This document provides an overview of quantum mechanics topics including:
1) The Schrödinger wave equation and its time-dependent and time-independent forms.
2) Expectation values and how they are used to calculate probabilities, momentum, position, and energy.
3) Specific quantum systems like infinite and finite square wells and simple harmonic oscillators. It also discusses quantization, degeneracy, and other concepts.
4) Barrier penetration and tunneling, where particles can pass through barriers that would be forbidden classically.
The document covers many fundamental aspects of quantum mechanics through examining various quantum systems and potentials.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
The document discusses the Schrodinger equation and its applications in quantum mechanics. It covers:
1. The postulates of quantum mechanics including that systems are described by wavefunctions and observables are represented by Hermitian operators.
2. Examples of operators for observables like position, momentum, energy.
3. The time-independent Schrodinger equation for a time-independent potential and its solution for an infinite square well potential.
4. Other examples like an infinite square well potential trapping an electron and calculating its energy levels and wavefunctions.
This course covers advanced quantum mechanics for graduate students. It aims to help students gain a deeper foundation in quantum mechanics through topics like the Schrödinger equation, particle in a box, harmonic oscillator, hydrogen atom, angular momentum, and approximation methods. The key objectives are to understand quantum theory and apply it to important physical systems, and recognize the necessity of quantum methods in atomic and nuclear physics. Some specific topics covered include the wave function, Born's statistical interpretation, probability, normalization, momentum, uncertainty principle, and the time-dependent and time-independent Schrödinger equations.
Unit 1 Quantum Mechanics_230924_162445.pdfSwapnil947063
1) Quantum mechanics is needed to explain phenomena at the microscopic level that classical physics cannot, such as the stability of atoms and line spectra of hydrogen.
2) According to de Broglie's hypothesis, all matter exhibits wave-particle duality - particles are associated with waves called matter waves. The wavelength of these matter waves is given by de Broglie's equation.
3) In quantum mechanics, the wave function ψ describes the wave properties of a particle. The probability of finding a particle in a region is given by the absolute square of the wave function |ψ|2 in that region.
Quantum chemistry is the application of quantum mechanics to solve problems in chemistry. It has been widely used in different branches of chemistry including physical chemistry, organic chemistry, analytical chemistry, and inorganic chemistry. The time-independent Schrödinger equation is central to quantum chemistry and can be used to model chemical systems like the particle in a box, harmonic oscillator, and hydrogen atom. Molecular orbital theory is also important in quantum chemistry for describing chemical bonding in molecules.
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan DashManmohan Dash
9 problems (part-I and II) and in depth the ideas of Quantum; such as Schrodinger Equation, Philosophy of Quantum reality and Statistical interpretation, Probability Distribution, Basic Operators, Uncertainty Principle.
This document summarizes key concepts about the particle in a rigid one-dimensional box:
1. It finds the energy eigenstates and discusses the wave functions and their properties like orthogonality.
2. It calculates the probability and expected values for the particle's position and discusses the physical interpretation of the wave function and coefficients when expanding an arbitrary function in the eigenstates.
3. It addresses several questions about normalized wave functions, time-dependent wave functions, energy measurements, and the wave function after a measurement.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
The document discusses the Schrodinger equation and its applications in quantum mechanics. It covers:
1. The postulates of quantum mechanics including that systems are described by wavefunctions and observables are represented by Hermitian operators.
2. Examples of operators for observables like position, momentum, energy.
3. The time-independent Schrodinger equation for a time-independent potential and its solution for an infinite square well potential.
4. Other examples like an infinite square well potential trapping an electron and calculating its energy levels and wavefunctions.
This course covers advanced quantum mechanics for graduate students. It aims to help students gain a deeper foundation in quantum mechanics through topics like the Schrödinger equation, particle in a box, harmonic oscillator, hydrogen atom, angular momentum, and approximation methods. The key objectives are to understand quantum theory and apply it to important physical systems, and recognize the necessity of quantum methods in atomic and nuclear physics. Some specific topics covered include the wave function, Born's statistical interpretation, probability, normalization, momentum, uncertainty principle, and the time-dependent and time-independent Schrödinger equations.
Unit 1 Quantum Mechanics_230924_162445.pdfSwapnil947063
1) Quantum mechanics is needed to explain phenomena at the microscopic level that classical physics cannot, such as the stability of atoms and line spectra of hydrogen.
