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Ladder Operators, Inherent Algebras and Associated Coherent States for Position-Dependent Mass Systems

Thesis Info

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Author

Naila Amir

Program

PhD

Institute

National University of Sciences & Technology

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2015

Thesis Completion Status

Completed

Subject

Natural Sciences

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/6710/1/Naila_Amir_Maths_NUST_2015.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726573941

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Position-dependent effective mass systems are of great significance due to the fact that these models have numerous applications in various areas of physics. The qualitative understanding of a complicated realistic system can be acquired by analyzing the exact solutions of a related simplified model. However, the quantization of position-dependent effective mass systems and finding their solutions, involve some conceptual and mathematical difficulties of a fundamental nature. The factorization method provides us with a powerful tool for obtaining solutions and the underlying algebraic structure of the exactly solvable systems. The underlying algebra of a system has vast applications in different areas of mathematics and physics, such as it plays an important role in the theory of coherent states. Coherent states are extremely useful in various areas such as quantum mechanics, quantum optics, quantum information and group theory. In this thesis, position-dependent effective mass systems are studied in the context of their quantization, finding the solutions, construction of the algebraic structure and associated coherent states. In the first part of the thesis we mainly focus on quantizing and obtaining the exact solutions of the systems with spatially varying mass. The next part deals with the construction of the ladder operators and the inherent algebra of the pertaining systems. The associated coherent states and their properties are presented in the final part of the thesis. Beside the traditional way of obtaining exact solutions by solving the Schrödinger equation there exists another elegant method to solve the systems algebraically by factorizing the corresponding Hamiltonian. This method is based on supersymmetric quantum mechanics and the integrability condition, commonly known as shape invariance. After quantizing the position-dependent effective mass system, this factorization technique is used to determine the energy spectrum and the corresponding wave functions. For the sake of completeness the iv Abstract v method of solving a time-independent Schrödinger equation with spatially varying mass is also discussed. Considering a non-linear harmonic oscillator as an illustrative example, it is shown that both the above procedures produce the same results. The property of shape invariance enables us to obtain the ladder operators of the confining system. A general recipe for the construction of the ladder operators and inherent algebra for the position-dependent effective mass systems is presented. In order to exemplify the general formalism, a non-linear harmonic oscillator together with several other examples of the shape invariant systems with position-dependent effective mass is considered. Explicit expressions for the ladder operators and the associated algebra are presented. Using the ladder operators and the underlying algebra, the coherent states for the positiondependent effective mass systems are constructed and their properties are analyzed. In particular, we emphasize on various kinds of coherent states for a non-linear harmonic oscillator with spatially varying mass. By realizing SU(1; 1) as the dynamic group of the system, the construction of Barut-Girardello coherent states is presented. In addition, an algebraic independent kind of coherent states, namely Gazeau-Klauder coherent states, are also constructed. The statistical properties of Barut-Girardello and Gazeau-Klauder coherent states are investigated by means of the Mandel parameter and the second order correlation function. Moreover, the temporal evolution of the Gazeau-Klauder coherent states is analysed by means of autocorrelation function. It is shown that these states mimic the phenomena of quantum revivals and fractional revivals during their time evolution.
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اہم تحقیقی و تدوینی اصطلاحات

اختلاف نسخ: تدوین متن میں میں مختلف نسخوں کے اختلافات
نسخہ: کسی قلمی یا مطبوعہ کتاب کی ایک جلد
اساسی نسخہ/بنیادی نسخہ: وہ نسخہ جسے تدوین میں اہم ترین مان کر متن دیا جائے۔
خطی نسخہ/قلمی نسخہ: ہاتھ سے لکھا ہوا نسخہ
دستخطی نسخہ: مصنف کے ہاتھ کا لکھا یا ٹائپ کیا ہوا نسخہ
وحید نسخہ: اگر کسی متن کا دنیا میں ایک ہی نسخہ ملتا ہو اور کوئی نقل نہ ہو تو اسے وحید نسخہ کہتے ہیں۔
آمیختہ نسخہ: وہ نسخہ جس کا متن پہلے کے دو نسخوں سے ملا کر تیار کیا گیا ہو۔
ماخذی نسخہ: جس نسخے سے کسی دوسرے نسخے کی نقل کی جائے
ماخذ: کتابیات کا پہلا مفہوم
مسودہ/کاپی ٹیکسٹ: جو صاف نسخہ تیارکرکے طباعت کے لیے دیا جاتا ہے
کلیات: کسی شاعر کا مجموعہ کلام (نثر /نظم) جو شاعر نے خود یا پھر کسی دوسرے شخص نے
مرتب کیا ہو۔
اشاریہ: کتاب کے آخر میں متن میں مذکورہ اشخاص، مقامات، کتب، اداروں وغیرہ
کی ہجائی ترتیب مع صفحہ نمبر
اوقاف: جملے، فقرے اور لفظ میں توقف اور تخصیص وغیرہ کے نشانات
رموزاوقاف: اوقاف کی علامتیں
بیاض: کسی کی ذاتی کاپی جس میں وہ اپنے یا دوسروں کے اشعار ، نظمیں یا غزلیں لکھ
لیتا ہے۔
تحریف: کسی شعر یا نثری جملیکے اصل متن میں تبدیلی کردینا
تخریج: کسی ادیب یا شاعر کے کلام میں دوسرے کلام کی نشاندہی کرنا
تحشیہ: کسی متن پر حاشیے لکھنا
ترقیمہ: مخطوطے کے آخر میں کاتب کی اختتامیہ عبارت جس میں کاتب کا نام، مالک
کتاب یا فرمائش کنندہ کا نام، زمان و مکان کتابت، اختتامی شعر وغیرہ میں
سے کچھ یا سب دیے ہوں
تسوید: کسی مضمو ن یا کتاب کا پہلا مسودہ لکھنا
تصحیح: متن میں اگر کچھ صریحا غلط ہے تو اس کو درست کرنا
حاشیہ:...

