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.
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.
This document presents a framework for determining the rotational-vibrational spectra and stability of diatomic molecules using quantum mechanics. It summarizes:
1) The framework models the diatomic molecule using a Hamiltonian that accounts for the kinetic energy of the nuclei and electrons, as well as the potential energies between nuclei, electrons, and nuclei-electron interactions.
2) It approximates the molecule as rigid to simplify the problem, treating nuclear motion as a perturbation. This allows separating the wavefunction into electronic and nuclear components.
3) It then solves for rotational states using a rigid rotor model that describes nuclear rotation, yielding energy levels dependent on the angular momentum quantum number. This provides insights into diatomic molecular stability and
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.
Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions that could help solve issues in the Standard Model such as the hierarchy problem. SUSY introduces new "quantum" dimensions beyond the usual 3 spatial and 1 time dimension. SUSY generators called Q transform fermions into bosons and vice versa. The SUSY algebra involves the generators Q satisfying anticommutation relations in addition to the usual commutation relations of generators like momentum P and angular momentum M. While experimental evidence for SUSY is still lacking, it is an attractive theoretical idea that may be discovered at energy scales below 1 TeV.
This document discusses atomic structure, beginning with the hydrogen atom and one-electron atoms. It then discusses the Hamiltonian and solutions of the Schrodinger equation for these systems. It introduces quantum numbers and describes the orbitals and energy levels. For polyelectronic atoms, it discusses separating the Schrodinger equation and introduces Hartree-Fock self-consistent field approximations. It describes Slater determinants which satisfy the Pauli exclusion principle for many-electron wavefunctions.
This document provides an overview of a molecular modeling course schedule and topics. The course will cover molecular properties, surfaces, electrostatics, electron microscopy, crystallography, NMR, molecular mechanics, sequence to structure relationships, visualization, molecular dynamics, ligand parameterization, and drug design. Key dates include a homework deadline of January 15th and a final exam on January 22nd. The instructor will discuss topics like classical forcefields, molecular dynamics simulations, solvation models, and hands-on exercises.
In this talk I will present real-time spectroscopy and different code to perform this kind of calculations.
This presentation can be download here:
http://www.attaccalite.com/wp-content/uploads/2022/03/RealTime_Lausanne_2022.odp
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.
This document presents a framework for determining the rotational-vibrational spectra and stability of diatomic molecules using quantum mechanics. It summarizes:
1) The framework models the diatomic molecule using a Hamiltonian that accounts for the kinetic energy of the nuclei and electrons, as well as the potential energies between nuclei, electrons, and nuclei-electron interactions.
2) It approximates the molecule as rigid to simplify the problem, treating nuclear motion as a perturbation. This allows separating the wavefunction into electronic and nuclear components.
3) It then solves for rotational states using a rigid rotor model that describes nuclear rotation, yielding energy levels dependent on the angular momentum quantum number. This provides insights into diatomic molecular stability and
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.
Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions that could help solve issues in the Standard Model such as the hierarchy problem. SUSY introduces new "quantum" dimensions beyond the usual 3 spatial and 1 time dimension. SUSY generators called Q transform fermions into bosons and vice versa. The SUSY algebra involves the generators Q satisfying anticommutation relations in addition to the usual commutation relations of generators like momentum P and angular momentum M. While experimental evidence for SUSY is still lacking, it is an attractive theoretical idea that may be discovered at energy scales below 1 TeV.
This document discusses atomic structure, beginning with the hydrogen atom and one-electron atoms. It then discusses the Hamiltonian and solutions of the Schrodinger equation for these systems. It introduces quantum numbers and describes the orbitals and energy levels. For polyelectronic atoms, it discusses separating the Schrodinger equation and introduces Hartree-Fock self-consistent field approximations. It describes Slater determinants which satisfy the Pauli exclusion principle for many-electron wavefunctions.
This document provides an overview of a molecular modeling course schedule and topics. The course will cover molecular properties, surfaces, electrostatics, electron microscopy, crystallography, NMR, molecular mechanics, sequence to structure relationships, visualization, molecular dynamics, ligand parameterization, and drug design. Key dates include a homework deadline of January 15th and a final exam on January 22nd. The instructor will discuss topics like classical forcefields, molecular dynamics simulations, solvation models, and hands-on exercises.
In this talk I will present real-time spectroscopy and different code to perform this kind of calculations.
This presentation can be download here:
http://www.attaccalite.com/wp-content/uploads/2022/03/RealTime_Lausanne_2022.odp
Electron-phonon coupling describes the interaction between electrons and phonons in materials. It can be calculated using density functional perturbation theory to obtain the electron-phonon matrix elements and phonon frequencies. This allows calculating temperature-dependent corrections to electronic band structures and optical properties within many-body perturbation theory. Yambo software implements these methods, calculating temperature renormalization of quasi-particle energies and broadening, as well as finite-temperature excitons and dielectric functions.
The document provides an introduction to computational quantum chemistry, including:
- Definitions of computational chemistry and computational quantum chemistry, which focuses on solving the Schrodinger equation for molecules.
- An overview of methods like ab initio quantum chemistry, density functional theory, and approximations like the Born-Oppenheimer approximation and basis set approximations.
- Descriptions of approaches like Hartree-Fock, configuration interaction, Møller-Plesset perturbation theory, and coupled cluster theory for including electron correlation effects.
In this talk I will discuss different approximations in DFT: pseduo-potentials, exchange correlation functions.
The presentation can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/03/dft_approximations.odp
Gaussian is capable of performing several quantum chemical calculations including molecular energies, geometry optimization, vibrational frequencies, NMR properties, potential energy surfaces, and reaction pathways. It takes a Gaussian input file specifying the calculation type, theory, basis set, coordinates, etc. Common calculation types include single point energy, geometry optimization, and vibrational frequency. The output file provides optimized geometry, frequencies, energies, and other molecular properties.
Probing spin dynamics from the Mott insulating to the superfluid regime in a ...Arijit Sharma
This document summarizes an experiment probing spin dynamics in a dipolar atomic Bose gas across the Mott insulating to superfluid transition in an optical lattice. Three key findings are:
1) In the Mott regime, spin dynamics shows complex oscillations that are well described by a model of intersite dipole-dipole interactions.
2) In the superfluid regime, spin dynamics is exponential and agrees with simulations including contact and dipolar interactions.
3) In the intermediate regime, oscillations survive with reduced amplitude, challenging theoretical descriptions accounting for dipolar interactions, contact interactions, and superexchange mechanisms.
This document is a physics problem set from MIT's 8.044 Statistical Physics I course in Spring 2004. It contains 5 problems related to statistical physics and probability distributions. Problem 1 considers the probability distribution and properties of the position of a particle undergoing simple harmonic motion. Problem 2 examines the probability distribution of the x-component of angular momentum for a quantum mechanical system. Problem 3 analyzes a mixed probability distribution describing the energy of an electron. Problem 4 involves finding and sketching the time-dependent probability distribution for the position of a particle given its wavefunction. Problem 5 concerns Bose-Einstein statistics and calculating properties of the distribution that describes the number of photons in a given mode.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
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.
This chapter discusses spin in strong interactions like pion-nucleon and nucleon-nucleon scattering. It introduces the density matrix and reaction matrix to describe mixed spin states. The density matrix is determined by the mean values of spin operators and completely characterizes the spin state. The reaction matrix relates the initial and final density matrices. This allows calculating observables in the final state given the initial state parameters and scattering matrix. Pauli matrices provide a complete spin operator basis for pion-nucleon reactions. The density matrix is expressed in terms of the target or beam polarization vector. Constraints from rotational, parity and time reversal symmetries on the nucleon-nucleon scattering matrix are also discussed.
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
Lecture 5: 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.
