1. The document discusses the application of the Schrödinger equation to the hydrogen atom. It separates the Schrödinger equation into radial, angular, and azimuthal parts to solve for the wave functions.
2. The solutions are quantified by three quantum numbers - the principal quantum number n, the orbital angular momentum quantum number l, and the magnetic quantum number ml. The values of these numbers are restricted based on the boundary conditions of the solutions.
3. The energy levels of the hydrogen atom solutions are quantified based on the principal quantum number n and have degeneracies determined by the other quantum numbers.
The chapter discusses the quantum mechanical model of the hydrogen atom.
It begins by applying the Schrödinger equation to the hydrogen atom potential and separating the equation into radial, angular, and azimuthal parts. This leads to the quantization of energy into discrete allowed energy levels.
The chapter then introduces the key quantum numbers - principal (n), angular momentum (l), and magnetic (ml) - that arise from the separated equations. The values of these numbers and their relationships are defined.
Finally, the chapter discusses how an external magnetic field interacts with the magnetic moments of orbiting electrons and intrinsic spin, causing the Zeeman effect of spectral line splitting into multiple energy levels.
The document summarizes key concepts about the hydrogen atom from quantum mechanics. It begins by introducing the Schrödinger equation and how it can be applied and solved for the hydrogen atom potential. The solution involves separation of variables into radial, angular, and azimuthal components. This leads to the identification of three quantum numbers - principal (n), angular momentum (l), and magnetic (ml) - that characterize the possible energy states. Higher sections discuss properties like orbital shapes, spin, and transition selection rules between energy levels and electron probability distributions.
The document discusses the calculation of vacuum polarization corrections to the energy levels of hydrogen atom. It presents numerical calculations of the one-photon vacuum polarization corrections to the 1s, 2s, 2p, 3s, 3p and 3d energy levels of hydrogen using the Uehling potential. The contribution of this correction decreases with increasing principal quantum number n and orbital quantum number l. It also calculates the relativistic effects on the 1s, 2s, 2p1/2 and 2p3/2 levels using Dirac wave functions. The results show the vacuum polarization effect is largest near the nucleus and decreases with n.
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
1) The Born-Oppenheimer approximation separates the molecular Schrodinger equation into electronic and nuclear parts based on the large mass difference between electrons and nuclei.
2) It assumes that over short time periods, electrons adjust instantaneously to nuclear motions. This allows treating electronic motions separately for fixed nuclear positions.
3) Solving the electronic Schrodinger equation for different nuclear configurations provides the potential energy surface for nuclear vibrations and rotations.
The document discusses the Schrodinger equation of the hydrogen atom. It shows how the Schrodinger equation can be separated into radial and angular variables using spherical coordinates. This results in three ordinary differential equations - one for the radial coordinate and two for the angular coordinates. The solutions of these equations involve quantum numbers such as the orbital angular momentum quantum number l and its magnetic quantum number ml.
The document discusses corrections to the simple hydrogen atom model. Relativistic corrections from the electron's kinetic energy and spin-orbit coupling lift the degeneracy of energy levels and introduce fine structure splitting. Spin-orbit coupling couples the electron's spin and orbital angular momentum, resulting in total angular momentum states. The three main corrections are from relativistic kinetic energy, spin-orbit coupling, and the Darwin term. Together these corrections depend only on the principal quantum number n and total angular momentum j. The Lamb shift is a further correction revealing the electron's self-interaction through quantum electrodynamics. Hyperfine
The document discusses the quantum mechanical description of the hydrogen atom. It begins by introducing the time-independent Schrodinger equation used to model hydrogen. It then discusses angular momentum and its quantum mechanical operators.
The main body of the document provides a step-by-step solution of the Schrodinger equation for hydrogen. This includes separating the equation into radial and angular components and solving each. The solutions provide the allowed energy levels and wavefunctions of hydrogen in terms of quantum numbers. The document also discusses other topics like spin, selection rules, probability distributions, and the Zeeman effect.
The chapter discusses the quantum mechanical model of the hydrogen atom.
It begins by applying the Schrödinger equation to the hydrogen atom potential and separating the equation into radial, angular, and azimuthal parts. This leads to the quantization of energy into discrete allowed energy levels.
The chapter then introduces the key quantum numbers - principal (n), angular momentum (l), and magnetic (ml) - that arise from the separated equations. The values of these numbers and their relationships are defined.
Finally, the chapter discusses how an external magnetic field interacts with the magnetic moments of orbiting electrons and intrinsic spin, causing the Zeeman effect of spectral line splitting into multiple energy levels.
The document summarizes key concepts about the hydrogen atom from quantum mechanics. It begins by introducing the Schrödinger equation and how it can be applied and solved for the hydrogen atom potential. The solution involves separation of variables into radial, angular, and azimuthal components. This leads to the identification of three quantum numbers - principal (n), angular momentum (l), and magnetic (ml) - that characterize the possible energy states. Higher sections discuss properties like orbital shapes, spin, and transition selection rules between energy levels and electron probability distributions.