2) According to de Broglie's hypothesis, all matter exhibits wave-particle duality - particles are associated with waves called matter waves. The wavelength of these matter waves is given by de Broglie's equation.
3) In quantum mechanics, the wave function ψ describes the wave properties of a particle. The probability of finding a particle in a region is given by the absolute square of the wave function |ψ|2 in that region.
Quantum chemistry is the application of quantum mechanics to solve problems in chemistry. It has been widely used in different branches of chemistry including physical chemistry, organic chemistry, analytical chemistry, and inorganic chemistry. The time-independent Schrödinger equation is central to quantum chemistry and can be used to model chemical systems like the particle in a box, harmonic oscillator, and hydrogen atom. Molecular orbital theory is also important in quantum chemistry for describing chemical bonding in molecules.
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan DashManmohan Dash
9 problems (part-I and II) and in depth the ideas of Quantum; such as Schrodinger Equation, Philosophy of Quantum reality and Statistical interpretation, Probability Distribution, Basic Operators, Uncertainty Principle.
This document summarizes key concepts about the particle in a rigid one-dimensional box:
1. It finds the energy eigenstates and discusses the wave functions and their properties like orthogonality.
2. It calculates the probability and expected values for the particle's position and discusses the physical interpretation of the wave function and coefficients when expanding an arbitrary function in the eigenstates.
3. It addresses several questions about normalized wave functions, time-dependent wave functions, energy measurements, and the wave function after a measurement.
- The atom consists of a small, dense nucleus surrounded by an electron cloud.
- Electrons can only exist in certain discrete energy levels around the nucleus. Their wavelengths are determined by the principal quantum number.
- The Bohr model improved on earlier models by introducing energy levels and quantization, but had limitations. The quantum mechanical model treats electrons as waves and uses Schrodinger's equation.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
The document summarizes the Vanderbilt pseudopotential method. It starts from all-electron calculations and defines a pseudo-wavefunction. The pseudopotential V'loc is constructed as the sum of the local potential Vloc, Hartree potential VH, and exchange-correlation potential VXC. Unlike other pseudopotentials which subtract VH and VXC, Vanderbilt's adds them. The method can produce both norm-conserving and ultra-soft pseudopotentials depending on whether the generalized norm-conserving condition is satisfied.
Charge Quantization and Magnetic MonopolesArpan Saha
This talk, given as a part of the Annual Seminar Weekend 2011, IIT Bombay, dealt with a homotopy-based
variant of the argument Dirac provided to show that the existence of a single magnetic monopole in the Universe
is a sufficient condition for the quantization of electric charge.
This document provides an overview of the Schrodinger wave equation, including:
1) It describes the derivation of the time-dependent Schrodinger wave equation based on what is known about the wave function of a particle.
2) It explains that the time-dependent equation can be used to derive the time-independent Schrodinger equation by assuming the wave function separates into spatial and temporal factors.
3) It notes that for the wave function to be physically acceptable, the energy values inserted into the time-independent equation must be quantized to satisfy normalization conditions on the wave function.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
1) The document discusses the exact solution to the Klein-Gordon shutter problem, finding that the wave function does not resemble the optical expression for diffraction but the charge density does show transient oscillations resembling a diffraction pattern.
2) It presents the exact solution for the Klein-Gordon shutter problem using discontinuous initial conditions, finding the wave function solution differs from Moshinsky's approximation.
3) When the exact relativistic charge density is plotted over time, it shows transient oscillations that resemble a diffraction pattern, despite some relativistic differences, demonstrating that diffraction in time exists in relativistic scenarios.
Fundamental Concepts on Electromagnetic TheoryAL- AMIN
The document summarizes key concepts from a presentation on electromagnetic theory. It discusses different types of fields, including scalar and vector fields. It also covers gradient, divergence, curl, coordinate systems, static electric and magnetic fields, Maxwell's equations, and other fundamental electromagnetic concepts. Multiple students contributed sections on topics including Coulomb's law, Biot-Savart law, Lorentz force, and Maxwell's equations in differential, integral and harmonic forms.
The document discusses key concepts in quantum mechanics including wavefunctions, operators, and the uncertainty principle. Some key points:
- A wavefunction Ψ(x,t) describes the probability of finding a particle at position x and time t. Operators like -iħ∂/∂x correspond to physical quantities like momentum.
