can be written as (11) We expand the equation and get the coupled equations (12) Then, (13) Pluging in to (12), we can obtain (14) (15) Here, we consider the harmonic term Using the operator relation (16) the equation becomes (17) à ¯Ã‚ ¼Ã…’ And we can get (18) With the method of [2], the energy of the equation (18) will be obtained . (19) If we set , the result becomes , which was well agree with non-relativistic quantum mechanics. And then we calculate the thermodynamic entities with the GUP corrected energy eigenvalue equation for the Dirac Oscillators. The partition function can be evaluated as . (20) We do this sum in a perturbative sense to distinguish the à ¯Ã‚ ¬Ã‚ rst term as the partition function of Dirac Oscillators, the equation can be rewritten as with. This equation guides us to write the GUP modified Helmholtz free energy as . (21) Simply, we write the expressions for the entropy internal energy as (22) where N is total number of Dirac oscillators. Summary In this paper, we studied an electron of mass m and spin 1/2 in relativistic quantum mechanics, which was confined in a box with the length L, We found that the eigenvalues were related to the length of box and the correction terms of the Helmholtz free energy tend to increase the thermodynamic quantities. We consider the Dirac oscillator and get the exact solution under a harmonic term, although the GUP corrected Hamiltonian of the harmonic oscillator has investigated[7-10]. We consider the different operator relation and get the partition function for the Dirac Oscillator. Moreover, we calculated the thermodynamic entities with the GUP corrected energy Eigen value equation for the Dirac Oscillator. So exploring relations in the basic foundations of the GUP is worth interesting [11]. This work was Supported by the Project of Guizhou Province Science and Technology OfficeNo. [2013]2255 and Guizhou Normal College project: 12YB005 . References [1] K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B 234 (1990) 276. [2] M. Maggiore, Phys. Lett. B 304 (1993) 65. [3] A. Kempf, G. Mangano, R.B. Mann, Phys. Rev. D 52 (1995) 1108. [4] Barun Majumder , Sourav Sen. Physics Letters B 717 (2012) 291–294 [5] H. Hassanabadi a, S. Zarrinkamar b, A.A. Rajabia. Physics Letters B 718 (2013) 1111–1113 [6] A. Kempf, J. Phys. A 30 (1997) 2093. [7] Pouria Pedram. Physics Letters B 710 (2012) 478–485 [8] P. Pedram, Phys. Rev. D 85 (2012) 024016, arXiv:1112.2327. [9] K. Nozari, T. Azizi, Gen. Rel. Grav 38 (2006) 735742; K. Nozari, H. Mehdipour, Chaos Solitons Fractals 32 (2007) 1637; K. Nozari and A.S. Sefidgar, Chaos, Solitons and Fractals 38 (2008) 339. [10] B. Majumder, Phys. Lett. B 701 (2011) 384. [11] S. Kalyana Rama, Phys. Lett. B 519 (2001) 103.

