JOINT MATHEMATICS & PHYSICS COLLOQUIUM
Non-Relativistic Fermionic Quantum Mechanics (Cracking Schrödinger Nutshell)
Young Lee
Department of Physics, Texas Southern University
Young.Lee@tsu.edu
Tuesday, September 25, 2018 2:30 pm – 3:30 pmTexas Southern University Science Building, Room 146
Abstract
The non-relativistic Schrodinger equation was formulated nearly 90 years ago and has been successfully applied to various quantum mechanical systems across multiple scientific fields producing the main developments to quantum theory. The primary importance of Schrodinger’s equation arises in the implementation of the analogue of Newton’s law and continuation of classical mechanics. In the development of relativistic quantum physics, Dirac expanded the theory combining Einstein’s special relativity, quantum mechanics and the so- called fermion particle of half integer spin. This was the first theory to consistently account for special relativity within the context of quantum mechanics. In this paper, we formulate a novel mathematical approach to solve Schrodinger’s equation within the context of particle and wave theory. In our examination, we present a similar, yet unique model of fermion structure, which is consistent with the present day Dirac description. We will introduce and discuss the 1 dimensional structure of a Harmonic oscillator and free particle fermionic state. Additionally, the 3+1 dimensional structure of the fermionic state will be briefly outlined. Finally, we will compare our model with Dirac’s theory for further study. This model is not Lorentz invariant; however, the connection of classical and non-relativistic quantum theory will be explained. We will also talk about the Hydrogen atom within the framework of bound state, fermionic quantum physics.
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TSU was recently awarded a National Science Foundation Targeted Infusion Grant under PI, Dr. Alamelu Sundaresan, with Co-PI Dr. Mark Harvey and CO-Is Drs. John Sapp, Azime Saydam and Bernnell Peltier-Glaze. This project combines structured peer-mentor infused curricula coupled with faculty mentored summer research for the following courses within COSET: MATH 136, CHEM 131, PHYS 237 and BIOL 131. The main objective of the proposal is to increase STEM retention of underrepresented students by offering dedicated peer mentoring and supervised faculty research opportunities. We are excited about this first full-fledged project of its kind at TSU consisting of an active learning concept and associated peer-mentoring, which will benefit both TSU students and the above four departments from June 2017 – May 2020. Peer mentors will be selected from upperclassmen in each of the departments and will be directly supervised by the PI, Co-PI and CO-Is. More to follow…
Dr. Perotti is currently working in collaboration with Dr. D. Bessis on the development of a new method in the spectral analysis of noisy time-series data for damped oscillators which makes use of J-matrices to evaluate Padé Approximations built on the time-series Z-transform. Applications of the method range from the detection of Gravitational Wave Bursts to the analysis of Magnetic Resonance data.
While in In Atlanta Dr. Perotti perfected codes which -starting from a suitable Pade’ approximant of the desired transmittance- reconstructed, through inverse scattering techniques, the corresponding monodimensional effective self-consistent potential having no bound states. In collaboration with Dr. A. Mezincescu he also wrote codes to calculate and subtract from the potential thus obtained the self-consistent field for a given density of donor dopant ions and finally –trough a change of coordinates- obtain the chemical composition profile (solution of the variable mass BenDaniel and Duke’s equation). The technique allows to efficiently design semiconductor heterostructures given the desired electron reflectance. They furthermore developed codes that, using double Darboux transformations on a given monodimensional potential with one bound state, maximize the asymmetry of the continuum function at a given energy having nonzero dipole matrix element with the bound state. The potential thus obtained optimizes the probability for an electron excited by incident light of a given frequency to be emitted in the desired direction. While In Munich Dr. Perotti developed -for comparison with laboratory experiments- semiclassical and full quantum Monte Carlo simulations of an ion moving in a one-dimensional optical lattice and a weak confining static electric field while at the same time being subject to a periodic driving force. The solution of the Fokker-Planck equation for the system in the diffusion limit, found in collaboration with Dr. V. Alekseev of the Lebedev Physical Institute in Moscow, allows to explain the apparent rise of the spatial diffusion coefficient when increasing the periodic force amplitude. In Pittsburgh Dr. Perotti worked on Classical numerical simulations and calculation of the Lyapunov exponents for a system consisting of an ensemble of (two-level) Rydberg atoms collectively interacting with its own selfconsistent field and an external microwave field in a cavity. By changing the strength of the external field and/or