- Applying these operators to Ψ yields the particle's momentum if Ψ is an eigenfunction, but not if Ψ is a superposition of momentum states.
- Heisenberg's uncertainty principle states the more precisely position is known, the less precisely momentum can be known, and vice versa. It is quantified as ΔxΔp ≥ ħ/
This document discusses wave equations and their applications. It contains 6 chapters that cover basic results of wave equations, different types of wave equations like scalar and electromagnetic wave equations, and applications of wave equations. The introduction explains that the document contains details about different types of wave equations and their applications in sciences. Chapter 1 discusses basic concepts like relativistic and non-relativistic wave equations. Chapter 2 covers the scalar wave equation. Chapter 3 is about electromagnetic wave equations. Chapter 4 is on applications of wave equations.
This document provides a summary of key concepts from a lecture on the finite square well model in quantum mechanics. It describes:
1) How the Schrodinger equation leads to either unbound or bound state solutions depending on the energy level E relative to the potential energy V0.
2) For bound states where 0 < E < V0, the wavefunction is oscillatory inside the well and exponentially decaying outside.
3) The energy levels and wavefunctions are quantized, with the number of possible energy levels increasing as the depth of the potential well V0 increases.
Zero Point Energy And Vacuum Fluctuations EffectsAna_T
The document discusses several topics related to zero point energy and vacuum fluctuations:
1) It explains how the Heisenberg uncertainty principle leads to zero point energy, which is the lowest possible energy of a quantum system in its ground state.
2) It describes vacuum fluctuations as quantum fluctuations of fields even in their lowest energy state. These fluctuations can be thought of as virtual particles being created and destroyed in the vacuum.
3) Several phenomena are discussed that demonstrate the effects of zero point energy and vacuum fluctuations, including spontaneous emission, the Casimir effect, and the Lamb shift.
Gravitational quantum mechanics: a theory for explaining spacetime. This a seminar on several scientific papers about quantum gravity Phenomenology which has been gathered several important outcomes.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
Bound states in 1d, 2d and 3d quantum wellsAhmed Aslam
This document discusses simulations of bound states in 1D, 2D, and 3D finite potential wells. For 1D wells, the finite difference method was used to solve the Schrodinger equation and obtain energy eigenvalues and eigenfunctions. For 2D wells, spline interpolation was employed. Bound states in 3D cubic and spherical wells were determined by solving the radial Schrodinger equation involving Bessel functions. The programs yielded plots of eigenfunctions and energy levels demonstrating bound states in wells of varying dimensions.
Dirac-delta function, Expectation values+ mathematical interpretation, Compatible observables, Incompatible observables, Difference between continuous spectra(unbound state) and line/discrete spectra(bound state), one example, including diagrams+ graphs.
- The document describes three common crystal structures: simple cubic, face-centered cubic, and body-centered cubic.
- It provides information on their unit cells, atomic packing factors, and the coordination number of atoms in each structure.
- Simple cubic has a coordination number of 6 and the lowest atomic packing factor at 0.52. Face-centered cubic has the highest atomic packing factor of 0.74.
1) The Schrodinger equation describes quantum mechanical systems and allows their behavior to be modeled as both particles and waves. It expresses physical parameters like position, momentum, and energy as operators.
2) The time-independent Schrodinger equation can be used to find the energy values of one-dimensional quantum systems like a particle in a box or quantum harmonic oscillator.
3) The Schrodinger equation in spherical coordinates is well-suited for modeling the hydrogen atom due to its spherical symmetry.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
- The atom consists of a small, dense nucleus surrounded by an electron cloud.
- Electrons can only exist in certain discrete energy levels around the nucleus. Their wavelengths are determined by the principal quantum number.
- The Bohr model improved on earlier models by introducing energy levels and quantization, but had limitations. The quantum mechanical model treats electrons as waves and uses Schrodinger's equation.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
The document summarizes the Vanderbilt pseudopotential method. It starts from all-electron calculations and defines a pseudo-wavefunction. The pseudopotential V'loc is constructed as the sum of the local potential Vloc, Hartree potential VH, and exchange-correlation potential VXC. Unlike other pseudopotentials which subtract VH and VXC, Vanderbilt's adds them. The method can produce both norm-conserving and ultra-soft pseudopotentials depending on whether the generalized norm-conserving condition is satisfied.