The document discusses the calculation of vacuum polarization corrections to the energy levels of hydrogen atom. It presents numerical calculations of the one-photon vacuum polarization corrections to the 1s, 2s, 2p, 3s, 3p and 3d energy levels of hydrogen using the Uehling potential. The contribution of this correction decreases with increasing principal quantum number n and orbital quantum number l. It also calculates the relativistic effects on the 1s, 2s, 2p1/2 and 2p3/2 levels using Dirac wave functions. The results show the vacuum polarization effect is largest near the nucleus and decreases with n.
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
1) The Born-Oppenheimer approximation separates the molecular Schrodinger equation into electronic and nuclear parts based on the large mass difference between electrons and nuclei.
2) It assumes that over short time periods, electrons adjust instantaneously to nuclear motions. This allows treating electronic motions separately for fixed nuclear positions.
3) Solving the electronic Schrodinger equation for different nuclear configurations provides the potential energy surface for nuclear vibrations and rotations.
The document discusses the Schrodinger equation of the hydrogen atom. It shows how the Schrodinger equation can be separated into radial and angular variables using spherical coordinates. This results in three ordinary differential equations - one for the radial coordinate and two for the angular coordinates. The solutions of these equations involve quantum numbers such as the orbital angular momentum quantum number l and its magnetic quantum number ml.
The document discusses corrections to the simple hydrogen atom model. Relativistic corrections from the electron's kinetic energy and spin-orbit coupling lift the degeneracy of energy levels and introduce fine structure splitting. Spin-orbit coupling couples the electron's spin and orbital angular momentum, resulting in total angular momentum states. The three main corrections are from relativistic kinetic energy, spin-orbit coupling, and the Darwin term. Together these corrections depend only on the principal quantum number n and total angular momentum j. The Lamb shift is a further correction revealing the electron's self-interaction through quantum electrodynamics. Hyperfine
The document discusses the quantum mechanical description of the hydrogen atom. It begins by introducing the time-independent Schrodinger equation used to model hydrogen. It then discusses angular momentum and its quantum mechanical operators.
The main body of the document provides a step-by-step solution of the Schrodinger equation for hydrogen. This includes separating the equation into radial and angular components and solving each. The solutions provide the allowed energy levels and wavefunctions of hydrogen in terms of quantum numbers. The document also discusses other topics like spin, selection rules, probability distributions, and the Zeeman effect.
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 document discusses Maxwell's equations and time-harmonic electromagnetic fields. It begins by presenting Maxwell's four equations describing electric and magnetic fields. It then discusses the constitutive relations relating the fields to material properties. Maxwell's equations can describe fields in linear and nonlinear media. The equations are also presented in integral form. Examples are provided on applying Maxwell's equations to derive the diffusion equation and skin depth. Poynting's theorem is then introduced, relating power flow to energy storage and dissipation. Finally, time-harmonic fields are discussed, with Maxwell's equations expressed using phasors. Power flow is defined for time-harmonic fields using complex conjugates.
Mie theory describes the scattering of electromagnetic radiation by a spherical particle. It provides an exact solution to Maxwell's equations for the scattering of a plane electromagnetic wave by a homogeneous sphere. Gustav Mie provided the mathematical description for the spectral dependence of scattering by a spherical nanoparticle. Mie theory can be used to calculate the absorption and scattering cross sections of nanoparticles and provides the basis for measuring particle size through light scattering. It is valid for particles ranging from much smaller to larger than the wavelength of light.
This document summarizes the key concepts of angular momentum of electrons and the Stern-Gerlach experiment. It discusses how electrons have both orbital angular momentum from their motion around the nucleus, as well as intrinsic spin angular momentum. The Stern-Gerlach experiment provided evidence of electron spin by observing the splitting of a silver atom beam in a non-homogeneous magnetic field. When combining multiple angular momenta, the total angular momentum and its quantum numbers are obtained by adding the angular momentum vectors or their corresponding quantum numbers. This allows predicting the energy levels and degeneracies of atomic orbitals.
The document discusses the wave functions of the hydrogen atom. It begins by presenting the Schrodinger wave equation in cartesian and polar coordinates. It then explains that the total wave function ψ can be separated into a radial wave function R(r) and an angular wave function Θ(θ)Φ(∅). R(r) depends on the distance r from the nucleus and describes the distribution of electron charge density, while the angular function depends on the direction and determines the shape of the orbital. Finally, it discusses important properties of the radial and angular wave functions for s, p, d and other orbitals.
This document presents a novel technique for solving the transcendental equations of selective harmonics elimination pulse width modulation (SHEPWM) inverters based on the secant method. The proposed algorithm uses the secant method to simplify the numerical solution of the nonlinear equations and solve them faster compared to other methods. Simulation results validate that the proposed method accurately estimates the switching angles to eliminate specific harmonics from the output voltage waveform and achieves near sinusoidal output current for various modulation indices and numbers of harmonics eliminated.