its detuning from the chosen atomic transition, the behavior of the system can be made to change from regular to weakly chaotic and finally to strongly chaotic. The system moreover offers a convenient way to change the total action by varying the number of interacting atoms. It is therefore considered a convenient system to explore the different dependence on the total action of the time at which quantum evolution begins to significantly deviate from the classical one in the case of regular and chaotic systems [G. P. Berman, E. N. Bulgakov and D. D. Holm, Los Alamos Report LA-UR-93_2187 (1993)]. In the first case this time -called “quantum break-time”- is expected to be a polynomial function of the action, in the latter a logarithmic one. His simulations were aimed at identifying the parameter ranges for the different regimes (regular, strongly chaotic, weakly chaotic) and finding their signature in the (averaged) quantities to be measured in the laboratory. He also conducted Classical and quantum simulations of the monodimensional hydrogen atom in a pulsed microwave field and a collinear constant static electric field, both with and without noise. These were compared with experimental results for atoms prepared in extreme Stark states in a static electric field and made to interact with a pulse of microwaves polarized collinear to the static field used to align the atoms. Furthermore he calculated the instantaneous quantum quasienergy (Floquet) levels for the above system and the Husimi functions of the corresponding states and of the projections on those states of the pulsed wavefunction to help interpret the results of the above quantum numerical simulations. In particular he was able to show the relevance of small classical phase space structures to pulsed quantum evolution. In Milano Dr. Perotti helped develop the codes used to numerically investigate the quantum dynamical localization in action (principal quantum number) of the electron probability for a mono (and bi-) dimensional hydrogen atom in a microwave field when the corresponding classical system is chaotic. He has applied similar techniques to the study of the stochastic ionisation of Alkali-metal atoms: microwave experiments with Alkali atoms are easier to perform than with Hydrogen; on the other hand, core effects cause significant deviations from the hydrogenic behaviour. The numerical simulations published by A. Buchleitner and his collaborators are in very good agreement with the laboratory experiment but are very time consuming; moreover only very qualitative arguments and no model has yet been advanced to interpret the regimes where Alkali ionisation deviates from that of Hydrogen. For regime II of A. Buchleitner and coworkers, where quantum effects are still dominant, Dr. Perotti developed an evaluation of the ionisation threshold based on the similarities between the phase space structures of Alkali in monochromatic microwave fields and Hydrogen in a bichromatic field. He has also shown that in the regime of very low microwave frequencies (regime III of A. Buchleitner and coworkers), a classical explanation is possible in terms of Chirikov’s criterion for transition to chaos. Recent research interests of Dr. Perotti also include the quantum double pendulum: a low dimensional autonomous system which in the classical limit displays chaotic behavior . It is simple enough to allow an extensive study which will be able to shed light on more complex chaotic autonomous systems being studied currently: e.g. the Hydrogen atom in crossed static electric and magnetic fields and the Helium atom. He has performed a study of energy curves and compared Husimi functions of eigenstates at important avoided crossings with the corresponding classical surfaces of section. He is now studying the behavior of suitable packets.
All students pursuing careers in the STEM as well as non-STEM areas must take one of the three service physics courses: Physical Sciences (for non-STEM-majors), College Physics (I & II, consisting of a comprehensive overview of laws of physics, without using calculus), and University Physics (I & II). The latter involves the use of calculus, which is the natural mathematical language of physics.
In addition to the lectures, there are accompanying Labs for College Physics and University Physics. These labs are intended to reinforce basic physics concepts through a “hands-on” approach emphasizing the statistical theory of measurements, and experiencing, first- hand, many of the fundamental laws of Newtonian physics as it impacts mechanics and gravity. These labs introduce students to basic topics such as fluids, waves, and thermodynamics (heat).
The more advanced Undergraduate Physics lab II introduces students to basic topics such as electricity and magnetism, optics, and modern physics. Students acquire a basic understanding of all these areas through experiments designed to reinforce the lectures. All of these topics are indispensable to the careers pursued by the diverse body of professional interests represented in our students, both majors and non-majors. Students with these skills and knowledge are in high demand in many professions (health sciences, pharmacy, education, engineering, aeronautics, etc.).