Charge Quantization and Magnetic MonopolesArpan Saha
This talk, given as a part of the Annual Seminar Weekend 2011, IIT Bombay, dealt with a homotopy-based
variant of the argument Dirac provided to show that the existence of a single magnetic monopole in the Universe
is a sufficient condition for the quantization of electric charge.
This document provides an overview of the Schrodinger wave equation, including:
1) It describes the derivation of the time-dependent Schrodinger wave equation based on what is known about the wave function of a particle.
2) It explains that the time-dependent equation can be used to derive the time-independent Schrodinger equation by assuming the wave function separates into spatial and temporal factors.
3) It notes that for the wave function to be physically acceptable, the energy values inserted into the time-independent equation must be quantized to satisfy normalization conditions on the wave function.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
1) The document discusses the exact solution to the Klein-Gordon shutter problem, finding that the wave function does not resemble the optical expression for diffraction but the charge density does show transient oscillations resembling a diffraction pattern.
2) It presents the exact solution for the Klein-Gordon shutter problem using discontinuous initial conditions, finding the wave function solution differs from Moshinsky's approximation.
3) When the exact relativistic charge density is plotted over time, it shows transient oscillations that resemble a diffraction pattern, despite some relativistic differences, demonstrating that diffraction in time exists in relativistic scenarios.
Fundamental Concepts on Electromagnetic TheoryAL- AMIN
The document summarizes key concepts from a presentation on electromagnetic theory. It discusses different types of fields, including scalar and vector fields. It also covers gradient, divergence, curl, coordinate systems, static electric and magnetic fields, Maxwell's equations, and other fundamental electromagnetic concepts. Multiple students contributed sections on topics including Coulomb's law, Biot-Savart law, Lorentz force, and Maxwell's equations in differential, integral and harmonic forms.
The document discusses key concepts in quantum mechanics including wavefunctions, operators, and the uncertainty principle. Some key points:
- A wavefunction Ψ(x,t) describes the probability of finding a particle at position x and time t. Operators like -iħ∂/∂x correspond to physical quantities like momentum.
- Applying these operators to Ψ yields the particle's momentum if Ψ is an eigenfunction, but not if Ψ is a superposition of momentum states.
- Heisenberg's uncertainty principle states the more precisely position is known, the less precisely momentum can be known, and vice versa. It is quantified as ΔxΔp ≥ ħ/
This document discusses wave equations and their applications. It contains 6 chapters that cover basic results of wave equations, different types of wave equations like scalar and electromagnetic wave equations, and applications of wave equations. The introduction explains that the document contains details about different types of wave equations and their applications in sciences. Chapter 1 discusses basic concepts like relativistic and non-relativistic wave equations. Chapter 2 covers the scalar wave equation. Chapter 3 is about electromagnetic wave equations. Chapter 4 is on applications of wave equations.
This document provides a summary of key concepts from a lecture on the finite square well model in quantum mechanics. It describes:
1) How the Schrodinger equation leads to either unbound or bound state solutions depending on the energy level E relative to the potential energy V0.
2) For bound states where 0 < E < V0, the wavefunction is oscillatory inside the well and exponentially decaying outside.
3) The energy levels and wavefunctions are quantized, with the number of possible energy levels increasing as the depth of the potential well V0 increases.
Zero Point Energy And Vacuum Fluctuations EffectsAna_T
The document discusses several topics related to zero point energy and vacuum fluctuations:
1) It explains how the Heisenberg uncertainty principle leads to zero point energy, which is the lowest possible energy of a quantum system in its ground state.
2) It describes vacuum fluctuations as quantum fluctuations of fields even in their lowest energy state. These fluctuations can be thought of as virtual particles being created and destroyed in the vacuum.
3) Several phenomena are discussed that demonstrate the effects of zero point energy and vacuum fluctuations, including spontaneous emission, the Casimir effect, and the Lamb shift.
Gravitational quantum mechanics: a theory for explaining spacetime. This a seminar on several scientific papers about quantum gravity Phenomenology which has been gathered several important outcomes.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
Bound states in 1d, 2d and 3d quantum wellsAhmed Aslam
This document discusses simulations of bound states in 1D, 2D, and 3D finite potential wells. For 1D wells, the finite difference method was used to solve the Schrodinger equation and obtain energy eigenvalues and eigenfunctions. For 2D wells, spline interpolation was employed. Bound states in 3D cubic and spherical wells were determined by solving the radial Schrodinger equation involving Bessel functions. The programs yielded plots of eigenfunctions and energy levels demonstrating bound states in wells of varying dimensions.
