The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

新刊全像三寶 (sic) 太監西洋記通俗演義Xin kan quan xiang san bao tai jian xi yang ji tong su yan yi.Histoire du voyage en Occident de l'eunuque San bao, nouvelle édition illustrée.

Par Luo Mu tang (alias deng), surnom Er nan li ren, avec préface de l'auteur (1597). Récit des expéditions maritimes de Zheng He, surnom San bao tai jian († 1431). Portraits au début de chaque livre. Gravé au pavillon Bu yue.20 livres (100 hui).

新刊全像三寶 (sic) 太監西洋記通俗演義Xin kan quan xiang san bao tai jian xi yang ji tong su yan yi.Histoire du voyage en Occident de l'eunuque San bao, nouvelle édition illustrée.

Par Luo Mu tang (alias deng), surnom Er nan li ren, avec préface de l'auteur (1597). Récit des expéditions maritimes de Zheng He, surnom San bao tai jian († 1431). Portraits au début de chaque livre. Gravé au pavillon Bu yue.20 livres (100 hui).

楊淇園先生超性事蹟Yang qi yuan xian sheng chao xing shi ji.Vie et conversion de Yang Qi yuan.

Ce personnage, postnom Ting jun, docteur en 1592, fut converti par le P. Ricci. Ouvrage de Ding Zhi lin (XVIIe ou XVIIIe siècle).

不得已辯Bu de yi bian.Réfutation d'un libelle contre le christianisme, de l'astronome musulman Yang Guang xian.

Incomplet du livre 20, qui est remplacé par le livre correspondant d'un exemplaire de l'édition précédente.Par le P. Verbiest. Préface de l'auteur, non datée. Table des matières. Texte et figures.

妄推吉凶辯Wang zhui ji xiong bian.Réfutation de Yang Guang xian et des devins.

Par le P. Verbiest (1669) ; texte avec annotations dans la marge supérieure. Gravé de nouveau à Canton, au Da yuan tang.

Collaborative reflection : supporting one practitioner's development of online learning communities /

This is a study of one participant's reflective practice as she worked to develop online communities in a face-to-face science course. Her process of reflective practice was examined in order to address factors that influenced her learning path, and the benefits and challenges of collaborative action research. These research goals were pursued using a collaborative action research methodology, initially chosen for its close match with Schon's (1983) model of reflective practice. The participant's learning fit vnth Mezirow's (1991) model of transformative learning. She began with beliefs that matched her goals, and she demonstrated significant learning in three areas. First, she demonstrated instrumental learning around the constraints of workload and time, and achieving online learning community indicators. Second, she demonstrated communicative learning that helped her to see her own needs for feedback and communication more clearly, and how other process partners had been a support to her. Third, her emancipatory learning saw her revisiting and questioning her goals. It was through the reflective conversation during the planned meetings and the researcher's reframing and interrogation of that reflection that the participant was able to clarify and extend her thinking, and in so doing, critically reflect on her practice as she worked to develop online learning communities. In this way, the collaborative action research methodology was an embodiment of co-constructivism through collaborative reflective practice. Schon's (1983) model of reflective practice positions a lone practitioners moving through cycles ofplan-act-observe-reflect. The results fi"om this study suggest that collaboration is an important piece of the reflective practice model.

Effects of reflection and social isolation on crayfish behaviour /

Visual stimuli and socialization influence exploratory behaviour in crayfish. The predominant components of spontaneous exploratory behaviour were determined by observing the activity of solitary adult crayfish (Procambarus clarkii) in a glass aquarium containing fresh water and no objects. Five distinct behaviours were observed: rearing up (climbing on the wall), turning around, cornering (facing the comer), backward walking, and crossing (crossing the midline of the aquarium). The frequency of rearing up, cornering and turning around decreased when reflection from the glass wall was blocked with black cardboard, black paint or non-reflective transparent plastic. In a tank containing mirrors on one side and non-reflective plastic on the other, crayfish cornered, reared up, and turned around more in front of the mirrors. Socialization was necessary for crayfish to respond to the reflection. Crayfish that were housed in pairs for two weeks exhibited more rearing up, turning around and cornering in front of the mirrors than in the non-reflective side. Crayfish isolated for two weeks did not show these differences. Socialized crayfish also exhibited more rearing up, turning around and cornering than did isolated crayfish. Thus, crayfish respond to visual stimuli provided by a glass tank, but the responds depends on socialization.