This document provides an analysis of quantizing energy levels for the Schwarzschild gravitational field using the prescriptions of Old Quantum Theory. It begins with introducing the Schwarzschild metric and deriving the Lagrangian for motion in a central gravitational field. It then applies the Bohr-Sommerfeld quantization rule to obtain expressions for angular momentum and energy as quantized values. Integrating these expressions provides a very simple formula for the quantized energy levels in the Schwarzschild field within the Old Quantum Theory framework.
The document discusses the structure of atoms, including:
1) Solving the Schrodinger equation for hydrogen-like atoms to determine allowed energies and wavefunctions.
2) The quantization of energy levels, orbital angular momentum, and other properties for hydrogen-like atoms.
3) How the concepts for hydrogen-like atoms can be applied to describe multi-electron atoms and molecules using approximations like the orbital model.
This document provides solutions to theoretical physics problems from the 1st Asian Physics Olympiad held in Karawaci, Indonesia in April 2000. The solutions include:
1) Deriving an expression for the relative angular velocity of Jupiter and Earth and calculating the relative velocity.
2) Calculating the detection limit of a radioactive source using an ionization chamber and determining the necessary voltage pulse amplifier gain.
3) Using Gauss' law to calculate the electric field and potential between the plates of a parallel plate capacitor and deriving an expression for the capacitance per unit length.
The document discusses including spin-orbit coupling in the author's model of photoassociation and rovibrational relaxation in NaCs. It presents the theoretical description, including the Hamiltonian with additional terms for spin-orbit interaction. The system is described by a wavefunction in the Born-Oppenheimer approximation. Equations are derived for the probability amplitudes of relevant rovibrational states including spin-orbit coupling between the A1Σ+ and b3Π electronic states. The initial condition of the scattering system at ultracold temperature is specified.
Quantum Theory. Wave Particle Duality. Particle in a Box. Schrodinger wave equation. Quantum Numbers and Electron Orbitals. Principal Shells and Subshells. A Fourth Quantum Number. Effective nuclear charge
The document discusses the Schrodinger equation and methods to approximate solutions to it. It begins by defining the time-independent Schrodinger equation and its components. It then provides examples of writing out the Schrodinger equation for different chemical systems with varying numbers of electrons and nuclei. Approximation methods are needed because the Schrodinger equation can only be exactly solved for a few simple systems. Two approximation methods discussed are the variational method and perturbation theory. The variational method uses a trial wavefunction to variationally minimize the energy.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
This document provides solutions to 15 problems related to magnetic materials and magnetism. It begins by deriving expressions for how the angular frequency and radius of a classical electron orbit change with the application of a magnetic field using the Lorentz force. It then calculates magnetic moments and susceptibilities for various systems. Other problems cover topics like Lenz's law, diamagnetism, paramagnetism, Bohr orbits, susceptibility measurements, and predicting magnetic behaviors in different materials. Detailed step-by-step workings are shown for each problem.
I am Peterson N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, University of Melbourne, Australia. I have been helping students with their homework for the past 8 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
This presentation discusses a computational chemistry study of the interaction energies of molecules bonded through chalcogen bonds. The study examined 40 molecules total where the substituent X bonded to selenium, which was also bonded to CH3 or H, was varied. The substituents included H, F, CH3, CF3, etc. Computational methods included DFT calculations at the MP2/aug-cc-pVTZ level to determine binding energies and natural bond orbital analysis. Key concepts explained include basis sets, geometry optimization, basis set superposition error and counterpoise correction.
This document discusses light-matter interaction using a two-level atom model. It describes how an atom with only two energy levels can be modeled as a two-dimensional quantum mechanical system. The interaction of such a two-level atom with an electromagnetic field is then derived, leading to Rabi oscillations between the atomic energy levels driven by the field. Dissipative processes require a statistical description using the density operator formalism.
The document discusses first order perturbation theory. It begins by introducing perturbation theory as an approximate method to solve the Schrodinger equation for complex quantum systems where the Hamiltonian cannot be solved exactly. It then presents the key equations of first order perturbation theory. The first order correction to the energy is given by the expectation value of the perturbation operator over the unperturbed ground state wavefunction. The first order correction to the wavefunction is expressed as a linear combination of unperturbed eigenstates.
Relativistic formulation of Maxwell equations.dhrubanka
This document discusses the relativistic formulation of Maxwell's equations. It begins by introducing the key concepts of special relativity that are needed, including Lorentz transformations and four-vectors. It then shows how the electric and magnetic fields transform under Lorentz transformations and how they can be combined into the electromagnetic field tensor. The document also discusses how charge and current densities transform and satisfy the continuity equation as a four-vector. Finally, it presents Maxwell's equations in their compact relativistic form in terms of the field tensor and its derivatives.
This document provides an overview of modern atomic theory and quantum mechanics. It discusses the key discoveries and models that led to our current understanding of atomic structure, including Dalton's atomic theory, the discovery of subatomic particles, and the development of quantum mechanics with Schrodinger's equation. The four quantum numbers - principal, angular momentum, magnetic, and spin - are introduced to describe the allowed states of electrons in atoms. Rules for writing electron configurations are also covered.