The Effects of High Glucose Intake

Glucose is a sugar that comes from the breakdown of all carbohydrates in the diet, and is very soluble in water. The chemical formula for glucose is C6H12O6. Glucose is a vital part of brain function as it is the only fuel that your brain can use. If you are consuming a low-carbohydrate diet, you may not be getting enough glucose for your brain to function properly. Because your brain ultimately controls all other functions within your body, giving your brain too little glucose can be dangerous to your overall health. Also, overconsumption of sugar in the body can lead to unpleasant phenomena. According to biochemistry Reginald Garrett, Ph. D. and Charles Grisham, Ph. D. the first major side effect of glucose overconsumption is a sugar high (hypoglycemia), which feels a bit like a caffeine buzz, because glucose absorption occurs really fast in the body when partaking too much sugary foods. Diabetes is a very scary and progressive disease, which can result a serious social problem and an economic burden on a person who has this kind of disease. Collective efforts are needed to improve both the level of knowledge about the disease and the outcomes of patient management. Therefore, the American Diabetes Association (ADA) aiming to increase the level of knowledge about diabetes and to determine both appropriate uses of blood glucose monitoring and appropriate measures required to accurately assess the performance of this practice. The first goal of ADA is to identify critical elements of a proper clinical trial of self-monitoring of blood glucose in type2 diabetes. According to American Diabetes Association, a person who has type 2 diabetes or a non-insulin-dependent has a lot of complications because their body can't use the insulin properly but their pancreas produces too much insulin. So, when the body can't use enough insulin and there is too much insulin in the body, the glucose or the sugar will build up in the blood instead of going to the cells. When these things happen, the cell of the body can't function properly because the cells need sugar for the energy. There are some problems found when the glucose build up to the blood and it include the following: first is damaging of the body, where the high glucose (sugar) levels in the blood damage the nerve and the small blood vessels in the eyes, kidneys, and the heart, and it can cause the hardening of a patients artery and it can lead into a sudden heart attack or maybe a stroke. Second is the malfunction of the kidney when the high amount of glucose in the blood of a patient, and it can cause a frequent urination which a large amount of water will be lost in the body, this can cause of dehydration. Third is when a person can't change the fluid that they lost or the person is not able to intake some fluids when they are very dehydrated, the patient becomes very sick and lead into diabetic come which is a life threatening complications. There some cases that the symptoms are very mild and can't be detected, in these cases the patients usually want to drink more liquid and they feel very thirsty, the patients feel exhausted at all time, they always feel nausea with the unclear vision, their skin is usually dry and flaky, when they get wounds it is very slow to get healed, and can't feel anything on their hands and feet because of losing sensation. From all these symptoms, only doctors can diagnose weather you are a hypoglycemic or hypoglycemic by conducting some test or measuring the glucose in the blood after instructing the patient not to eat for eight hours before the testing. This test is called glucose tolerance. There were an experimental studies and observational studies that too much intake of sugar in the body is more likely to develop type 2 diabetes, and too much sugar intake is associated with the development of obesity. Although there's no established RDA (Recommended Daily Allowance) for sugar, because as far as it's known, sugar is not really needed for nutrition, compare to salt that has restrictions in taking it. All foods have some natural sugars, but sugar itself, the white or brown but the stuff in the cookies, is a purely optional taste sensation. The USDA (United States Department of Agriculture) recommends that sugar make up no more than 8% of the daily intake of calories, but most. American adults take in twice that much. If you still consume sugar, it should be 12 teaspoons added sugar for a 2000 calorie per day diet, that is, no more than 40 grams per day for a 2000 calorie diet. Remember that the daily limit of 40 grams refers to refined and processed sugars only. There seems to be no limit on natural sugars. In these studies using the models called econometric of cross-sectional data on diabetes which is repeated and there are 175 countries participate for the nutritional components of food, these studies found that one can of soda is 150 kcal per person per day and it is increasing the availability of sugar and there is a relation with increasing of diabetes cases. When they're done with this testing and tried to control for other food types like; some meat, some fruits, some oils, and some cereals, monitoring the total calories intake, the overweight person and some obesity cases, some period effects, and some social economic like their income, their crowded places and a very low income. There was no other food typed that was yielding, important individual association with the total number of cases of a disease in a given population at a specific time after they were controlling the case for obesity. According to this study, the sugar on diabetes was the biggest impact and it was not dependent of tending to sit about without taking much exercise and the usage of alcohol, there was an effect which is modified but it's not that clear for obesity or for the diabetes. After all these reviews, the high sugar intake and the proportion of a population found to have diabetes, and the bearing of high sugar intake on overweight are being discussed. The evidence is not accurate for the proportion of a population found to have diabetes was because of high sugar intake, but it doesn't mean that taking too much sugar in the body is not important. Since the population of obesity in the world, and the case of diabetes is growing rapidly, there should be restrictions of sugar in the diet, and it is very important like the other restrictions of diet. In conclusion the requirement of screening is the screener need to show them that they like the benefit of screening. It didn't show that there is a clear benefit from the screening of the hyperglycemia or they called it glucose intolerance, mostly was women who are pregnant, and pregnant women who are at risk of caesarean section, and the disadvantage which is the acquired of the status of this disease. That screening was being argued on how to identify somebody who is at risk of succeeding non-insulin-dependent (type 2 diabetes) or a patient who is suspected to have that disease and undiagnosed is a good thing.

Asl And The American Sign Language - 880 Words

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The most of the deaf people are usually comfortable communicating with hearing people and people are happy that the hearing people are attempted using American Sign Language to communicate and instead ignoring them. American Sign Language has overcome the communication barrier for hard of hearing, hearing people, infants, toddlers and elderly grandparents who could not hear. American Sign Language is a visual language, for example, the facial expression, body movements and hand gesture. The deaf people priority was in person interacting and eye contactsShow MoreRelatedAmerican Sign Language ( Asl )1686 Words   |  7 Pages Classifiers are handshapes we use in American sign language (ASL) to show the movement, placement, orientation, size, and shape of a noun. Since ASL is a rule-governed language when using classifiers you must first identify the noun, then you can use the classifier to show how the object moves or is placed in relationship to other objects (Aron). 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As the name describes, this language relies solely on visible movements and facial expressions with no auditoryRead MoreDeaf Culture1589 Words   |  7 PagesDeaf Culture In mainstream American society, we tend to approach deafness as a defect. Helen Keller is alleged to have said, Blindness cuts people off from things; deafness cuts people off from people. (rnib.org) This seems a very accurate description of what Kellers world must have been. We as hearing people tend to pity deaf people, or, if they succeed in the hearing world, admire them for overcoming a severe handicap. We tend to look at signing as an inferior substitute for real communication

Aging Population A Growing Challenge - 1018 Words

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