This document discusses the quantum harmonic oscillator model. It introduces harmonic oscillators, Hermite polynomials, and the Schrodinger equation as it relates to the harmonic oscillator potential. The solution of the Schrodinger equation for a harmonic oscillator yields the energy levels and vibrational wave functions, which are expressed in terms of Hermite polynomials. References for further reading on quantum chemistry and the quantum harmonic oscillator are also provided.
This document provides an overview of statistical mechanics. It defines microstates and macrostates, and explains that statistical mechanics studies systems with many microstates corresponding to a given macrostate. The Boltzmann distribution is derived, which gives the probability of finding a system in a particular microstate as being proportional to the exponential of the negative of the energy of that microstate divided by the temperature. Maxwell-Boltzmann statistics are described as applying to classical distinguishable particles, yielding the Maxwell-Boltzmann distribution. References for further reading are also included.
1. DFT+U is a method that adds Hubbard corrections to DFT to better account for localized electrons and electronic correlations in transition metal oxides that LDA/GGA cannot describe accurately.
2. It introduces an on-site Coulomb repulsion term U to the energy functional that favors electron localization and integer orbital occupations.
3. The U parameter can be computed using linear response theory by perturbing occupation matrices and evaluating screened response matrices in a supercell calculation.
Rigid rotators are two rotating atoms with a fixed bond length that can be used to model diatomic molecules. They allow calculation of rotational energy classically using moment of inertia and quantum mechanically using the Schrodinger equation. Rotational energy is proportional to the rotational quantum number J and the rotational constant B, following the equation Ej = BJ(J+1). Transitions between rotational energy levels obey the selection rule that the change in J is ±1. Bond lengths can be calculated from the moment of inertia using the relation I = μr^2, where μ is the reduced mass.
This document discusses using a master equation approach to simulate electron spin resonance (ESR) spectral lineshapes. It compares using 6-state and 48-state stochastic models to represent rotational diffusion, an important relaxation process in ESR. Simulated spectra from both models capture the main spectral features but the 48-state model provides more detail, especially at higher dispersion. The results help establish criteria for selecting appropriate models to faithfully reproduce ESR lineshapes over a wide range of transition rates.
This document discusses statistical mechanics and the distribution of energy among particles in a system. It provides 3 main types of statistical distributions based on the properties of identical particles: Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Maxwell-Boltzmann statistics applies to distinguishable particles, while Bose-Einstein and Fermi-Dirac apply to indistinguishable particles (bosons and fermions respectively), with the key difference being that fermions obey the Pauli exclusion principle. The document also discusses applications of these distributions, including the Maxwell-Boltzmann distribution law for molecular energies in an ideal gas.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
This document introduces key concepts in statistical mechanics, including the idea that macroscopic properties are thermal averages of microscopic properties. It discusses common statistical ensembles like the microcanonical ensemble (isolated systems with constant energy) and the canonical ensemble (systems in equilibrium with a heat reservoir). The canonical partition function Z relates microscopic quantum mechanics to macroscopic thermodynamics and can be used to calculate thermodynamic variables. Properties like heat capacity can be derived from fluctuations in energy calculated from the partition function.
Accelerated electric charges radiate electromagnetic radiation. The amount and properties of the radiation depend on the acceleration of the charge over time. For non-relativistic charges, the power radiated is proportional to the square of the acceleration. The spectrum of the radiation is proportional to the square of the Fourier transform of the charge's dipole moment. For relativistic charges, the power radiated has additional terms depending on the velocity and components of the acceleration parallel and perpendicular to the velocity. Relativistic aberration affects the observed direction of radiation emitted by a moving charge.
This dissertation by Stéphane Valladier examines photoassociation and rovibrational cooling of sodium cesium molecules using chirped laser pulses and stimulated Raman adiabatic passage (STIRAP). It was submitted in partial fulfillment of the requirements for a Doctor of Philosophy degree from the University of Oklahoma. The document acknowledges the advisors and committee members who oversaw the work and thanks friends and family for their support. It includes a dedication to family members and an abstract in both English and French.
This document provides an overview of near-dissociation expansion (NDE) theory for quantizing energy levels of diatomic molecules near dissociation. Section 1 introduces the topic and outlines subsequent sections. Section 2 discusses the historical motivation of using the Birge-Sponer method to determine dissociation energies from spectroscopic data before NDE theory. Section 3 outlines the assumptions of NDE theory, including the Wentzel-Kramers-Brillouin approximation and representing the long-range potential as a dispersion expansion involving inverse powers of the internuclear separation.
Electron-phonon coupling describes the interaction between electrons and phonons in materials. It can be calculated using density functional perturbation theory to obtain the electron-phonon matrix elements and phonon frequencies. This allows calculating temperature-dependent corrections to electronic band structures and optical properties within many-body perturbation theory. Yambo software implements these methods, calculating temperature renormalization of quasi-particle energies and broadening, as well as finite-temperature excitons and dielectric functions.
The document provides an introduction to computational quantum chemistry, including:
- Definitions of computational chemistry and computational quantum chemistry, which focuses on solving the Schrodinger equation for molecules.
- An overview of methods like ab initio quantum chemistry, density functional theory, and approximations like the Born-Oppenheimer approximation and basis set approximations.
- Descriptions of approaches like Hartree-Fock, configuration interaction, Møller-Plesset perturbation theory, and coupled cluster theory for including electron correlation effects.
In this talk I will discuss different approximations in DFT: pseduo-potentials, exchange correlation functions.
The presentation can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/03/dft_approximations.odp
Gaussian is capable of performing several quantum chemical calculations including molecular energies, geometry optimization, vibrational frequencies, NMR properties, potential energy surfaces, and reaction pathways. It takes a Gaussian input file specifying the calculation type, theory, basis set, coordinates, etc. Common calculation types include single point energy, geometry optimization, and vibrational frequency. The output file provides optimized geometry, frequencies, energies, and other molecular properties.
Probing spin dynamics from the Mott insulating to the superfluid regime in a ...Arijit Sharma
This document summarizes an experiment probing spin dynamics in a dipolar atomic Bose gas across the Mott insulating to superfluid transition in an optical lattice. Three key findings are:
1) In the Mott regime, spin dynamics shows complex oscillations that are well described by a model of intersite dipole-dipole interactions.
2) In the superfluid regime, spin dynamics is exponential and agrees with simulations including contact and dipolar interactions.
3) In the intermediate regime, oscillations survive with reduced amplitude, challenging theoretical descriptions accounting for dipolar interactions, contact interactions, and superexchange mechanisms.
This document is a physics problem set from MIT's 8.044 Statistical Physics I course in Spring 2004. It contains 5 problems related to statistical physics and probability distributions. Problem 1 considers the probability distribution and properties of the position of a particle undergoing simple harmonic motion. Problem 2 examines the probability distribution of the x-component of angular momentum for a quantum mechanical system. Problem 3 analyzes a mixed probability distribution describing the energy of an electron. Problem 4 involves finding and sketching the time-dependent probability distribution for the position of a particle given its wavefunction. Problem 5 concerns Bose-Einstein statistics and calculating properties of the distribution that describes the number of photons in a given mode.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
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.
This chapter discusses spin in strong interactions like pion-nucleon and nucleon-nucleon scattering. It introduces the density matrix and reaction matrix to describe mixed spin states. The density matrix is determined by the mean values of spin operators and completely characterizes the spin state. The reaction matrix relates the initial and final density matrices. This allows calculating observables in the final state given the initial state parameters and scattering matrix. Pauli matrices provide a complete spin operator basis for pion-nucleon reactions. The density matrix is expressed in terms of the target or beam polarization vector. Constraints from rotational, parity and time reversal symmetries on the nucleon-nucleon scattering matrix are also discussed.
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
Lecture 5: 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.