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
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
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Similar to lecture_11-chapter7HydrogenAtom-1new.pdf
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 document discusses Maxwell's equations and time-harmonic electromagnetic fields. It begins by presenting Maxwell's four equations describing electric and magnetic fields. It then discusses the constitutive relations relating the fields to material properties. Maxwell's equations can describe fields in linear and nonlinear media. The equations are also presented in integral form. Examples are provided on applying Maxwell's equations to derive the diffusion equation and skin depth. Poynting's theorem is then introduced, relating power flow to energy storage and dissipation. Finally, time-harmonic fields are discussed, with Maxwell's equations expressed using phasors. Power flow is defined for time-harmonic fields using complex conjugates.
Mie theory describes the scattering of electromagnetic radiation by a spherical particle. It provides an exact solution to Maxwell's equations for the scattering of a plane electromagnetic wave by a homogeneous sphere. Gustav Mie provided the mathematical description for the spectral dependence of scattering by a spherical nanoparticle. Mie theory can be used to calculate the absorption and scattering cross sections of nanoparticles and provides the basis for measuring particle size through light scattering. It is valid for particles ranging from much smaller to larger than the wavelength of light.
This document summarizes the key concepts of angular momentum of electrons and the Stern-Gerlach experiment. It discusses how electrons have both orbital angular momentum from their motion around the nucleus, as well as intrinsic spin angular momentum. The Stern-Gerlach experiment provided evidence of electron spin by observing the splitting of a silver atom beam in a non-homogeneous magnetic field. When combining multiple angular momenta, the total angular momentum and its quantum numbers are obtained by adding the angular momentum vectors or their corresponding quantum numbers. This allows predicting the energy levels and degeneracies of atomic orbitals.
The document discusses the wave functions of the hydrogen atom. It begins by presenting the Schrodinger wave equation in cartesian and polar coordinates. It then explains that the total wave function ψ can be separated into a radial wave function R(r) and an angular wave function Θ(θ)Φ(∅). R(r) depends on the distance r from the nucleus and describes the distribution of electron charge density, while the angular function depends on the direction and determines the shape of the orbital. Finally, it discusses important properties of the radial and angular wave functions for s, p, d and other orbitals.
This document presents a novel technique for solving the transcendental equations of selective harmonics elimination pulse width modulation (SHEPWM) inverters based on the secant method. The proposed algorithm uses the secant method to simplify the numerical solution of the nonlinear equations and solve them faster compared to other methods. Simulation results validate that the proposed method accurately estimates the switching angles to eliminate specific harmonics from the output voltage waveform and achieves near sinusoidal output current for various modulation indices and numbers of harmonics eliminated.
This document provides an analysis of quantizing energy levels for the Schwarzschild gravitational field using the prescriptions of Old Quantum Theory. It begins with introducing the Schwarzschild metric and deriving the Lagrangian for motion in a central gravitational field. It then applies the Bohr-Sommerfeld quantization rule to obtain expressions for angular momentum and energy as quantized values. Integrating these expressions provides a very simple formula for the quantized energy levels in the Schwarzschild field within the Old Quantum Theory framework.
The document discusses the structure of atoms, including:
1) Solving the Schrodinger equation for hydrogen-like atoms to determine allowed energies and wavefunctions.
2) The quantization of energy levels, orbital angular momentum, and other properties for hydrogen-like atoms.
3) How the concepts for hydrogen-like atoms can be applied to describe multi-electron atoms and molecules using approximations like the orbital model.
This document provides solutions to theoretical physics problems from the 1st Asian Physics Olympiad held in Karawaci, Indonesia in April 2000. The solutions include:
1) Deriving an expression for the relative angular velocity of Jupiter and Earth and calculating the relative velocity.
2) Calculating the detection limit of a radioactive source using an ionization chamber and determining the necessary voltage pulse amplifier gain.
3) Using Gauss' law to calculate the electric field and potential between the plates of a parallel plate capacitor and deriving an expression for the capacitance per unit length.
The document discusses including spin-orbit coupling in the author's model of photoassociation and rovibrational relaxation in NaCs. It presents the theoretical description, including the Hamiltonian with additional terms for spin-orbit interaction. The system is described by a wavefunction in the Born-Oppenheimer approximation. Equations are derived for the probability amplitudes of relevant rovibrational states including spin-orbit coupling between the A1Σ+ and b3Π electronic states. The initial condition of the scattering system at ultracold temperature is specified.