These labs are the cornerstone of a student’s immersion into, and understanding of, physics. These Labs will prepare students not only in the technical preparation of physics, but also in the scientific method of empirical research impacting all the sciences. Students will learn team work, sharpen their advanced mathematics skills (i.e. vectors, trigonometry, algebra, and statistical analysis), and develop critical thinking skills based on direct interaction with experiments.
The Medical and Nuclear Physics Teaching Lab builds upon the successful implementation of a Health Physics program at TSU begun in 2008. Health Physics concerns the understanding of nuclear processes as it affects safe working environments for living organisms (i.e. human beings in particular), within medical facilities, nuclear power plants, government research facilities, food processing centers, etc. Some of the equipment resources in this lab include:
Students enrolled in the traditional health physics program at TSU take two semesters of laboratory classes, which focus on unique hands- on experiments involving various radiation detection apparatuses, analyses, and techniques. In the first laboratory component, students learn how to use a Geiger-Mueller system to measure radiation emitted for various
radionuclide species. In the second installment of the laboratory sequence, students learn how to use a sodium iodide detector system to measure spectroscopic signatures from gamma-emitting radioisotopes. The final component in this sequence is a nuclear electronics laboratory, in which students learn fundamental signal pulse analysis using standard instrumentation commonly employed in nuclear and radiation physics.
Given that Houston hosts the world’s largest Medical Consortium, The Texas Medical Center (TMC), it is natural to expand upon proven Health Physics capabilities that overlap Medical Physics. This concerns the understanding of nuclear physics as applied to medical diagnosis and treatment for illnesses such as cancer. To this extent, the Medical/Health – Nuclear Physics Teaching Laboratory will also expand into this area. TSU- Physics has produced exceptional students working or interning as radiation safety specialists. A list of these facilities included the Nuclear Regulatory Commission, UT MD Anderson Cancer Center, UT Health Science Center at Houston, Texas Commision on Environmental Quality (Radiation Division), UT Southwestern Medical Center. Additionally, TSU-physics graduates have gone on to attend either graduate school or medical school at the following institutions: Texas A&M University at College Station, University of Texas at Austin, School of Medicine-Texas Tech Health Sciences, Howard University, Morgan State University, Tulane University.
Funding for the radiation health physics laboratories have been provided by grants from the Nuclear Regulatory Commission.
h3>DEPARTMENT OF PHYSICS
The Medical and Nuclear Physics Laboratory is a faculty research lab and also and a training facility for student researchers. This laboratory focuses on low energy nuclear physics, in which fundamental techniques are applied that are effective for detection and analysis of radiation levels impacting people in many workplace environments including nuclear power plants, medical diagnostic and treatment facilities, food processing plants, homeland security checkpoints, etc.
This Laboratory is also concerned with basic and applied research in nuclear physics. The “applied” thrust pertains to understanding related issues in the context of (1) medically related, radiation physics based, therapy and diagnostic applications impacting diseases; and (2) understanding the radiation thresholds in various workplace settings impacting organisms and the environment.
Selected students will participate in fundamental research projects emphasizing these two areas specified above. In addition, students will have an opportunity to participate in (1) high energy nuclear physics research at the Brookhaven National Laboratory and (2) nuclear astrophysics at the Facility for Rare Isotope Beams (FRIB) on the campus of Michigan State University. Besides mastering basic dosimetry and radiation detection skills developed within the curriculum, participating students will get to apply these skills in the prosecution of basic and applied research. Additional capabilities in Monte Carlo simulation studies in nuclear collision physics, radiation biophysics, radiation shielding and therapeutic applications of radiation will also be developed.