علماء اہلسنت (بریلوی) کی تفاسیر کا اسلوب بیان: ایک جائزہ

The Holy Quran is the first and very important source of Islamic Law. This book was revealed in Arabic language. When Islam spread over the Subcontinent with the passage of time then It was needed to translate this book in the regional languages for the better understanding of Allah's message. The people of this era could not understand the real teaching of Quran without its translation. The Ulema e Ahlesunat (Barelvi) school of thought took participation in the field of Tafseer to convey this holy message to the people of Subcontinent. The intellectual efforts of the scholars of the Ahlesunat (Barelvi) thoughts can be found in every field of Islamic teaching. These Ulema extended their contributions in Islamic teaching through illustration of the Quran. They wrote translations of the Quran in different ages and tried to solve the problems which were raised in this era about Islamic teaching. In this research article the authors analyzed the style of selected Mofasereen of the Ahlesunat( Barelvi's )school of thought

Synthesis and Characterization of Nanoparticles Added Cutl-Based Superconductors

Low anisotropic (Cu 0.5 Tl 0.5 )Ba 2 Ca 2 Cu 3 O 10-δ (CuTl-1223) phase of Cu 1-x Tl x Ba 2 Ca 2 Cu 3 O 10- δ [CuTl-12(n-1)n]; n=1, 2, 3, ... high temperature superconducting (HTSC) family was synthesized by solid-state reaction method. Nanoparticles (NPs) were prepared separately by different techniques (i.e. co-precipitation, sol-gel, and colloidal solution methods). Different kinds and sizes of NPs such as non-magnetic metallic (Ag, Au), non-magnetic metallic oxide (Al 2 O 3 ) and magnetic ferrite (CoFe 2 O 4 ) were added with different wt.% during the final sintering process of CuTl-1223 superconducting matrix to get the required (NPs) x /CuTl-1223 nanoparticles-superconducting composites. These composites were characterized by different experimental techniques such as x-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), energy dispersive x-ray (EDX) spectroscopy, resistivity versus temperature measurements, critical current density (J c ) measurements, magnetic ac-susceptibility measurements, Fourier transform infrared (FTIR) spectroscopy etc. The experimental data were analyzed by different well established theoretical models such as Aslamazov-Larkin (AL), Lawerence-Doniach (LD) and Maki-Thompson (MT) models. It was observed that tetragonal structure and stoichiometry of the host CuTl-1223 superconducting phase remained unaltered after the addition of these NPs, which indicates about the occupancy of these NPs at the inter-granular spaces. Non-magnetic metallic NPs (Ag, Au) have improved the superconducting transport properties (T c , J c , etc) up to certain optimum concentration of these NPs in CuTl-1223 matrix. The improvement in the superconducting properties can be attributed to an increase in the inter-grains connectivity by healing up the inter-grains voids and pores after the addition of these NPs. The improved inter-grains connections can facilitate the carriers transport processes across the inter-crystallite sites due to their metallic nature. But the superconducting volume fraction starts to be decreased after certain optimum inclusion level of these non-magnetic non-superconducting metallic NPs, which causes the suppression of superconductivity parameters. The effects of highly coercive cobalt ferrite (CoFe 2 O 4 ) nanoparticles addition on superconducting properties of CuTl-1223 matrix were also explored. The magnetic behavior of CoFe 2 O 4 nanoparticles was determined by MH-loops with the help of superconducting quantum interference device (SQUID). The resistivity versus temperature measurements showed an increase in T c (0), which could be most probably due to improvement of weak-links by the addition of these nanoparticles. The increase of mass density with xxviincreasing content of these NPs can also be an evidence of filling up the voids in the matrix. But the addition of these NPs beyond an optimum level caused the agglomeration and produced additional stresses in material and suppressed the superconductivity. We observed non- monotonic variation of superconducting properties after the inclusion of nano-Al 2 O 3 particles, which can be associated with inhomogeneous distribution of these NPs at the grain-boundaries in CuTl-1223 matrix. But overall suppression of superconducting properties was attributed to a pair-breaking mechanism caused by reflection/scattering of carriers across these insulating nano- Al 2 O 3 particles present at the grain-boundaries of the host CuTl-1223 matrix. The presence of Al 2 O 3 nanoparticles at the grain-boundaries possibly reduced the number of flux pinning centers, which were present in the form of weak-links in pure CuTl-1223 superconducting matrix. The suppression of activation energy {U (eV)} may be due to weak flux pinning in the nano-Al 2 O 3 particles added samples. The superconducting microscopic parameters (i.e. zero temperature coherence length along c-axis {xc (0)}, inter-layer coupling (J), inter-grain coupling (a) etc.) deduced from fluctuation induced conductivity (FIC) analysis with the help of above mentioned theoretical models explained the experimental findings very well. For example, the increase in the values of inter-grain coupling (a) deduced from FIC analysis is a theoretical evidence of improved inter-grain coupling in the host CuTl-1223 matrix with the increased contents of these nanoparticles. We also calculated the activation energy {U (eV)} of (NPs) x /CuTl-1223 nanoparticles-superconducting composites. The increase in T c (0), J c , U (eV) etc and decrease in normal state resistivity {ρ 300 K (Ω-cm)} were observed after the addition Ag, Au and CoFe 2 O 4 NPs in CuTl-1223 superconducting phase. The suppression of superconducting properties (i.e. T c (0), J c , U (eV) etc) after the addition of Al 2 O 3 nanoparticles in host CuTl-1233 superconducting matrix was observed.