Quantum Theory. Wave Particle Duality. Particle in a Box. Schrodinger wave equation. Quantum Numbers and Electron Orbitals. Principal Shells and Subshells. A Fourth Quantum Number. Effective nuclear charge
The document discusses the Schrodinger equation and methods to approximate solutions to it. It begins by defining the time-independent Schrodinger equation and its components. It then provides examples of writing out the Schrodinger equation for different chemical systems with varying numbers of electrons and nuclei. Approximation methods are needed because the Schrodinger equation can only be exactly solved for a few simple systems. Two approximation methods discussed are the variational method and perturbation theory. The variational method uses a trial wavefunction to variationally minimize the energy.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
This document provides solutions to 15 problems related to magnetic materials and magnetism. It begins by deriving expressions for how the angular frequency and radius of a classical electron orbit change with the application of a magnetic field using the Lorentz force. It then calculates magnetic moments and susceptibilities for various systems. Other problems cover topics like Lenz's law, diamagnetism, paramagnetism, Bohr orbits, susceptibility measurements, and predicting magnetic behaviors in different materials. Detailed step-by-step workings are shown for each problem.
I am Peterson N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, University of Melbourne, Australia. I have been helping students with their homework for the past 8 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
This presentation discusses a computational chemistry study of the interaction energies of molecules bonded through chalcogen bonds. The study examined 40 molecules total where the substituent X bonded to selenium, which was also bonded to CH3 or H, was varied. The substituents included H, F, CH3, CF3, etc. Computational methods included DFT calculations at the MP2/aug-cc-pVTZ level to determine binding energies and natural bond orbital analysis. Key concepts explained include basis sets, geometry optimization, basis set superposition error and counterpoise correction.
This document discusses light-matter interaction using a two-level atom model. It describes how an atom with only two energy levels can be modeled as a two-dimensional quantum mechanical system. The interaction of such a two-level atom with an electromagnetic field is then derived, leading to Rabi oscillations between the atomic energy levels driven by the field. Dissipative processes require a statistical description using the density operator formalism.
The document discusses first order perturbation theory. It begins by introducing perturbation theory as an approximate method to solve the Schrodinger equation for complex quantum systems where the Hamiltonian cannot be solved exactly. It then presents the key equations of first order perturbation theory. The first order correction to the energy is given by the expectation value of the perturbation operator over the unperturbed ground state wavefunction. The first order correction to the wavefunction is expressed as a linear combination of unperturbed eigenstates.
Relativistic formulation of Maxwell equations.dhrubanka
This document discusses the relativistic formulation of Maxwell's equations. It begins by introducing the key concepts of special relativity that are needed, including Lorentz transformations and four-vectors. It then shows how the electric and magnetic fields transform under Lorentz transformations and how they can be combined into the electromagnetic field tensor. The document also discusses how charge and current densities transform and satisfy the continuity equation as a four-vector. Finally, it presents Maxwell's equations in their compact relativistic form in terms of the field tensor and its derivatives.
This document provides an overview of modern atomic theory and quantum mechanics. It discusses the key discoveries and models that led to our current understanding of atomic structure, including Dalton's atomic theory, the discovery of subatomic particles, and the development of quantum mechanics with Schrodinger's equation. The four quantum numbers - principal, angular momentum, magnetic, and spin - are introduced to describe the allowed states of electrons in atoms. Rules for writing electron configurations are also covered.
Similar to lecture_11-chapter7HydrogenAtom-1new.pdf (20)
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
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
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.
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 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.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
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)”
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.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
ESR spectroscopy in liquid food and beverages.pptx
lecture_11-chapter7HydrogenAtom-1new.pdf
1. ◼ 7.1 Application of the Schrödinger Equation to the
Hydrogen Atom
◼ 7.2 Solution of the Schrödinger Equation for Hydrogen
◼ 7.3 Quantum Numbers
◼ 7.4 Magnetic Effects on Atomic Spectra – The Normal
Zeeman Effect
CHAPTER 7
The Hydrogen Atom
This spherical system has very high symmetry causing
very high degeneracy of the wavefunctions
2. ◼ 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
Laplace operator
4. ( ) ( )
2 2
2 2 2 2 2 2
1 2 3 0 1 2 3
2
2
E n n n E n n n
mL
= + + = + + where
2 2
0 2
2
E
mL
= . Then the second, third, fourth,
and fifth levels are
1. :
( )
2 2 2
2 0 0
2 1 1 6 (degenerate)
E E E
= + + =
( )
2 2 2
3 0 0
2 2 1 9 (degenerate)
E E E
= + + =
( )
2 2 2
4 0 0
3 1 1 11 (degenerate)
E E E
= + + =
( )
2 2 2
5 0 0
2 2 2 12 (not degenerate)
E E E
= + + =
Problem6.26
Find the energies of the second, third, fourth, and fifth levels for the three dimensional cubical box. Which
energy levels are degenerate?
A given state is degenerate when there is more
than one wave function for a given energy
5. 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.
◼ Use the Schrödinger wave equation for molecules
◼ We can remove the degeneracy by applying a magnetic field to the
atom or molecule
6. 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 .
The pendulum is a simple harmonic oscillator , Foucault pendulum(see miscellaneous on SIBOR )
8. 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.
9. 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.
10. ◼ 7.1 Application of the Schrödinger Equation to the
Hydrogen Atom
◼ 7.2 Solution of the Schrödinger Equation for Hydrogen
◼ 7.3 Quantum Numbers
◼ 7.4 Magnetic Effects on Atomic Spectra – The Normal
Zeeman Effect
CHAPTER 7
The Hydrogen Atom
11.
