Browsing by Subject "Schrodinger equation"
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Item An investigation into the possibility of an integral solution to the radical Schrodinger equation.(Texas Tech University, 1964-05) Walker, John DavidNot availableItem The dynamics of bose gases(2015-05) Taliaferro, Kenneth William; Chen, Thomas; Maggi, Francesco; Pavlovic, Natasa; Tzirakis, Nikolaos; Vasseur, AlexisWe study the Gross-Pitaevskii (GP) hierarchy, which is an infinite sequence of coupled partial differential equations that models the dynamics of Bose gases and arises in the derivation of the cubic and quintic nonlinear Schrödinger equations from an N-body linear Schrödinger equation. In Chapter 2, we consider the cubic case in R³ and derive the GP hierarchy in the strong topology corresponding to the spaces used by Klainerman and Machedon in (82). We also prove that positive semidefiniteness of solutions is preserved over time and use this result to prove global well-posedness of solutions to the GP hierarchy. This is based on a joint work with Thomas Chen (24). In Chapters 3 and 4, we prove uniqueness of solutions to the GP hierarchy in R[superscript d] in a low regularity Sobolev type space in the cubic and quintic cases, respectively. These chapters are an extension of the work of Chen-Hainzl-Pavlović-Seiringer (17) and are based on joint works with Younghun Hong and Zhihui Xie (70,71).Item The dynamics of bose gases(2015-05) Taliaferro, Kenneth William; Chen, Thomas; Maggi, Francesco; Pavlovic, Natasa; Tzirakis, Nikolaos; Vasseur, AlexisWe study the Gross-Pitaevskii (GP) hierarchy, which is an infinite sequence of coupled partial differential equations that models the dynamics of Bose gases and arises in the derivation of the cubic and quintic nonlinear Schrödinger equations from an N-body linear Schrödinger equation. In Chapter 2, we consider the cubic case in R³ and derive the GP hierarchy in the strong topology corresponding to the spaces used by Klainerman and Machedon in (82). We also prove that positive semidefiniteness of solutions is preserved over time and use this result to prove global well-posedness of solutions to the GP hierarchy. This is based on a joint work with Thomas Chen (24). In Chapters 3 and 4, we prove uniqueness of solutions to the GP hierarchy in R[superscript d] in a low regularity Sobolev type space in the cubic and quintic cases, respectively. These chapters are an extension of the work of Chen-Hainzl-Pavlović-Seiringer (17) and are based on joint works with Younghun Hong and Zhihui Xie (70,71).Item Finite element solution of the S-limit Schrodinger equation of helium(Texas Tech University, 1979-12) Keller, John WilliamThe Schroedinger equation, for all but the simplest systems, is an elliptic partial differential equation. Almost every method of solution is based on the expansion of the unknown solution in terms of a set of known global functions. Only a few calculations, using strictly numerical techniques based on the finite difference method, have been reported. Shorter cycle times and the increasing memory of computer hardware along with ease of programing provide the major impetus in the investigation of strictly numerical techniques in quantum mechanics. Recently, the Finite Element Method CFEM), which has been used extensively in engineering fields, has been applied to equations of quantum chemistry. However, only three of these calculations involved solution of a partial differential equation (PDE). This paper reports the application of the FEM to a 2-dimensional problem, that of the S-wave limit of the He atom. This problem has also been treated by the numerical finite difference method . The purpose of this study is not to establish that the FEM is an efficient method for solving quantum mechanical problems, but merely to explore the procedure to learn what is involved in its application. This problem has been chosen because its simplicity allows examination of the details of the FEM.Item On von Neumann's hypothesis of collapse of the wave function and quantum Zeno paradox in continuous measurement(2011-05) Kim, Dongil; Sudarshan, E. C. G.; Petrosky, Tomio Y.; Bohm, Arno R.; Schieve, William C.; Wyatt, Robert E.The experiment performed by Itano, Heinzen, Bollinger and Wineland on the quantum Zeno effect is analyzed in detail through a quantum map derived by conventional quantum mechanics based on the Schrodinger equation. The analysis shows that a slight modification of their experiment leads to a significantly different result from the one that is predicted through von Neumann's hypothesis of collapse of the wave function in the quantum measurement theory. This may offer a possibility of an experimental test of von Neumann's quantum measurement theory.Item The effects of the variations of the nuclear diffuseness and range parameters on the stability of 126 [superscript x] 184,228(Texas Tech University, 1978-05) Yen, Yi-Yih WilliamNot available