This document discusses the quantum harmonic oscillator model. It introduces harmonic oscillators, Hermite polynomials, and the Schrodinger equation as it relates to the harmonic oscillator potential. The solution of the Schrodinger equation for a harmonic oscillator yields the energy levels and vibrational wave functions, which are expressed in terms of Hermite polynomials. References for further reading on quantum chemistry and the quantum harmonic oscillator are also provided.
This document provides an overview of statistical mechanics. It defines microstates and macrostates, and explains that statistical mechanics studies systems with many microstates corresponding to a given macrostate. The Boltzmann distribution is derived, which gives the probability of finding a system in a particular microstate as being proportional to the exponential of the negative of the energy of that microstate divided by the temperature. Maxwell-Boltzmann statistics are described as applying to classical distinguishable particles, yielding the Maxwell-Boltzmann distribution. References for further reading are also included.
1. DFT+U is a method that adds Hubbard corrections to DFT to better account for localized electrons and electronic correlations in transition metal oxides that LDA/GGA cannot describe accurately.
2. It introduces an on-site Coulomb repulsion term U to the energy functional that favors electron localization and integer orbital occupations.
3. The U parameter can be computed using linear response theory by perturbing occupation matrices and evaluating screened response matrices in a supercell calculation.
Rigid rotators are two rotating atoms with a fixed bond length that can be used to model diatomic molecules. They allow calculation of rotational energy classically using moment of inertia and quantum mechanically using the Schrodinger equation. Rotational energy is proportional to the rotational quantum number J and the rotational constant B, following the equation Ej = BJ(J+1). Transitions between rotational energy levels obey the selection rule that the change in J is ±1. Bond lengths can be calculated from the moment of inertia using the relation I = μr^2, where μ is the reduced mass.
This document discusses using a master equation approach to simulate electron spin resonance (ESR) spectral lineshapes. It compares using 6-state and 48-state stochastic models to represent rotational diffusion, an important relaxation process in ESR. Simulated spectra from both models capture the main spectral features but the 48-state model provides more detail, especially at higher dispersion. The results help establish criteria for selecting appropriate models to faithfully reproduce ESR lineshapes over a wide range of transition rates.
This document discusses statistical mechanics and the distribution of energy among particles in a system. It provides 3 main types of statistical distributions based on the properties of identical particles: Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Maxwell-Boltzmann statistics applies to distinguishable particles, while Bose-Einstein and Fermi-Dirac apply to indistinguishable particles (bosons and fermions respectively), with the key difference being that fermions obey the Pauli exclusion principle. The document also discusses applications of these distributions, including the Maxwell-Boltzmann distribution law for molecular energies in an ideal gas.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
This document introduces key concepts in statistical mechanics, including the idea that macroscopic properties are thermal averages of microscopic properties. It discusses common statistical ensembles like the microcanonical ensemble (isolated systems with constant energy) and the canonical ensemble (systems in equilibrium with a heat reservoir). The canonical partition function Z relates microscopic quantum mechanics to macroscopic thermodynamics and can be used to calculate thermodynamic variables. Properties like heat capacity can be derived from fluctuations in energy calculated from the partition function.
Accelerated electric charges radiate electromagnetic radiation. The amount and properties of the radiation depend on the acceleration of the charge over time. For non-relativistic charges, the power radiated is proportional to the square of the acceleration. The spectrum of the radiation is proportional to the square of the Fourier transform of the charge's dipole moment. For relativistic charges, the power radiated has additional terms depending on the velocity and components of the acceleration parallel and perpendicular to the velocity. Relativistic aberration affects the observed direction of radiation emitted by a moving charge.
This dissertation by Stéphane Valladier examines photoassociation and rovibrational cooling of sodium cesium molecules using chirped laser pulses and stimulated Raman adiabatic passage (STIRAP). It was submitted in partial fulfillment of the requirements for a Doctor of Philosophy degree from the University of Oklahoma. The document acknowledges the advisors and committee members who oversaw the work and thanks friends and family for their support. It includes a dedication to family members and an abstract in both English and French.
This document provides an overview of near-dissociation expansion (NDE) theory for quantizing energy levels of diatomic molecules near dissociation. Section 1 introduces the topic and outlines subsequent sections. Section 2 discusses the historical motivation of using the Birge-Sponer method to determine dissociation energies from spectroscopic data before NDE theory. Section 3 outlines the assumptions of NDE theory, including the Wentzel-Kramers-Brillouin approximation and representing the long-range potential as a dispersion expansion involving inverse powers of the internuclear separation.
This document provides a strategic communication plan for Cougar Success, a website created by Washington State University to help students succeed. It includes a situation analysis, research plan, methodology, results, implications, and communication plan. The team conducted research through surveys, focus groups, and interviews to understand student needs and how to improve the website's communication and impact on student retention. Their recommendations include revamping the website, targeting communication methods, connecting students to the site through other WSU resources, tailoring content for all class years, focusing on academic resources, featuring student involvement, and regularly updating content. The overall goal is to increase website viewership by 40% to strengthen usability and minorly improve retention.
This document presents a method for solving the coupled-channels time-independent Schrödinger equation for bound states of the A1Σ+ − b3Π0 electronic states in NaCs, which are coupled by spin-orbit interaction. The method expands the coupled-channel eigenstates over a basis of rovibrational eigenstates of the uncoupled potentials. This leads to a system of equations for the expansion coefficients that can be solved by diagonalizing a 260x260 matrix. Plots of the bound-state matrix elements of the spin-orbit coupling operator show they decrease for more highly-excited vibrational states. Based on this, the method approximates the problem by neglecting couplings to continuum states
Мы открыыли первый свой коливинг. Это презентация - краткое описание что мы делаем и условия на которых можно псовместно проживать в нашем коливинге. ДОБРО ПОЖАЛОВАТЬ!!!
This dissertation examines photoassociation and rovibrational cooling of sodium cesium (NaCs) molecules using chirped laser pulses and stimulated Raman adiabatic passage (STIRAP). The author develops a theoretical model and equations to describe the system and interactions, including Coulomb, rotational, spin-orbit and light-matter interactions. Potential energy curves and electric dipole moments for NaCs between electronic states are provided. The goal is to transfer population between molecular states adiabatically using STIRAP to produce ultracold NaCs molecules.
O documento compara as teorias do desenvolvimento humano de Jean Piaget, Sigmund Freud e Erik Erikson. Apresenta as fases cognitivas de Piaget, as fases psicosssexuais de Freud e as fases psicossociais de Erikson, mostrando como cada um enxerga o desenvolvimento em diferentes faixas etárias, desde o nascimento até a vida adulta.
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.
Nonlinear Electromagnetic Response in Quark-Gluon PlasmaDaisuke Satow
The document discusses nonlinear electromagnetic response in quark-gluon plasma, specifically focusing on quadratic induced currents. It first outlines collision-dominant and collisionless cases. For the collision-dominant case, it lists possible forms of quadratic currents using CP symmetry properties and derives the Boltzmann equation in relaxation time approximation to calculate induced currents order-by-order in electromagnetic fields. The linear terms reproduce known results while quadratic terms are most sensitive to quark chemical potential at high temperature.
The Propagation and Power Deposition of Electron Cyclotron Waves in Non-Circu...IJERA Editor
The document summarizes a numerical study of the propagation and power deposition of electron cyclotron waves in non-circular HL-2A tokamak plasmas. The ray trajectories and power deposition were simulated by solving the plasma equilibrium equation, ray equations, and quasi-linear Fokker-Planck equation. The results show that shaping effects and temperature profiles have little influence on ECRH, while plasma density significantly affects propagation and power deposition. When ordinary mode EC waves are launched from the mid-plane and low-field side, ray trajectories bend as the parallel refractive index increases and can even recurve to the low-field side when the index reaches a certain value. Single absorption decreases with increasing both poloidal and toroidal
This document describes a method for simulating electromagnetic wave propagation in two-level dielectric media using time-domain transmission line modeling (TLM). The technique incorporates a semi-classical model of a two-level medium to simulate its quantum properties within the TLM method. The approach is validated by showing it corresponds to the classical Lorentz oscillator model for small signals and excitations. Results demonstrate absorption, amplification, self-induced transparency and lasing in the two-level medium.