The objective is to develop a strong cohort of students that can pursue these studies at high powered facilities, eventually transitioning into professionals in the diverse areas of nuclear physics; e.g., heavy ion collision physics, nuclear astrophysics, radiation safety and protection, radiation therapy, diagnostic imaging and radiation biophysics. The advanced training received by selected students, developed within the context of prosecuting basic and applied research in medical, health, and nuclear physics, respectively will produce uniquely trained students able to enter the workforce in diverse areas such as nuclear facilities, government laboratories, medical treatment and diagnostic facilities, food processing plants, etc. Students can also pursue advanced training in health physics as a stepping stone to careers as radiation oncologists, radiation therapists, etc., within the medical profession. Houston hosts the world’s largest medical facility, the Texas Medical Center, with tremendous employment opportunities for such students. Some students can also elect to pursue a Ph.D. in medical physics or health physics – or even fundamental nuclear physics, in which the smallest constituents of matter are probed at the world’s most elite high energy physics facilities. In all cases, our physics students will have an abundance of career opportunities.
This is a research and student research training laboratory, emphasizing ongoing departmental research capabilities in noise filtering, atomic physics, and medical physics based research. Students learn modern Nuclear Magnetic Resonance (NMR) techniques which require an understanding of the underlying quantum physics associated with the spinning of atomic nuclei in an ambient, static, magnetic field and the spectrum of the electromagnetic radiation given off by these excitations. NMR is an important physical process with many applications including nuclear waste disposal and cancer detection in soft tissues. In the second case, Magnetic Resonance Imaging (MRI) is the conventional terminology encountered by many in cancer detection protocols. The NMR lab is equipped with the following equipment:
A typical experiment involves probe preparation, field configuration, pulse design, data acquisition and data processing. Students will become familiar with contemporary electronics and digital signal and image processing. Participating students will also learn how to do research as part of a team, each contributing a particular skill. The ultimate objective of the underlying research is to develop more cost effective techniques, requiring low level magnetic fields, for effective cancer detection.
This Laboratory represents a link to the outside world of expanded learning opportunities for our students. Students and faculty members will be able to participate in Distance Learning courses in specialized areas not represented within the Department.
This facility currently supports TSU-Physics’ participation within the Texas Physics Consortium (TPC) which includes eight other campuses across the State, pooling their curriculum resources and expanded faculty specializations (i.e. Astronomy, Cosmology, Geophysics, etc.). The other physics programs participating in the TPC include Midwestern State University, Prairie View A & M University, Texas A&M University-Corpus Christi, Texas A&M University-Kingsville, Tarleton State University, Texas A & M University – Commerce, West Texas A & M University, and Texas A & M University-Central Texas.
Expanding the course offerings available to TSU students will make them more competitive in their post-graduate career objectives. The production of well trained students is essential for sustaining TSU’s research activities. This classroom contributes to the learning process in a very modern way, ensuring the production of the best trained TSU students possible, able to impact ongoing, faculty led, research.
There are key experiments in the history of physics that revolutionized our way of thinking about the fundamental structure of nature and the ensuing laws that characterize physical processes. Some of these experiments can be done by upper level physics majors. These
experiments also serve to expose the student to advanced skills indispensable in today’s advancing
technological evolution impacting many industries. Some of the more traditional experiments include:
Michelson Morley interferometer Experiment which confirmed the invariance of the speed of light in a vacuum. This experiment formed the basis for the famous E = mc^{2 }>relation which is at the heart of nuclear fission (i.e. atomic bombs, nuclear power plants, etc.) and nuclear fusion (the energy from the sun, and potentially future energy machines which produce less radioactive bi-products tha
n conventional nuclear power plants).
Millikan Oil Drop Experiment – that measured the charge on an electron.
X-Ray Diffraction for discovering the crystal structure of compounds.
Davisson Germer Experiment which revealed the duality principle of matter (i.e. what we call a particle can sometimes behave as a wave) – “matter waves.”
Nuclear Magnetic Resonance – atomic nuclei, when placed in magnetic fields, will occupy different energy levels that can be excited and de-excited through electromagnetic radiation. This is relevant in many fields, particularly medicine. Other important experiments impact nuclear physics, laser physics, semiconductors, superconductivity, nanostructures, etc. These labs are the cornerstone of a student’s immersion into, and understanding of, physics. Physics lab courses are the stepping stones into student’s participation in faculty led research. This Lab ensures the production of high caliber students able to assist faculty in their research efforts.