An analysis of decoupling procedures in generating thermal Green's functions of O([lambda]â ´) by the Zubarev equation of motion method

The Zubarev equation of motion method has been applied to an anharmonic crystal of O( ,,4). All possible decoupling schemes have been interpreted in order to determine finite temperature expressions for the one phonon Green's function (and self energy) to 0()\4) for a crystal in which every atom is on a site of inversion symmetry. In order to provide a check of these results, the Helmholtz free energy expressions derived from the self energy expressions, have been shown to agree in the high temperature limit with the results obtained from the diagrammatic method. Expressions for the correlation functions that are related to the mean square displacement have been derived to 0(1\4) in the high temperature limit.

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metals

We have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the Lennard-Jones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the Lennard-Jones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

A qualitative analysis of reflection in undergraduate and graduate programs

The purpose ofthis study was to explore various types ofreflection and to explore reflection on action, reflection as a practice, and reflection as a process. In doing this, the intent was to discover the perceived benefits of reflection in the classroom and to provide guidelines for future use at the undergraduate and graduate level. The qualitative components in this study included the data collection strategy of semistructured interviews with 2 undergraduate students, 2 graduate students, 1 undergraduate studies professor, and 1 graduate studies professor. The data analysis strategies included a within-case analysis and a cross-case analysis. Through the interviews participants discussed their experiences with the use ofreflection in the classroom. Through the completion ofthis analysis the researcher expected to discover the benefits ofreflection at this level of education, as well as provide suggestions for future use. Both undergraduate and graduate students and professors were found to benefit from the use of reflection in the classroom. The use ofreflection in the undergraduate and graduate classroom was found to improve student/teacher and student/peer relationships, foster critical thinking, allow for connections between learned theory and life experience, and improve students' writing abilities. Based on the results ofthe study the implications ofreflection for the undergraduate and graduate classroom and for further research are provided.

Dialogue journal writing: a tool for critical reflection in the adult ESL learner

This study attempted to determine whether or not dialogue journal writing encouraged critical reflection in the adult ESL (English as a Second Language) learner. According to research in adult education and anecdotal evidence, the process ofdialogue journal writing can facilitate critical reflection in the adult learner. However, little research has been conducted to examine whether or not journal writing can facilitate critical reflection in the second language learner. As a result, ten low-intermediate level adult ESL students from Brock University's Intensive English Language Programme participated in a dialogue journal writing programme in their writing class. The participants wrote journal entries over a 10-week period, and were interviewed once throughout the process to determine their perceptions ofthe journal writing experience. They also were observed by the researcher throughout the journal writing sessions to establish whether any behaviours or intrusions might affect the participants' writing processes. After the content ofthe journals and the interviews, and the observations made by the researcher were analysed, it was confirmed that, for these participants, dialogue journal writing did not necessarily encourage critical reflection. Moreover, the participants' perceptions ofjournal writing were that it helped them to practise the syntax, vocabulary, and rhetorical patterns ofEnglish; nevertheless, it did not foster critical reflection or thinking.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

On the path integral formulation and the evaluation of quantum statistical averages

Four problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .

Optimization of trial wave functions for use in quantum Monte Carlo with application to LiH

Methods for both partial and full optimization of wavefunction parameters are explored, and these are applied to the LiH molecule. A partial optimization can be easily performed with little difficulty. But to perform a full optimization we must avoid a wrong minimum, and deal with linear-dependency, time step-dependency and ensemble-dependency problems. Five basis sets are examined. The optimized wavefunction with a 3-function set gives a variational energy of -7.998 + 0.005 a.u., which is comparable to that (-7.990 + 0.003) 1 of Reynold's unoptimized \fin ( a double-~ set of eight functions). The optimized wavefunction with a double~ plus 3dz2 set gives ari energy of -8.052 + 0.003 a.u., which is comparable with the fixed-node energy (-8.059 + 0.004)1 of the \fin. The optimized double-~ function itself gives an energy of -8.049 + 0.002 a.u. Each number above was obtained on a Bourrghs 7900 mainframe computer with 14 -15 hrs CPU time.