1) The document discusses dynamics modeling for robotic manipulators using the Denavit-Hartenberg representation and Lagrangian mechanics. It describes using the Euler-Lagrange method to derive equations of motion for robotic links by computing kinetic and potential energy terms.
2) As an example, dynamics equations are derived for a simple 1 degree-of-freedom robotic arm. Kinetic and potential energy expressions are written and the Lagrangian is computed to obtain the equation of motion.
3) State-space modeling basics are reviewed using the example of a damped spring-mass system, showing how to write the system dynamics as state-space matrices to evaluate responses like step response.
This document discusses a computational study of MAX phases using density functional theory. MAX phases are a group of materials that exhibit both metallic and ceramic properties. The study uses the WIEN2k software to calculate the electronic structure and properties of MAX phases like Cr2AlC and Cr2GaC from their density of states and band structure plots. Manganese is incorporated into the structures at varying concentrations to study their magnetic properties.
This document presents a theoretical treatment of charge exchange processes that can occur during the scattering of positively charged lithium ions (Li+) from the surface of a narrow band insulator (KF) in the presence of a laser field. Equations are derived to describe the dynamics and describe how the laser field can be incorporated into the system Hamiltonian. The treatment is then applied to model charge exchange during the scattering of Li+ from KF surfaces. The key conclusions are that the charge state of the scattering species can be controlled by adjusting parameters of the applied laser field, such as frequency and intensity.
This document contains instructions and problems related to a physics exam on relativistic particles and superconducting magnets. It includes 4 problems:
1) Describing the motion of a relativistic particle subject to an attractive central force, including graphs of position vs time and momentum vs position.
2) Modeling a meson as two quarks with a central attractive force, and graphing their motion.
3) Transforming the motion graphs from problem 2 into a different reference frame moving at 0.6c.
4) Calculating the energy of a meson moving at 0.6c as observed in the lab frame.
The document provides answer sheets for the problems and specifies the
Phonon frequency spectrum through lattice dynamics and normal coordinate anal...Alexander Decker
The document discusses the lattice dynamics and normal coordinate analysis of the high-temperature superconductor Tl2Ca3Ba2Cu4O12. It presents the following key points:
1. Lattice dynamics calculations using the three-body force shell model reproduce observed Raman and infrared phonon frequencies reasonably well.
2. Normal coordinate analysis using Wilson's F-G matrix method yields vibrational frequencies in good agreement with experimental values and lattice dynamics calculations.
3. Potential energy distribution calculations confirm that the chosen vibrational frequencies make the maximum contribution to the potential energy of the material's normal coordinate frequencies.
Using the two forms of Fish-Bone potential (I and II), a self-consistent calculations are carried out to perform the analysis of binding energies, root mean square radii and form factors using different configuration symmetries of 20Ne nucleus. A computer simulation search program has been introduced to solve this problem. The Hilbert space was restricted to three and four dimensional variational function space spanned by single spherical harmonic oscillator orbits. A comparison using Td and D3h configuration symmetries are carried out.
Alexei Starobinsky - Inflation: the present statusSEENET-MTP
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Study on the dependency of steady state response on the ratio of larmor and r...eSAT Journals
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Study on the dependency of steady state response on the ratio of larmor and r...
SV-InclusionSOcouplinginNaCs
1. Including spin-orbit coupling of the first excited electronic states in the
photoassociation and rovibrational relaxation of NaCs
v. 2.6
Goal To find equations for the probability amplitudes of the relevant rovibrational states in my process
(see PhD proposal) when including the spin-orbit coupling interaction between the A1
Σ+ state and the
b3
Π state of NaCs.
1 Introduction
In my PhD proposal [1, §4.2], I mentioned the necessity to include the effect of spin-orbit (SO) coupling
between the A1
Σ+ and the b3
Π electronic states of NaCs into my calculation. Zaharova et al. [2] deter-
mined experimentally the strength of the SO interaction for the A1
Σ+ − b3
Π manifold of NaCs. After
learning about the fundamentals of SO coupling in diatomic molecules1, I now have the necessary under-
standing to include the SO interaction in the model Hamiltonian of my problem.
Section 2 presents the major steps involved in the derivation. Section 3 discusses how the inclusion of
the SO coupling affects the various operators appearing in the total Hamiltonian, and proposes ways to
deal with the consequences. Section 4 outlines the next steps to take toward a solution.
2 Theoretical Description
2.1 The problem
Given a gaseous mixture of Na and Cs atoms at ultracold temperature (T = 200µK), I study the transfer of
population from a scattering wave packet |χX0 above the asymptote of the X1Σ+electronic state (JX = 0)
of NaCs, to a low-lying rovibrational state |X,vX,JX of the X1Σ+ state (JX = 0 or 2), using high-lying
rovibrational state(s) |A ∼ b in the A1
Σ+—b3
Π manifold as intermediate state(s).
The key point of the present notes is to find a way to specify the actual physical content of the state(s)
|A ∼ b : which high-lying states are useful, whether they are rather of singlet or triplet character, or given
the energy these states have, whether the singlet/triplet distinction is meaningful.
One gaussian laser pulse linearly-polarized (the pump pulse) binds the scattering atoms into a di-
atomic molecule in a superposition of rovibrational states of the A1
Σ+ −b3
Π manifold. A second gaussian
laser pulse linearly-polarized (the Stokes pulse) allows stimulated emission of a photon from the molecule.
The Stokes pulse is designed so that the molecule ends up in a rovibrational state of the X1Σ+ state with
a rovibrational energy as low as possible.
1The references I used are [3, §9.3], [4, §3.1 & 3.4], and [5].
1
2. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
2.2 Approximations
The intensities of the two laser pulses are sufficiently low (≤ 300kW.cm−2
) to treat the laser fields semi-
classically. Also, the wavelengths of the lasers range from ≈850nm to ≈900nm, and the spatial extension
of the molecule remains below 3nm, justifying the application of the Long Wave Approximation (LWA):
the spatial dependence of the laser electromagnetic field can be neglected. The lasers interact with the
molecules and the gas of atoms through the electric dipole interaction (EDA): other electric moments of
the molecule are also neglected.
When using wave functions in this problem, I will use the Born-Oppenheimer approximation: the
dynamics of the nuclei are decoupled from the dynamics of the electrons. This decoupling has two
consequences2[4, p. 90]:
1. For a given set α of quantum numbers describing the electrons, the wave function of the molecule
can be written as a product of two wave functions, one for the nuclei and one for the electrons
ψBO
α,v = χv(R,θ,ϕ)Φel
α (#r ;R),
where #r denotes the set of coordinates of all electrons, and the semi-colon indicates that R is a
parameter,
2. The internuclear separation R being a parameter for the electronic wave function, the radial part
Tn(R) of the nuclear kinetic energy operator Tn does not act upon the electronic wave function
Φel
α Tn(R) Φel
α = Φel
α |Φel
α Tn(R) = δαα Tn(R) .
2.3 The model
2.3.1 Hamiltonian of the system
The system consists of Na and Cs atoms governed by the Hamiltonian
H (t) = Tn + Vnn + Vne + Vee + Te
He
+HSO −
#
d ·
#
E (t), (1)
where
Tn – kinetic energy operator for the nuclei,
Vnn – nucleus-nucleus Coulomb interaction,
Vne – nucleus-electron Coulomb interaction,
Vee – electron-electron Coulomb interaction,
2See my notes on the Born-Oppenheimer approximation, Hund’s cases, and adiabaticity
SV-InclusionSOcouplinginNaCs.tex 2
3. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
Te – kinetic energy operator for the electrons,
HSO ≡ i ˆai
#
i · #si – spin-orbit interaction (see [4, Eq. (3.4.3) p. 182]. Also the sum runs only over
open shells electrons, [4, last sentence p. 183])
#
d – electric dipole operator,
#
E (t) ≡
#
EP (t) +
#
ES(t) – electric field operator (P: pump pulse, S: Stokes pulse).