12. 7.1: Application of the Schrödinger
Equation to the Hydrogen Atom
◼ The approximation of the potential energy of the electron-proton
system is electrostatic:
◼ Rewrite the three-dimensional time-independent Schrödinger
Equation.
For Hydrogen-like atoms (He+ or Li++)
◼ Replace e2 with Ze2 (Z is the atomic number)
◼ Use appropriate reduced mass μ
13. Application of the Schrödinger Equation
◼ The potential (central force) V(r) depends on the distance r
between the proton and electron.
Transform to spherical polar
coordinates because of the
radial symmetry.
Insert the Coulomb potential
into the transformed
Schrödinger equation.
14. Application of the Schrödinger Equation
◼ The wave function ψ is a function of r, θ, .
Equation is separable.
Solution may be a product of three functions.
◼ We can separate Equation 7.3 into three separate differential
equations, each depending on one coordinate: r, θ, or .
Equation 7.3
Divide and conquer !!
15. 7.2: Solution of the Schrödinger Equation
for Hydrogen
◼ Substitute Eq (7.4) into Eq (7.3) and separate the resulting
equation into three equations: R(r), f(θ), and g( ).
Separation of Variables
◼ The derivatives from Eq (7.4)
◼ Substitute them into Eq (7.3)
◼ Multiply both sides of Eq (7.6) by r2 sin2 θ / Rfg
16. Solution of the Schrödinger Equation
◼ Only r and θ appear on the left side and only appears on the right
side of Eq (7.7)
◼ The left side of the equation cannot change as changes.
◼ The right side cannot change with either r or θ.
◼ Each side needs to be equal to a constant for the equation to be true.
Set the constant −mℓ
2 equal to the right side of Eq (7.7)
◼ It is convenient to choose a solution to be .
-------- azimuthal equation
17. Solution of the Schrödinger Equation
◼ satisfies Eq (7.8) for any value of mℓ.
◼ The solution be single valued in order to have a valid solution for
any , which is
◼ mℓ to be zero or an integer (positive or negative) for this to be
true.
◼ If Eq (7.8) were positive, the solution would not be realized.
◼ Set the left side of Eq (7.7) equal to −mℓ
2 and rearrange it.
◼ Everything depends on r on the left side and θ on the right side of
the equation.
18. Solution of the Schrödinger Equation
◼ Set each side of Eq (7.9) equal to constant ℓ(ℓ + 1).
◼ Schrödinger equation has been separated into three ordinary
second-order differential equations [Eq (7.8), (7.10), and (7.11)],
each containing only one variable.
----Radial equation
----Angular equation
19. Solution of the Radial Equation
◼ The radial equation is called the associated Laguerre equation
and the solutions R that satisfy the appropriate boundary
conditions are called associated Laguerre functions.
◼ Assume the ground state has ℓ = 0 and this requires mℓ = 0.
Eq (7.10) becomes
◼ The derivative of yields two terms.
Write those terms and insert Eq (7.1)
20. Solution of the Radial Equation
◼ Try a solution
A is a normalization constant.
a0 is a constant with the dimension of length.
Take derivatives of R and insert them into Eq (7.13).
◼ To satisfy Eq (7.14) for any r is for each of the two expressions in
parentheses to be zero.
Set the second parentheses equal to zero and solve for a0.
Set the first parentheses equal to zero and solve for E.
Both equal to the Bohr result
21. Quantum Numbers
◼ The appropriate boundary conditions to Eq (7.10) and (7.11)
leads to the following restrictions on the quantum numbers ℓ
and mℓ:
❑ ℓ = 0, 1, 2, 3, . . .
❑ mℓ = −ℓ, −ℓ + 1, . . . , −2, −1, 0, 1, 2, . ℓ . , ℓ − 1, ℓ
❑ |mℓ| ≤ ℓ and ℓ < 0.
◼ The predicted energy level is
22. 1. The wave function given is ( ) 0
/
100 , , r a
r Ae
−
= so *
is given by 0
2 /
* 2
100 100
r a
A e
−
= .
To normalize the wave function, compute the triple integral over all space
0
2
2 /
* 2 2
0 0 0
sin r a
dV A r e drd d
−
=
. The integral yields 2 , and the
integral yields 2. This leaves
( )
0
2 /
* 2 2 2 3 2
0
3
0
0
2
4 4
2 /
r a
dV A r e dr A a A
a
−
= = =
This integral must equal 1 due to normalization which leads to 3 2
0 1
a A
= so
3
0
1
A
a
= .
Problem7.8
The wave function for the ground state of hydrogen is given by
100(r,,) = A e-r/ao
Find the constant A that will normalize this wave function over all space.