Dirac-delta function, Expectation values+ mathematical interpretation, Compatible observables, Incompatible observables, Difference between continuous spectra(unbound state) and line/discrete spectra(bound state), one example, including diagrams+ graphs.
- The document describes three common crystal structures: simple cubic, face-centered cubic, and body-centered cubic.
- It provides information on their unit cells, atomic packing factors, and the coordination number of atoms in each structure.
- Simple cubic has a coordination number of 6 and the lowest atomic packing factor at 0.52. Face-centered cubic has the highest atomic packing factor of 0.74.
1) The Schrodinger equation describes quantum mechanical systems and allows their behavior to be modeled as both particles and waves. It expresses physical parameters like position, momentum, and energy as operators.
2) The time-independent Schrodinger equation can be used to find the energy values of one-dimensional quantum systems like a particle in a box or quantum harmonic oscillator.
3) The Schrodinger equation in spherical coordinates is well-suited for modeling the hydrogen atom due to its spherical symmetry.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
3D Hybrid PIC simulation of the plasma expansion (ISSS-14)
TR-6.ppt
1. 1
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling
CHAPTER 6
Quantum Mechanics II
I think it is safe to say that no one understands quantum mechanics. Do
not keep saying to yourself, if you can possibly avoid it, “But how can it
be like that?” because you will get “down the drain” into a blind alley from
which nobody has yet escaped. Nobody knows how it can be like that.
- Richard Feynman
2. 2
6.1: The Schrödinger Wave Equation
The Schrödinger wave equation in its time-dependent form for a
particle of energy E moving in a potential V in one dimension is
The extension into three dimensions is
where is an imaginary number.
3. 3
General Solution of the Schrödinger
Wave Equation
The general form of the wave function is
which also describes a wave moving in the x direction.
In general the amplitude may also be complex.
The wave function is also not restricted to being real.
Notice that the sine term has an imaginary number. Only
the physically measurable quantities must be real.
These include the probability, momentum and energy.
4. 4
Normalization and Probability
The probability P(x) dx of a particle being between x and X + dx
was given in the equation
The probability of the particle being between x1 and x2 is given
by
The wave function must also be normalized so that the
probability of the particle being somewhere on the x axis is 1.
5. 5
Properties of Valid Wave Functions
Boundary conditions
1) In order to avoid infinite probabilities, the wave function must be finite
everywhere.
2) In order to avoid multiple values of the probability, the wave function
must be single valued.
3) For finite potentials, the wave function and its derivative must be
continuous. This is required because the second-order derivative term
in the wave equation must be single valued. (There are exceptions to
this rule when V is infinite.)
4) In order to normalize the wave functions, they must approach zero as x
approaches infinity.
Solutions that do not satisfy these properties do not generally
correspond to physically realizable circumstances.
6. 6
Time-Independent Schrödinger Wave
Equation
The potential in many cases will not depend explicitly on time.
The dependence on time and position can then be separated in the
Schrödinger wave equation. Let ,
which yields:
Now divide by the wave function:
The left side of Equation (6.10) depends only on time, and the right side
depends only on spatial coordinates. Hence each side must be equal to
a constant. The time dependent side is
7. 7
We integrate both sides and find:
where C is an integration constant that we may choose to be 0. Therefore
This determines f to be
This is known as the time-independent Schrödinger wave equation, and it is a
fundamental equation in quantum mechanics.
Time-Independent Schrödinger Wave
Equation Continued
8. 8
Stationary State
The wave function can be written as:
The probability density becomes:
The probability distributions are constant in time. This is a standing
wave phenomena that is called the stationary state.
9. 9
Momentum Operator
To find the expectation value of p, we first need to represent p in terms
of x and t. Consider the derivative of the wave function of a free particle
with respect to x:
With k = p / ħ we have
This yields
This suggests we define the momentum operator as .
The expectation value of the momentum is
10. 10
The position x is its own operator as seen above.
The time derivative of the free-particle wave function is
Substituting ω = E / ħ yields
The energy operator is
The expectation value of the energy is
Position and Energy Operators
11. 11
Comparison of Classical and Quantum
Mechanics
Newton’s second law and Schrödinger’s wave equation are
both differential equations.