The Hamiltonian H (t) governs 2 nuclei and a total of 11 + 55 = 66 electrons and is written in the center
of mass frame. The kinetic energy operator of the nuclei is defined as:
Tn(
#
R) ≡ −
2
2µ
1
R2
∂
∂R
R2 ∂
∂R
+
R2
2µR2
(2)
with the rotational energy operator defined as:
R2
≡ − 2 1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin2
θ
∂2
∂ϕ2
. (3)
Figure 1 defines the angles θ and ϕ in the center of mass frame.
ˆX
ˆY
ˆZ ˆz
Cs
Na
θ
ϕ
Figure 1: Definition of angles θ and ϕ in the center of mass frame. The ˆz axis is the
molecular axis. The cesium atom being heavier than the sodium atom, the center of mass
of the diatomic molecule is closer to Cs than to Na.
2.3.2 Descriptor of the system
There are two ways to describe the system: either using a wave function or a density operator. The
treatment via the density operator is the best way to treat the initial condition (i.e. a gaseous mixture in
thermal equilibrium at ultracold temperature T = 200µK), but requires to solve the quantum Liouville-
von Neumann equation. If the density operator is expressed in a basis of the relevant Hilbert space
SV-InclusionSOcouplinginNaCs.tex 3
4. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
of dimension N, then solving the quantum Liouville-von Neumann equation means solving N2 coupled
partial differential equations. When using a wave function formalism, solving the problem means solving
only N coupled partial differential equations.
The density operator also allows for the appropriate treatment of spontaneous emission, which I am
not considering in my problem.
Furthermore, there exists a way to express the initial condition for the system in the density operator
formalism using an expansion over wave packets [6, p. 013412-3]. To facilitate my understanding of the
underlying physics in my problem, I think it is best to examine the dynamics of the process for a given
wave packet, and then see how the initial spatial width, central position, and energy of said wave packet
affect the dynamics. Doing so allows to describe the system simply by a wave function, and thus to solve
only N coupled partial differential equations.
Therefore I describe the system with a wave function:
|Ψ (t) =
α
∞
J=0
J
MJ =−J
1
R
Γ α
JMJ Ω(R,t)|JMJΩ |Φel
α , (4)
where
• R ≡ internuclear separation,
• α ≡ set of quantum numbers labeling a particular electronic state,
• |Φel
α ≡ electronic wave function for electronic state α, satisfying the Born-Oppenheimer approxi-
mation (§2.2),
• The rotational energy operator R2 acts on the kets |JMJΩ as
R2
|J,MJ,Ω = (J 2
− J 2
z + L 2
− L 2
z + S 2
− S 2
z )|JMJΩ . (5)
The rotational perturbations discussed in Sec. 3.1.2.3 p. 96 and Sec. 3.2.1.1 p. 107-108 of [4] are
neglected in this calculation because the lasers involved in the problem are far off-resonance from
any transition that the rotational perturbations would allow.
• Γ α
JMJ Ω(R,t) is a superposition of rovibrational and continuum states of the electronic state α with
rotational quantum numbers J, MJ, Ω:
Γ α
JMJ Ω(R,t) =
v
a
αJ
v (t)|αvJ +
+∞
E∞
α
a
αJ
E (t)|χ
αJ
E dE (6)
where the |αvJ s are the rovibrational states in electronic state α with vibrational quantum number
v and rotational quantum number J, E∞
α the asymptotic value of the potential energy for electronic
state α (here, E∞
X = 0), and |χ
αJ
E the energy-normalized stationary scattering state with energy E
above the asymptote of the electronic state α with rotational quantum number J.
SV-InclusionSOcouplinginNaCs.tex 4
5. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
The goal of these notes is to find a justification for which states should be involved in the discrete sum in
Eq. (6).
Plugging the wave function (4) into the Time-Dependent Schr¨odinger Equation
i
∂
∂t
|Ψ (t) = H (t)|Ψ (t) , (7)
and using the fact that the lasers are linearly polarized along the ˆZ-axis (see Fig. 1) yields equations for
the Γ ’s:
∀α, J, MJ, Ω,
i
∂
∂t
Γ α
JMJ Ω(R,t) = −
2
2µ
∂2
∂R2
−
1
2R2
Φel
α | JMJΩ R2
JMJΩ |Φel
α + V BO
α Γ α
JMJ Ω(R,t)
+
α
Φel
α | JMJΩ H SO
J MJ Ω |Φel
α Γ α
J MJ Ω (R,t)
+ (−1)MJ +1
2J + 1E (t)
α
dαα (R)
2J + 3
J 1 J + 1
−MJ 0 MJ
J 1 J + 1
0 0 0
Γ α
J+1MJ Ω(R,t)
+ 2J − 1
J 1 J − 1
−MJ 0 MJ
J 1 J − 1
0 0 0
only 0 if J 0 and MJ ±J
Γ α
J−1MJ Ω(R,t)
(8)
where the 2 × 3 matrices are Wigner 3-j symbols.
2.3.3 Initial condition
At t = 0, the system is simply a pair of atoms scattering above the asymptote of the X1Σ+electronic state,
for which Ω = 0. Thus ∀α X1Σ+, ∀{J,MJ,Ω}, Γ α
JMJ Ω(R,t = 0) = 0. The gaseous mixture of atoms is in
thermal equilibrium at T = 200µK. I calculated3 that at this temperature only the JX = 0 rotational state
is occupied, and therefore MJX
= 0. Consequently
Γ X1Σ+
000 (R,t = 0) = 1, and ∀J 0, Γ X1Σ+
JMJ 0 (R,t = 0) = 0. (9)
I will translate the initial condition Eq. (9) into conditions for the expansion coefficients—probability
amplitudes— a’s that appear in Eq. (6) later in these notes.
3See Mathematica notebook Rotational level probability of occupation.nb.
SV-InclusionSOcouplinginNaCs.tex 5
6. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
2.3.4 Method of solution
To solve the problem, notice that the Hamiltonian H (t) can be split into a time-independent term H0 and
a time-dependent term Vext(t). The idea here is to first find the eigenbasis of H0, which solves for the
spatial dependence of the descriptor |Ψ (t) , and then expanding the Γ ’s over the eigenbasis of H0 will
yield equations for the time-dependent expansion coefficients a’s of Eq. (6).
2.4 Rules for the calculation
2.4.1 Choice of basis
To start the derivation, I need to choose a basis for the electronic states |Φel
α . The research published
in Zaharova et al. [2], where Hund’s case (a) potentials and spin-orbit coupling functions are reported,
suggests to begin deriving equations in the Hund’s case (a) basis.
Since we initially decided to model the population transfer using the A1
Σ+ state as an intermediate
state, and given the results of [2], the electronic states involved in the problem are the X1Σ+ state, the
A1
Σ+ state, and the b3
Π state.
2.4.2 Selection rules and allowed transitions
Electric dipole Electric dipole transitions between a singlet and a triplet electronic state are forbid-
den, so dXb = dAb = 0. Moreover, although rotational transitions within the same electronic states are
rigorously allowed, the lasers used in this project are far off resonance from any rotational transition
within the same electronic state. Therefore, for the purpose of the derivation I can safely assume that
dXX = dAA = dbb = 0.