23. Hydrogen Atom Radial Wave Functions
◼ First few radial wave functions Rnℓ
◼ Subscripts on R specify the values of n and ℓ
24. Solution of the Angular and Azimuthal
Equations
◼ The solutions for Eq (7.8) are
◼ Solutions to the angular and azimuthal equations are linked
because both have mℓ
◼ Group these solutions together into functions
---- spherical harmonics
26. Solution of the Angular and Azimuthal
Equations
◼ The radial wave function R and the spherical harmonics Y
determine the probability density for the various quantum
states. The total wave function depends on n, ℓ,
and mℓ. The wave function becomes
27. 7.3: Quantum Numbers
The three quantum numbers:
❑ n Principal quantum number
❑ ℓ Orbital angular momentum quantum number
❑ mℓ Magnetic quantum number
The boundary conditions:
❑ n = 1, 2, 3, 4, . . . Integer
❑ ℓ = 0, 1, 2, 3, . . . , n − 1 Integer
❑ mℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ Integer
The restrictions for quantum numbers:
❑ n > 0
❑ ℓ < n
❑ |mℓ| ≤ ℓ
28. 1) For what levels in the hydrogen atom will we not find
l=2 states??
a) n = 4, 5
b) n = 3, 4
c) n = 2, 1
d) n = 5, 6
Clicker - Questions
29. 2) Which of the following states of the hydrogen atom is
allowed?
a) n = 6, l = 2, ml = 0
b) n = 2, l = 2, ml = 0
c) n = 5, l = 2, ml = 3
d) n = 1, l = 2, ml = 1
Clicker - Questions
30. 1. It is required that 5
and m .
4: 0, 1, 2, 3, 4
m
= = ; 3: 0, 1, 2, 3
m
= = ;
2: 0, 1, 2
m
= = 1: 0, 1
m
= = 0: 0
m
= =
Problem7.11
List all quantum numbers (n,l,ml) for the n=5 level in atomic hydrogen.
31. Principal Quantum Number n
◼ It results from the solution of R(r) in Eq (7.4) because R(r) includes
the potential energy V(r).
The result for this quantized energy is
◼ The negative means the energy E indicates that the electron and
proton are bound together.
32. Orbital Angular Momentum Quantum
Number ℓ
◼ It is associated with the R(r) and f(θ) parts of the wave function.
◼ Classically, the orbital angular momentum with L =
mvorbitalr.
◼ ℓ is related to L by .
◼ In an ℓ = 0 state, .
It disagrees with Bohr’s semi-classical “planetary” model of
electrons orbiting a nucleus L = nħ.
33. Orbital Angular Momentum Quantum
Number ℓ
◼ A certain energy level is degenerate with respect to ℓ when the
energy is independent of ℓ.
◼ Use letter names for the various ℓ values
❑ ℓ = 0 1 2 3 4 5 . . .
❑ Letter = s p d f g h . . .
◼ Atomic states are referred to by their n and ℓ
◼ A state with n = 2 and ℓ = 1 is called a 2p state
◼ The boundary conditions require n > ℓ
35. ◼ The relationship of L, Lz, ℓ, and
mℓ for ℓ = 2.
◼ is fixed.
◼ Because Lz is quantized, only
certain orientations of are
possible and this is called space
quantization.
Magnetic Quantum Number mℓ
◼ The angle is a measure of the rotation about the z axis.
◼ The solution for specifies that mℓ is an integer and related to
the z component of L.
36. Magnetic Quantum Number mℓ
◼ Quantum mechanics allows to be quantized along only one
direction in space. Because of the relation L2 = Lx
2 + Ly
2 + Lz
2 the
knowledge of a second component would imply a knowledge of the
third component because we know .
◼ We expect the average of the angular momentum components
squared to be
Use a math table for the summation result
Since the sum
38. Honda 600RR
Who races this bike?
Why can anybody race it, if he just
dares to go fast?
The oval track of the Texas
World Speedway allows speeds
of 250 mph.
39.
40. ◼ The Dutch physicist Pieter Zeeman showed the spectral lines
emitted by atoms in a magnetic field split into multiple energy
levels. It is called the Zeeman effect.
Normal Zeeman effect:
◼ A spectral line is split into three lines.
◼ Consider the atom to behave like a small magnet.
◼ The current loop has a magnetic moment μ = IA and the period T =
2πr / v.
◼ Think of an electron as an orbiting circular current loop of I = dq / dt
around the nucleus.
◼ where L = mvr is the magnitude of the orbital
angular momentum
7.4: Magnetic Effects on Atomic Spectra—The
Normal Zeeman Effect
41. ◼ The angular momentum is aligned with the magnetic moment, and
the torque between and causes a precession of .
Where μB = eħ / 2m is called a Bohr magneton.
◼ cannot align exactly in the z direction and
has only certain allowed quantized orientations.
◼ Since there is no magnetic field to
align them, point in random
directions. The dipole has a
potential energy
The Normal Zeeman Effect
43. The Normal Zeeman Effect
◼ The potential energy is quantized due to the magnetic quantum
number mℓ.
◼ When a magnetic field is applied, the 2p level of atomic hydrogen
is split into three different energy states with energy difference of
ΔE = μBB Δmℓ.
mℓ Energy
1 E0 + μBB
0 E0
−1 E0 − μBB
50. ◼ An atomic beam of particles in the ℓ = 1 state pass through a
inhomogeneous magnetic field along the z direction.