Newton’s second law can be derived from the Schrödinger
wave equation, so the latter is the more fundamental.
Classical mechanics only appears to be more precise because
it deals with macroscopic phenomena. The underlying
uncertainties in macroscopic measurements are just too small
to be significant.
12. 12
6.2: Expectation Values
The expectation value is the expected result of the average of
many measurements of a given quantity. The expectation value
of x is denoted by <x>
Any measurable quantity for which we can calculate the
expectation value is called a physical observable. The
expectation values of physical observables (for example,
position, linear momentum, angular momentum, and energy)
must be real, because the experimental results of
measurements are real.
The average value of x is
13. 13
Continuous Expectation Values
We can change from discrete to
continuous variables by using the
probability P(x,t) of observing the
particle at a particular x.
Using the wave function, the
expectation value is:
The expectation value of any
function g(x) for a normalized wave
function:
14. 14
Some expectation values are sharp some
others fuzzy
Since there is scatter in the actual positions
(x), the calculated expectation value will
have an uncertainty, fuzziness (Note that x
is its own operator.)
15. 15
Some expectation values are sharp some
others fuzzy, continued I
For any observable, fuzzy or not
If not fuzzy, ΔQ = 0
Because <Q2>= <Q>2
x may as well stand
for any kind of
operator Q
16. 16
Some expectation values are sharp some
others fuzzy, continued II
Eigenvalues of operators are always sharp (an actual – physical -
measurement may give some variation in the result, but the
calculation gives zero fuzziness
Say Q is the Hamiltonian operator
A wavefunction that solves this
equation is an eigenfunction of this
operator, E is the corresponding
eigenvalue, apply this operator
twice and you get E2 – which sure
is the same as squaring to result of
applying it once (E)
17. 17
6.3: Infinite Square-Well Potential
The simplest such system is that of a particle trapped in a box with
infinitely hard walls that the particle cannot penetrate. This potential
is called an infinite square well and is given by
Clearly the wave function must be zero where the potential is
infinite.
Where the potential is zero inside the box, the Schrödinger wave
equation becomes where .
The general solution is .
18. 18
Quantization
Boundary conditions of the potential dictate that the wave function must
be zero at x = 0 and x = L. This yields valid solutions for integer values
of n such that kL = nπ.
The wave function is now
We normalize the wave function
The normalized wave function becomes
These functions are identical to those obtained for a vibrating string with
fixed ends.
19. 19
Quantized Energy
The quantized wave number now becomes
Solving for the energy yields
Note that the energy depends on the integer values of n. Hence the
energy is quantized and nonzero.
The special case of n = 0 is called the ground state energy.
20. 20
6.4: Finite Square-Well Potential
The finite square-well potential is
The Schrödinger equation outside the finite well in regions I and III is
or using
yields . Considering that the wave function must be zero at
infinity, the solutions for this equation are
21. 21
Inside the square well, where the potential V is zero, the wave equation
becomes where
Instead of a sinusoidal solution we have
The boundary conditions require that
and the wave function must be smooth where the regions meet.
Note that the
wave function is
nonzero outside
of the box.
Finite Square-Well Solution
22. 22
Penetration Depth
The penetration depth is the distance outside the potential well where
the probability significantly decreases. It is given by
It should not be surprising to find that the penetration distance that
violates classical physics is proportional to Planck’s constant.
23. 23
The wave function must be a function of all three spatial coordinates.
We begin with the conservation of energy
Multiply this by the wave function to get
Now consider momentum as an operator acting on the wave
function. In this case, the operator must act twice on each dimension.
Given:
The three dimensional Schrödinger wave equation is
6.5: Three-Dimensional Infinite-Potential Well
24. 24
Degeneracy
Analysis of the Schrödinger wave equation in three dimensions
introduces three quantum numbers that quantize the energy.
A quantum state is degenerate when there is more than one wave
function for a given energy.
Degeneracy results from particular properties of the potential energy
function that describes the system. A perturbation of the potential
energy can remove the degeneracy.
25. 25
6.6: Simple Harmonic Oscillator
Simple harmonic oscillators describe many physical situations: springs,
diatomic molecules and atomic lattices.
Consider the Taylor expansion of a potential function:
Redefining the minimum potential and the zero potential, we have
Substituting this into the wave equation:
Let and which yields .