Because the lasers used in this problem are linearly polarized and given Eq. (9), the pump pulse can
only populate rovibrational levels of the A1
Σ+ − b3
Π manifold with J = 1. The Stokes pulse can then
populate only the rotational states of the X1Σ+ state with JX = 0 or 2. Moreover, because the lasers are
linearly polarized, the selection rule for MJ is ∆MJ = 0. Since the system starts with MJ = 0, all states
involved in the problem will have MJ = 0. Therefore, I will no longer specify MJ anymore, and remember
that it remains equal to 0 throughout the whole problem.
Spin-Orbit The spin-orbit operator HSO couples only electronic states that dissociate to the same asymp-
tote, hence X1Σ+ H SO A1
Σ+ = X1Σ+ H SO b3
Π = 0. Moreover 1Σ+ states are not affected by diag-
onal spin-orbit coupling: X1Σ+ H SO X1Σ+ = A1
Σ+ H SO A1
Σ+ = 0. The only remaining non-zero
terms of the spin-orbit operator are:
A1
Σ+ H SO
b3
Π = −
√
2ξSO
Ab (R) = −
√
2ξ(R) and b3
Π H SO
b3
Π = −ηSO
b0b1
(R) = −η(R),
where the notation from [2] has been modified for clarity.
SV-InclusionSOcouplinginNaCs.tex 6
7. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
Rotational operator R2 matrix elements Following the recommendations of [4, pp. 96, 107-108], I
merge the electronic orbital angular momentum matrix element X1Σ+| 000 L 2 000 |X1Σ+ with the
potential V BO
X (R). Note that using the asymptotic electronic wave functions for the X1Σ+ state given in
Eq. (12) of Katˆo [5] yields
X1Σ+| 000 L 2 000 |X1Σ+ = 0. However, this is only an estimate of the matrix element at large in-
ternuclear separation: in diatomic molecules the electronic orbital angular momentum never commutes
with the Hamiltonian, and thus the quantum number L is not a constant of the motion. Zaharova et al.
[2, p. 012508-6] applied the van Vleck pure precession hypothesis4 to estimate the matrix elements5
A1
Σ+| 100 L 2 100 |A1
Σ+ = b3
Π| 100 L 2 100 |b3
Π = 2.
Since I neglect the rotational perturbations when defining R2 in Eq. (5), all operators in the Hamil-
tonian H (t) defined in Eq. (1) have the same selection rule for Ω, ∆Ω = 0. The system starts as a pair of
scattering atoms above the asymptote of the X1Σ+ state. Therefore Ω, like MJ, starts as 0, and keeps the
same value throughout the whole process. Thus in what follows, I will no longer specify the quantum
numbers MJ and Ω, and remember that they are always equal to zero.
3 An apparently smaller system of equations
Using all the information from Sec. 2.4 and plugging it into Eq. (8) yields the system of equations:
i
∂
∂t
Γ X
0
Γ X
2
Γ A
1
Γ b
1
=
−
2
2µ
∂2
∂R2 + V BO
X (R) 0 −dXA(R)
√
3
3 E (t) 0
0 −
2
2µ
∂2
∂R2 − 6
R2 + V BO
X (R) −dXA(R)2
√
15
15 E (t) 0
−dAX(R)
√
3
3 E (t) −dAX(R)2
√
15
15 E (t) −
2
2µ
∂2
∂R2 − 4
R2 + V BO
A (R) −
√
2ξ(R)
0 0 −
√
2ξ(R) −
2
2µ
∂2
∂R2 − 4
R2 + V BO
b (R) − η(R)
Γ X
0
Γ X
2
Γ A
1
Γ b
1
(10)
The above system of equations looks like a 4×4 system of coupled partial differential equations, which
is already not a trivial thing to solve. If I was to use the method outlined in Sec. 2.3.4 without more
input, I would have to remember that the X1Σ+ state with JX = 0 supports 87 rovibrational states, the
JX = 2 supports 86 rovibrational states, the A1
Σ+ state (JA = 1) supports 147 rovibrational states, and
the b3
Π state (JA = 1) supports 106 rovibrational states. Therefore, I would be faced with a system of
coupled, no-longer-partial, differential equations (the only remaining variable being time, t) of dimension
426×426.
Analysing transition dipole moment matrix elements (TDMME) allows to reduce drastically the num-
ber of rovibrational states to involve in the problem. Because of the spin-orbit coupling between the
4See my report 20121101-20121108-SVweeklyReport.pdf.
5 The analytic potentials reported in [2] do not contain the L 2 term, which is explicitly separated (see [2, Eqs. (4) (5)
p. 012508-6]). The OU12 potentials that were constructed for the A1Σ+ and b3Π states from the experimental results of [2], are
also devoid of the L 2 term, which is accounted for explicitly in the equations below.
SV-InclusionSOcouplinginNaCs.tex 7
8. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
A1
Σ+ state and the b3
Π state, it is no longer valid to decide on which rovibrational states to include
based on examination of the TDMME between the X1Σ+ state and the A1
Σ+ state. What follows exposes
the necessary preliminary steps that lead to the relevant quantities to analyse in order to pick the proper
rovibrational state(s) in the A1
Σ+ − b3
Π manifold.
3.1 The hybrid basis
I can split the 4×4 matrix in Eq. (10) in 4 terms: the nuclear kinetic energy T, the rotational energy R, the
electric dipole-electric field interaction D, and the electronic and spin-orbit term Hel. In the Hund’s case
(a) basis A defined by the 4 kets {|X1Σ+,J = 0 ,|X1Σ+,2 , |A1
Σ+,1 , |b3
Π0,1 }, these matrices are:
TA = −
2
2µ
∂2
∂R2 0 0 0
0 ∂2
∂R2 0 0
0 0 ∂2
∂R2 0
0 0 0 ∂2
∂R2
A
RA = −
2
2µ
0 0 0 0
0 − 6
R2 0 0
0 0 − 4
R2 0
0 0 0 − 4
R2
A
(11a)
DA = −E (t)
0 0 dXA(R)
√
3
3 0
0 0 dXA(R)2
√
15
15 0
dAX(R)
√
3
3 dAX(R)2
√
15
15 0 0
0 0 0 0
A
Hel
A =
V BO
X (R) 0 0 0
0 V BO
X (R) 0 0
0 0 V BO
A (R) −
√
2ξ(R)
0 0 −
√
2ξ(R) V BO
b (R) − η(R)
A
(11b)
Diagonalizing Hel provides a new hybrid6 basis H. Expressing the eigenvectors of Hel in the basis A gives
the passage matrix U from basis H to basis A.
The eigenvalues of Hel are
V BO
X (R) (doubly degenerate) (12a)
V1/2(R) =
1
2
VA + Vb0 − (VA − Vb0)2 + 8ξ2 (12b)
V3/2(R) =
1
2
VA + Vb0 + (VA − Vb0)2 + 8ξ2 (12c)
where all quantities are R-dependent, V BO
A = VA, and Vb0(R) = V BO
b (R) − η(R) to simplify the notation.
Looking at Fig. 2 explains the choice of labels for the eigenvalues: V1/2(R) dissociates to the Na(32S1/2)+Cs(62P1/2)
asymptote while V3/2(R) dissociates to Na(32S1/2)+Cs(62P3/2), which is consistent with the spin-orbit cou-
pling function accounting for fine structure. Note that asymptotically, the PECs V1/2(R) and V3/2(R)
should merge with the corresponding Hund’s case (c) PECs, respectively (2)0+ and (3)0+.
Expressing the eigenvectors of Hel in basis A gives the passage matrix U from H to A. The diagonal-
6This basis is hybrid because it does not correspond to any pure Hund’s case (a), neither is it diabatic or adiabatic since Tn is
almost diagonal in H for some ranges of R and definitely non-diagonal in other ranges.