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orcid.org/0000-0002-8820-9073, scholar.google.com, research gate
Dr. Vrinceanu was born and raised in Romania. As a student at the “Mihai Viteazul” High School in Ploiesti, he was awarded the bronze medal at the International Physics Olympiad held in London. This scientific competition is the equivalent of the Olympic Games for sports and considers the very best students that represent their own countries. Immediately after obtaining his University degree in Mathematical Physics, he was offered the position of Assistant Professor in the Department of Theoretical Physics at the Bucharest University.
He obtained his doctoral degree in Theoretical Atomic Physics from Georgia Institute of Technology in Atlanta. The problem treated in his PhD Thesis made a significant theoretical contribution providing an elegant and efficient solution to an outstanding problem incapable of being solved for more than 40 years. In recognition, he was awarded the Sigma-Xi award of Best Ph.D Thesis 2001 and he was selected for Thesis Prize of the American Physical Society of Division of Atomic, Molecular and Optical Physics (DAMOP).
At the Institute for Theoretical Atomic and Molecular Physics at the Harvard-Smithsonian Center for Astrophysics Dr. Vrinceanu was the leader in a theoretical investigation of strongly magnetized antihydrogen atoms, aimed to clarify understanding and interpretation of the recent experiments at the European Center for Nuclear Reseach (CERN) by Harvard scientists. Their results on formation and electric field ionization of highly excited anti-hydrogen atoms are ground breaking and completely changed the way the experimental data is analyzed. For the upmost importance of the research, their work published in Physical Review Letter was featured on the journal cover.
Dr. Vrinceanu was granted the prestigious Director Fellowship at Los Alamos National Laboratory where he worked on large scale Monte Carlo and Molecular Dynamics Simulations of cold magnetized and un-magnetized plasmas.
Theoretical studies of:
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• CONACyT PDRA-Fellowship for Mexican Young Researcher #250356 : US$50,000.0, 2015-2017
• Western Alliance to Expand Student opportunities – US$2,378.0, 2014
• Graduate Studies, Brigham Young University – US$90,000.0 for 3 years to support a
graduate student, 2012
• Western Alliance to Expand Student opportunities – US$2,256.0, 2012
• Western Alliance to Expand Student opportunities – US$10,630.0, 2011
• Federal Government Grant – Ecology Institute of Gto, US$170,000.0, 2010
• Guanajuato State Government Funding, US$350,000.0, 2008-2010
Dr. Mark C. Harvey received a B.S. degree in physics from Virginia State University (Petersburg, VA). He completed his graduate studies in physics at Hampton University (Hampton, VA) where he earned his M.S. in physics and Ph.D. in experimental nuclear physics. Dr. Harvey’s doctoral research experiment in medium energy nuclear physics was carried out at the National Institute for Nuclear and High Energy Physics in Amsterdam, The Netherlands. After earning his doctoral degree, Dr. Harvey became a postdoctoral research assistant in the PHENIX collaboration at the Brookhaven National Laboratory (Upton, NY) in relativistic heavy ion collision physics. After completing his postdoc at Brookhaven, Dr. Harvey taught introductory physics courses for one year at the Rochester Institute of Technology (Rochester, NY), before returning to research as a postdoctoral fellow at the University of Texas M. D. Anderson Cancer Center (Houston, TX). He has been awarded federally funded grants from the U.S. Nuclear Regulatory Commission, National Science Foundation and the National Institutes of Health. Currently, Dr. Harvey’s research interests include (1) radiation dose assessment in environmental and medical health physics and (2) high-energy nuclear physics; using advanced data analysis methods, radiation detection apparatuses and Monte Carlo techniques. He has mentored underrepresented STEM students on research involving both computational modeling and/or measurement of stray radiation at various facilities including the National Synchrotron Light Source II and the McGovern School of Medicine at The University of Texas Health Science Center at Houston. Dr. Harvey is an associate professor of physics and has served as both the director and primary instructor of the health physics program at Texas Southern University.