26. 26
Parabolic Potential Well
If the lowest energy level is zero, this violates the uncertainty principle.
The wave function solutions are where Hn(x) are Hermite
polynomials of order n.
In contrast to the particle in a box, where the oscillatory wave function is a
sinusoidal curve, in this case the oscillatory behavior is due to the polynomial,
which dominates at small x. The exponential tail is provided by the Gaussian
function, which dominates at large x.
27. 27
Analysis of the Parabolic Potential Well
The energy levels are given by
The zero point energy is called the Heisenberg
limit:
Classically, the probability of finding the mass is
greatest at the ends of motion and smallest at the
center (that is, proportional to the amount of time
the mass spends at each position).
Contrary to the classical one, the largest probability
for this lowest energy state is for the particle to be
at the center.
28. 28
6.7: Barriers and Tunneling
Consider a particle of energy E approaching a potential barrier of height V0 and
the potential everywhere else is zero.
We will first consider the case when the energy is greater than the potential
barrier.
In regions I and III the wave numbers are:
In the barrier region we have
29. 29
Reflection and Transmission
The wave function will consist of an incident wave, a reflected wave, and a
transmitted wave.
The potentials and the Schrödinger wave equation for the three regions are
as follows:
The corresponding solutions are:
As the wave moves from left to right, we can simplify the wave functions to:
30. 30
Probability of Reflection and Transmission
The probability of the particles being reflected R or transmitted T is:
The maximum kinetic energy of the photoelectrons depends on the
value of the light frequency f and not on the intensity.
Because the particles must be either reflected or transmitted we have:
R + T = 1.
By applying the boundary conditions x → ±∞, x = 0, and x = L, we arrive
at the transmission probability:
Notice that there is a situation in which the transmission probability is 1.
31. 31
Tunneling
Now we consider the situation where classically the particle does not have
enough energy to surmount the potential barrier, E < V0.
The quantum mechanical result, however, is one of the most remarkable features
of modern physics, and there is ample experimental proof of its existence. There
is a small, but finite, probability that the particle can penetrate the barrier and
even emerge on the other side.
The wave function in region II becomes
The transmission probability that
describes the phenomenon of tunneling is
32. 32
Uncertainty Explanation
Consider when κL >> 1 then the transmission probability becomes:
This violation allowed by the uncertainty principle is equal to the
negative kinetic energy required! The particle is allowed by quantum
mechanics and the uncertainty principle to penetrate into a classically
forbidden region. The minimum such kinetic energy is:
33. 33
Analogy with Wave Optics
If light passing through a glass prism reflects from an
internal surface with an angle greater than the critical
angle, total internal reflection occurs. However, the
electromagnetic field is not exactly zero just outside
the prism. If we bring another prism very close to the
first one, experiments show that the electromagnetic
wave (light) appears in the second prism The situation
is analogous to the tunneling described here. This
effect was observed by Newton and can be
demonstrated with two prisms and a laser. The
intensity of the second light beam decreases
exponentially as the distance between the two prisms
increases.
34. 34
Potential Well
Consider a particle passing through a potential well region rather than through a
potential barrier.
Classically, the particle would speed up passing the well region, because K = mv2 / 2 =
E + V0.
According to quantum mechanics, reflection and transmission may occur, but the
wavelength inside the potential well is smaller than outside. When the width of the
potential well is precisely equal to half-integral or integral units of the wavelength, the
reflected waves may be out of phase or in phase with the original wave, and
cancellations or resonances may occur. The reflection/cancellation effects can lead to
almost pure transmission or pure reflection for certain wavelengths. For example, at the
second boundary (x = L) for a wave passing to the right, the wave may reflect and be
out of phase with the incident wave. The effect would be a cancellation inside the well.
35. 35
Alpha-Particle Decay
The phenomenon of tunneling explains the alpha-particle decay of heavy,
radioactive nuclei.
Inside the nucleus, an alpha particle feels the strong, short-range attractive
nuclear force as well as the repulsive Coulomb force.
The nuclear force dominates inside the nuclear radius where the potential is
approximately a square well.
The Coulomb force dominates
outside the nuclear radius.
The potential barrier at the nuclear
radius is several times greater than
the energy of an alpha particle.
According to quantum mechanics,
however, the alpha particle can
“tunnel” through the barrier. Hence
this is observed as radioactive decay.