SV-InclusionSOcouplinginNaCs.tex 8
9. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
ization of Hel yields:
U =
|X1
Σ+
,0 |X1
Σ+
,2 |V1/2,1 |V3/2, 1
X1
Σ+
,0| 1 0 0 0
X1
Σ+
,2| 0 1 0 0
A1
Σ+
,1| 0 0 cosγ −sinγ
b3
Π0,1| 0 0 sinγ cosγ
(13)
where
cosγ =
√
2ξ(R)
(2ξ2(R) + (VA − V1/2)2)1/2
sinγ =
(VA − V1/2)
(2ξ2(R) + (VA − V1/2)2)1/2
(14)
A similar situation and set of definitions can be found in Londo˜no et al. [7].
NaCs
b3
0
A1
V3 2
V1 2
4 6 8 10 12 14 16
10 000
11 000
12 000
13 000
14 000
15 000
16 000
17 000
Internuclear Separation
Energycm
1
Figure 2: NaCs Hund’s case (a) potential energy curves (PECs) for the b3
Π and A1
Σ+
state, coupled by spin-orbit interactions to yield hybrid PECs V1/2 and V3/2. Note the
double-well of the V1/2 curve with a local maximum around 4.25 ˚A, and the smooth
step of the V3/2 adiabatic curve for internuclear separations around 9.27 ˚A. The PECs
are drawn using OU12 potentials.
Looking at Fig. 2 combined with Eq. (14), I notice that when V1/2 = VA, then cosγ = 1 and sinγ = 0 i.e.
|V1/2, 1 = |A1Σ+, 1 : the |V1/2, 1 state has singlet character, and conversely, |V3/2, 1 = |b3Π0, 1 , i.e. the
|V3/2, 1 state has triplet character.
SV-InclusionSOcouplinginNaCs.tex 9
10. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
3.2 Rotational matrix in the hybrid basis
The particular shape of the matrices U and RA renders the transformation of RA into the hybrid basis H
rather trivial:
RH = U−1
RAU = U†
RAU = −
2
2µ
0 0 0 0
0 − 6
R2 0 0
0 0 − 4
R2 0
0 0 0 − 4
R2
H
= RA. (15)
3.3 The dipole-field interaction in the hybrid basis
The point of theses notes is to obtain the relevant TDMME to determine which rovibrational states should
be used in my process. The electric transition dipole moment function between the X1Σ+ state and the
A1
Σ+ state published by Aymar and Dulieu [8] is a real function, and so I can simplify the notation by
defining d(R) ≡ dAX(R) = dXA(R). The transformation of the dipole-field interaction matrix from basis A
to basis H gives
DH = U−1
DAU = U†
DAU =
|X1
Σ+
,0 |X1
Σ+
,2 |V1/2,1 |V3/2, 1
X1
Σ+
,0| 0 0 −
√
3
3 cosγ
√
3
3 sinγ
X1
Σ+
,2| 0 0 −
√
3
3 cosγ
√
3
3 sinγ
V1/2,1| −
√
3
3 cosγ −2
√
15
15 cosγ 0 0
V3/2,1|
√
3
3 sinγ 2
√
15
15 sinγ 0 0
H
d(R)E (t). (16)
Thus the relevant transition dipole moment functions to consider when accounting for the spin-orbit
interaction are
dX1Σ+↔V1/2
(R) = d(R)cosγ and dX1Σ+↔V3/2
(R) = d(R)sinγ. (17)
(The Clebsch-Gordan coefficients are left out of the definitions, as they are not necessary for a qualitative
discussion.) Technically, the TDMME I need to examine are the matrix elements vX d(R)cosγ v1/2
and vX d(R)sinγ v3/2 , where |v1/2 and |v3/2 are bound states of the V1/2(R) and the V3/2(R) adiabatic
PECs respectively. The next section explains how to obtain the |v1/2 ’s and |v3/2 ’s.
SV-InclusionSOcouplinginNaCs.tex 10
11. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
3.4 The kinetic energy operator expressed in the adiabatic basis
Because the transformation U depends on the internuclear separation R, the kinetic energy matrix T is no
longer diagonal in the adiabatic basis H (as expected, see [4, p. 94]):
TH = U−1
TAU = U†
TAU = −
2
2µ
∂2
∂R2 0 0 0
0 ∂2
∂R2 0 0
0 0 −
dγ
dR
2
+ ∂2
∂R2 −
d2
γ
dR2 − 2
dγ
dR
∂
∂R
0 0
d2
γ
dR2 + 2
dγ
dR
∂
∂R
−
dγ
dR
2
+ ∂2
∂R2
H
. (18)
Let’s examine the functions
dγ
dR ,
dγ
dR
2
, and
d2
γ
dR2 , plotted in figures 3, 4, and 5 respectively.
All three figures show that except in two rather narrow regions, the derivatives of γ with respect to R
are essentially 0. In the regions where the derivatives are significantly different than zero, the high-lying
rovibrational wave functions these derivatives act upon are likely to have small amplitudes and oscillate
a lot [7, Fig. 1]. Therefore, I expect diagonal and off-diagonal matrix elements of the derivatives of γ
with respect to R between the rovibrational wave functions of V1/2(R) and V3/2(R) to be small. Thus I
am inclined to solve for the rovibrational bound states of V1/2(R) and V3/2(R) by considering the various
derivatives of γ with respect to R as perturbations to the problem. Once I obtain such rovibrational
bound states, I will effectively calculate the matrix elements of the perturbation and quantify whether
the perturbative treatment is justified in the first place.
4 What’s next?
There are n things that need to be done7 in light of the current notes:
1. Compare the mixing angle matrix elements
vi
dγ
dR
2
vi , i = 1/2, 3/2, (19a)
v1/2
d2
γ
dR2
v3/2 , (19b)
v1/2
dγ
dR
∂
∂R
v3/2 , and v3/2
dγ
dR
∂
∂R
v1/2 , (19c)
7As of 15 May 2013
SV-InclusionSOcouplinginNaCs.tex 11
12. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
4 6 8 10 12 14
4
3
2
1
0
Internuclear Separation R
dΓ
dR
1
Figure 3: First derivative of the mixing angle γ(R) with respect to the internuclear separation.
The extrema occur at R ≈ 4.25 ˚A and R ≈ 9.27 ˚A with respective values (dγ/dR)max ≈ −4.59 ˚A
−1
and
(dγ/dR)min ≈ 0.35 ˚A
−1
to the radial kinetic energy matrix elements
v1/2
∂2
∂R2
v1/2 and v3/2
∂2
∂R2
v3/2 . (20)
If the matrix elements of Eq. (19) are small compared to those of Eq. (20), then the perturbation
treatment is justified, and I can proceed to the following step, otherwise, I need to find a way to
solve the coupled differential equation for the rovibrational wave functions of the J = 1 V1/2 and
V3/2 PECs
Appendix A Getting the derivative of the mixing angle from its tangent
It is easy to obtain the tangent of γ from Eq. (14):
tanγ =
sinγ
cosγ
=
VA − V1/2
√
2ξ
.
SV-InclusionSOcouplinginNaCs.tex 12
13. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
4 6 8 10 12 14
0
5
10
15
20
Internuclear Separation R
dΓ
dR
2
2
Figure 4: Square of the first derivative of the mixing angle γ(R) with respect to the internuclear
separation. The extrema occur at the same R values as in Fig. 3.
Defining u(R) =
VA − V1/2
√
2ξ
, then γ = arctanu. Remembering now that
d
dR
arctanu =
u
1 + u2
,
one gets
dγ
dR
=
d
dR
arctanu =
1
1 +
VA−V1/2√
2ξ
2
d
dR
VA − V1/2
√
2ξ
,
which is an expression for dγ/dR without ever calculating γ explicitly. Substituting the definition for
V1/2 from Eq. (12) leads to
dγ
dR
=
1
1 +
VA−Vb0
2
√
2ξ
+ VA−Vb0
2
√
2ξ
2
+ 1
2
d
dR
VA − Vb0
2
√
2ξ
+
VA − Vb0
2
√
2ξ
2
+ 1
,
showing that the derivative of the mixing angle can be expressed solely from the Hund’s case (a) potentials
and the relevant spin-orbit coupling terms. This latter expression was used to obtain figures 3, 4 and 5.