Principal Investigator, Secondary Neutron Exposures in Pediatric Proton Radiotherapy, National Institutes of Health F32, 2008
Co-Investigator, Faculty Support for Developing Radiation Dosimetry Research in Health Physics using Monte Carlo Techniques at Texas Southern University, Nuclear Regulatory Commission HR-FN-1009-NED02, 2010
Principal Investigator, Medical Health Physics Scholarship Program at Texas Southern University, U.S. Nuclear Regulatory Commission NRC-HQ-84-14-FOA-0003, 2014
Co-Investigator, Unified Approach to Increase STEM Undergraduate Students Employment in the Department of the Navy, Office of Naval Research ONR FOA 14-002, 2014
Principal Investigator, Medical Health Physics Scholarship Program at Texas Southern University, U.S. Nuclear Regulatory Commission NRC-HQ-84-16-FOA-0001, 2016
Co-Investigator, Undergraduate Peer-Mentor Project-Based Program in STEM at Texas Southern University, National Science Foundation HRD-1719318, 2017
Principal Investigator, Contract – Texas Southern University, Texas Commission on Environmental Quality (TCEQ) 582-20-10298, 2019
Co-Investigator, HBCU Collision Collaboration – J/psi Peripheral Collision Analysis and Detection, National Science Foundation EiR-1901420, 2019
Dr. Handy received all of his four degrees from Columbia University, concentrating in mathematics and physics, at the undergraduate level, and theoretical physics (i.e. path integral formulations of non-abelian gauge theories) at the graduate level, working under Dr. A. Mueller and Dr. John Klauder (AT&T). As a freshman he assisted Dr. Martin Gutzwiller (IBM) in his computational investigations of quantum chaos. He was the first participant and graduate of the AT&T Cooperative Research Fellowship Program which has produced many minority and women Ph.D’s in diverse academic disciplines. From 1978 – 81 he was a postdoctoral fellow at the Los Alamos National Laboratory. Following a brief appointment in industry, Dr. Handy joined the faculty of (presently) Clark Atlanta University (1983 – 2005) where he co-founded the Center for Theoretical Studies of Physical Systems, one of the first successful HBCU-research and student mentoring centers in the nation. In 2005 he became the chair of the new physics program at Texas Southern University, in Houston, Texas. While at LANL, he focused on the development of singular perturbation, multi-scale, methods for strong coupling field theory problems. This led to a fascination with the Moment Problem (i.e. reconstruction of a positive signal through a hierarchy of, successively decreasing, scale related information) in pure mathematics and its incorporation into quantum physics. One branch of these investigations would eventually lead to the theory of Contiuous Wavelets as derived, through other means, by Grossman and Morlet. A second branch, funded through an NSF "Creativity Award" (1983 – 88), was the discovery that the moment problem leads to powerful new computational methods (within the spirit of control theory) for reliably predicting certain features of singular perturbation/strongly coupled systems. From 1985 – 1988 Dr. Handy pioneered these methods, together with D. Bessis. This approach, The "Eigenvalue Moment Method (EMM)", is now recognized as one of the first applications of Semidefinite Programming (SDP) in quantum physics, anticipating the importance of these methods by at least a decade. More recently, SDP has made a tremendous impact both in pure mathematics (i.e. combinatorics) and quantum chemistry through the N-Body problem reduction methods advocated by D. Mazziotti (U. Chicago), and others. Dr. Handy continues his research in this general area, with close to seventy publications, developing new ways of exploiting positivity constraints to computationally solve various quantum physics problems through the generation of rapidly converging lower and upper bounds to the physical parameters.
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The Vocano Function #1
The Vocano Function #2
C. R. Handy, D. Khan, S. Okbagabir, and T. Yarahmad " Moment Problem Quantization within A Generalized Scalet-Wigner (Auto-Scaling) Transform Representation,” J. Phys. A: Math. Gen. 36, 1623 (2003).
C. R. Handy and X. Q. Wang, "Spectral Bounds for the PT-breaking Hamiltonian $p^2 + x^4 + iax$, J. Phys. A.: Math. & Gen. 36, 11513 (2003).
C. R. Handy "Positivity Representations for non-Hermitian Hamiltonians," Czechoslovak J. of Physics 54, 57 (2004).
A. Rodriguez, C. R. Handy, and C. Trallero-Giner, Reply to Comment on "Excited States in the Infinite Quantum Lens Potential: Conformal Mapping and Moment Quantization Methods,” J. Phys.: Condenced Matter 16, 2945 (2004).