◼
◼
◼ The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and the
mℓ = 0 state will be undeflected.
◼ If the space quantization were due to the magnetic quantum number
mℓ, mℓ states is always odd (2ℓ + 1) and should have produced an odd
number of lines.
The Normal Zeeman Effect
51.
52. 7.5: Intrinsic Spin
◼ Samuel Goudsmit and George Uhlenbeck in Holland proposed that
the electron must have an intrinsic angular momentum and
therefore a magnetic moment.
◼ Paul Ehrenfest showed that the surface of the spinning electron
should be moving faster than the speed of light!
◼ In order to explain experimental data, Goudsmit and Uhlenbeck
proposed that the electron must have an intrinsic spin quantum
number s = ½.
53. Intrinsic Spin
◼ The spinning electron reacts similarly to the orbiting electron in a
magnetic field.
◼ We should try to find L, Lz, ℓ, and mℓ.
◼ The magnetic spin quantum number ms has only two values,
ms = ±½.
The electron’s spin will be either “up” or
“down” and can never be spinning with its
magnetic moment μs exactly along the z axis.
The intrinsic spin angular momentum
vector .
54. Intrinsic Spin
◼ The magnetic moment is .
◼ The coefficient of is −2μB as with is a consequence of theory
of relativity.
◼ The gyromagnetic ratio (ℓ or s).
◼ gℓ = 1 and gs = 2, then
◼ The z component of .
◼ In ℓ = 0 state
◼ Apply mℓ and the potential energy becomes
no splitting due to .
there is space quantization due to the
intrinsic spin.
and
55. Space quantization of the electron
spin angular momentum
In the frame of the
electron there is
an internal
magnetic field
created by the
orbiting proton=
doubled splitting
Doublet splitting due to the
electron spin magnetic moment
56. 1. For the 4f state n = 4 and 3
= . The possible m values are 0, 1, 2,
and 3
with
1/ 2
s
m = for each possible m value. The degeneracy of the 4f state is then (with 2 spin
states per m ) equal to 2(7) = 14.
Problem7.29
Use all four quantum numbers (n,l.ml,ms) to write down all possible sets of
quantum numbers for the 4f state of atomic hydrogen. What is the total
degeneracy?
57. 1. For the 5d state n = 5 and 2
= . The possible m values are 0, 1,
and 2,
with
1/ 2
s
m = for each possible m value. The degeneracy of the 5d state is then (with 2 spin
states per m ) equal to 2(5) = 10.
Problem7.32
Use all four quantum numbers (n,l.ml,ms) to write down all possible sets of
quantum numbers for the 5d state of atomic hydrogen. What is the total
degeneracy?
58. 7.6: Energy Levels and Electron Probabilities
◼ For hydrogen, the energy level depends on the principle quantum
number n.
◼ In ground state an atom cannot emit
radiation. It can absorb
electromagnetic radiation, or gain
energy through inelastic
bombardment by particles.
Forbidden transitions: 3P-2P, 3d-2S,4F-3S,
etc
59. Selection Rules
◼ We can use the wave functions to calculate transition
probabilities for the electron to change from one state to another.
Allowed transitions:
◼ Electrons absorbing or emitting photons to change states when
Δℓ = ±1.
Forbidden transitions:
◼ Other transitions possible but occur with much smaller
probabilities when Δℓ ≠ ±1.
Conservation of angular momentum: photon carries one unit of angular momentum.
The atom changes by one unit of angular momentum in the radiation process
60. 3-D Probability Distribution Functions
◼ We must use wave functions to calculate the probability
distributions of the electrons.
◼ The “position” of the electron is spread over space and is not
well defined.
◼ We may use the radial wave function R(r) to calculate radial
probability distributions of the electron.
◼ The probability of finding the electron in a differential volume
element dτ is .
61. 3-D Probability Distribution Functions
◼ The differential volume element in spherical polar coordinates is
Therefore,
◼ We are only interested in the radial dependence.
◼ The radial probability density is P(r) = r2|R(r)|2 and it depends
only on n and l.
72. The SPHERES Tether Slosh investigation combines fluid dynamics equipment with
robotic capabilities aboard the station. In space, the fuels used by spacecraft can
slosh around in unpredictable ways making space maneuvers difficult. This
investigation uses two SPHERES robots tethered to a fluid-filled container covered in
sensors to test strategies for safely steering spacecraft such as dead satellites that
might still have fuel in the tank.
74. 1. If we determine the thermal energy that equals the energy required for the spin-flip
transition, we have ( )
6 5
3 3
5.9 10 eV 8.617 10 eV/K
2 2
kT T
− −
= = . This gives
0.0456 K
T = .
Problem 7.31
The 21-cm line transition of atomic hydrogen results from a spin-flip transition for
the electron in the parallel state of the n=1 state. What temperature in interstellar
space gives a hydrogen atom enough energy (5.9x10-6eV) to excite another
hydrogen atom in a collision?