SV-InclusionSOcouplinginNaCs.tex 13
14. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
4 6 8 10 12 14
20
10
0
10
20
Internuclear Separation R
d2
Γ
dR2
2
Figure 5: Second derivative of the mixing angle γ(R) with respect to the internuclear separation.
What looks like a discontinuity around R ≈ 4.25 ˚A is not, d2
γ/dR2 just varies very rapidly around
R ≈ 4.25 ˚A, but remains smooth and continuous.
Appendix B Checking hermicity of the kinetic energy operator
All operators defined in Eqs. (11a) are hermitian. This property is obvious for all operators that do not
involve a derivative with respect to R: R, D, and Hel. A hermitian operator remains hermitian under a
unitary transformation. Thus the change of basis defined by U conserves the hermicity of R, D, and Hel
whether they are expressed in basis A or H.
However it is not trivial that the kinetic energy operator T is hermitian in the first place, and remains
so after the transformation U. Let’s prove that T is indeed hermitian, no matter what basis it is expressed
in.
First consider matrix elements of the form
vα −
2
2µ
d2
dR2
vα ,
where α denotes any of the electronic states, and |vα is any rovibrational state belonging to the electronic
state |Φel
α . The rovibrational state |vα satisfies the time-independent Schr¨odinger equation (TISE):
−
2
2µ
d2
dR2
|vα + V total
α |vα = Evα
|vα ,
SV-InclusionSOcouplinginNaCs.tex 14
15. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
where V total
α is the sum of the rotational energy and all other potential energies. Then
vα −
2
2µ
d2
dR2
vα = Evα
δvαvα
− vα V total
α vα (21a)
= Evα
δvαvα
− vα V total
α vα (21b)
= vα −
2
2µ
d2
dR2
vα , (21c)
since V total
α is purely multiplicative and given the properties of the Kronecker δ. Matrix elements of the
type described in the previous equation occur both in the A and H basis. Equations 21 show that TA and
the parts of TH that contain d2
/dR2 are indeed hermitian.
The function dγ/dR is purely multiplicative, therefore
vα
dγ
dR
2
vα = vα
dγ
dR
2
vα ,
so all diagonal blocks of TH are hermitian.
Let’s focus now on the off-diagonal blocks of TH. To finish proving that TH is hermitian, I need to
prove that
v3/2
d2
γ
dR2
+ 2
dγ
dR
∂
∂R
v1/2 = v1/2 −
d2
γ
dR2
− 2
dγ
dR
∂
∂R
v3/2 . (22)
Let’s recall the rule of integration by parts for the product of three well-behaved functions f ,g, and h:
b
a
f gh dR = [f gh]b
a −
b
a
f ghdR −
b
a
f g hdR,
and apply this expression to
f (R) = R|v3/2 = ψv3/2
(R) = ψv3/2
, g(R) =
dγ
dR
, h(r) = R|v1/2 = ψv1/2
(R) = ψv1/2
.
Starting from part of the matrix element on the left hand side of Eq. (22):
v3/2
dγ
dR
∂
∂R
v1/2 =
+∞
0
ψv3/2
dγ
dR
ψv1/2
dR (23a)
= ψv3/2
dγ
dR
ψv1/2
R=+∞
R=0
−
+∞
0
ψv3/2
dγ
dR
ψv1/2
dR −
+∞
0
ψv3/2
dγ
dR
ψv1/2
dR, (23b)
where the quantity between square brackets is zero, since the wave functions are zero at R = 0 and R = +∞.
SV-InclusionSOcouplinginNaCs.tex 15
16. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
Permuting the order of the products in the remaining integrals yields
v3/2
dγ
dR
∂
∂R
v1/2 = −
+∞
0
ψv1/2
dγ
dR
ψv3/2
dR −
+∞
0
ψv1/2
dγ
dR
ψv3/2
dR (23c)
= − v1/2
dγ
dR
∂
∂R
v3/2 − v1/2
d2
γ
dR2
v3/2 (23d)
Let’s now combine Eq. (23d) with Eq. (22)
v3/2
d2
γ
dR2
+ 2
dγ
dR
∂
∂R
v1/2 = v3/2
d2
γ
dR2
v1/2 + 2 v3/2
dγ
dR
∂
∂R
v1/2 (24a)
= v1/2
d2
γ
dR2
v3/2 − 2 v1/2
dγ
dR
∂
∂R
v3/2 − 2 v1/2
d2
γ
dR2
v3/2 (24b)
= v1/2 −
d2
γ
dR2
− 2
dγ
dR
∂
∂R
v3/2 , (24c)
which completes the proof that TH is hermitian, as it should.
First, verifying that TH is hermitian allows to check whether I did any algebraic mistake when passing
from basis A to basis H8. Second, notice that the V1/2 state holds 146 rovibrational states, and the V3/2
holds 114. If I did not recall that T is hermitian, I would have had to calculate (146 + 114)2 = 67600
matrix elements. Thanks to hermicity, I now only have to calculate
146 × (146 + 1)/2 = 10731 elements of the form v1/2
dγ
dR
2
−
2
2µ
∂2
∂R2
v1/2 ,
114 × (114 + 1)/2 = 6555 v3/2
dγ
dR
2
−
2
2µ
∂2
∂R2
v3/2 ,
114 × 146 = 16644 v3/2
d2
γ
dR2
+ 2
dγ
dR
∂
∂R
v1/2 ,
that is 33930 matrix elements, about half what I was about to calculate before I remembered (and
checked!) the hermicity of T.
8 In versions of these notes prior to 2.4, TH was not hermitian, because I dropped a minus sign along the way.
SV-InclusionSOcouplinginNaCs.tex 16
17. St´ephane
Valladier
Inclusion of Spin-Orbit coupling in my PhD project
v. 2.6
22nd December, 2013
References
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Adiabatic Passage (2011).
[2] J. Zaharova, M. Tamanis, R. Ferber, A. N. Drozdova, E. A. Pazyuk, and A. V. Stolyarov, Solution of the fully-
mixed-state problem: Direct deperturbation analysis of the A1Σ+–b3Π complex in a NaCs dimer, Physical Review A,
79(1), 012508 (2009).
[3] P. F. Bernath, Spectra of atoms and molecules, Oxford University Press, New York, 2nd edition (2005).
[4] H. Lefebvre-Brion and R. W. Field, The spectra and dynamics of diatomic molecules, Elsevier Academic Press,
Amsterdam; Boston (2004).
[5] H. Katˆo, Energy Levels and Line Intensities of Diatomic Molecules. Application to Alkali Metal Molecules, Bulletin
of the Chemical Society of Japan, 66(11), 3203 (1993).
[6] J. Vala, O. Dulieu, F. Masnou-Seeuws, P. Pillet, and R. Kosloff, Coherent control of cold-molecule formation through
photoassociation using a chirped-pulsed-laser field, Phys. Rev. A, 63, 013412 (2000).
[7] B. E. Londo˜no, J. E. Mahecha, E. Luc-Koenig, and A. Crubellier, Resonant coupling effects on the photoassociation
of ultracold Rb and Cs atoms, Phys. Rev. A, 80, 032511 (2009).
[8] M. Aymar and O. Dulieu, Calculations of transition and permanent dipole moments of heteronuclear alkali dimers
NaK, NaRb and NaCs, Molecular Physics, 105(11-12), 1733 (2007).
SV-InclusionSOcouplinginNaCs.tex 17