0 + E (1) and the first correction to the ground state energy is given by

 Conrad Stewart
 7 months ago
 Views:
Transcription
1 1 Problem set 9 Handout: 1/24 Due date: 1/31 Problem 1 Prove that the energy to first order for the lowestenergy state of a perturbed system is an upper bound for the exact energy of the lowestenergy state of the perturbed system, that is, that E () + E (1) E. The ground state energy is given by E =< ψ H ψ > (1) and the first correction to the ground state energy is given by E 1 =< ψ H 1 ψ > (2) where the total Hamiltonian is given by H = H + H 1. We know from the variational theorem that using the zerothorder wavefunction as a trial function for the total Hamiltonian will give an upper bound to the energy < ψ H ψ > E (3) since ψ is not the wavefunction associated with the Hamiltonian. Therefore, we can prove that < ψ H ψ > E (4) < ψ H + H 1 ψ > E (5) < ψ H ψ > + < ψ H 1 ψ > E (6) E + E1 E (7) Problem 2 An electron moves in a harmonic potential, V = 1 2 x2. What is the effect, to first order, on the energies of a perturbing electric field, Ĥ 1 = Fx? Explain your reasoning. [hint: you do not need to evaluate any integrals to answer the question.]. (8) 1
2 Since the particle moves in a Harmonic potential the wavefunction squared is symmetric. An electric field perturbation V 1 = Ex which is antisymmetric will have no effect since < ψ V 1 ψ >= dx ψ 2 V 1 = dxsymmetric antisymmetric = (9) Problem 3 Calculate the energy to first order of He + in its lowestenergy state. Use the hydrogen atom in its ground state as your zerothorder approximation. Use atomic units. The zeroth order Hamiltonian is for the hydrogen atom H = r (1) whereas the Hamiltonian for the He + is given by H = r (11) Thus, the total Hamiltonian can be split as The energy to first order is then H = R = H 1 r = H + H 1 (12) E = E + E 1 = < 1s 1 1s > (13) r Evaluating the firstorder correction to the energy we get E 1 = < 1s 1 r 1s >= 4π π exp( 2r)rdr = 4 1 = 1 (14) 22 The energy to firstorder is then E = 1/2 1 = 3/2. Problem 4 A hydrogen atom in its ground state is perturbed by applying a uniform electric field, Fz, along the zdirection. 2
3 1. What is the firstorder change in the energy? 2. For the firstorder correction to the groundstate wavefunction, we can consider a 2s,1s and a 2pz,1s to be the coefficients for mixing in 2s and 2p z character. Write down the expression for these two coefficients. 3. Predict the signs of these coefficients. Explain your reasoning. The ground state wavefunction for the hydrogen atom is 1s = 1 π exp( r). a) The firstorder change in the energy due to the electric field is then given by = F 1 π = F 1 π b) The two cofficients are r 3 exp( 2r)dr < 1s H 1 1s >= F < 1s r cosθ 1s > (15) r 3 exp( 2r)dr π π cosθ sin θdθ sym. antisym. 2π 2π dφ (16) dφ = (17) c (1) 2s,1s = < 2s H1 1s > E 2s E 1s = (18) c (1) 2p z,1s = < 2p z H 1 1s > E 2pz E 1s = (19) The first coefficient are zero due to symmetry, i.e. it cannot improve the energy due to perturbation along the Z since it is symmetric in all directions. The second coefficient is positive since the 2p z function is oriented along the direction of the perturbation and will therefore improve the wavefunction. Problem 5 The HellmanFeynman theorem establishes a connection between the change in the energy and the change in the Hamiltonian of a system experiences an electric field perturbation. The theorem is given by de df = Ĥ (2) where E is the energy, F is the electric field strength and Ĥ is the total hamiltonian for the system. If we apply an electric field perturbation along the zdirection we can write the firstorder hamiltonian as Ĥ (1) = ˆµ z F (21) 3
4 where ˆµ z is the electric dipole moment operator in the zdirection. 1) Use the HellmanFeynman theorem to express the change in the energy due to an electric field in the zdirection in terms of the dipole moment operator. 2) Expand the energy of a system in a Taylor expansion relative to the energy, E, in the absence of the field. The expectation value of the dipole moment operator can also be written as < ˆµ z >= µ z + α zzf + (22) where µ z is the permanent dipole moment of the system and α zz is the polarizability. 3) Use this expansion to express the permanent dipole moment and polarizability in terms of derivatives of the energy with respect to the electric field. 4) Use nondegenerate perturbation theory to obtain an expression for the energy of the system in terms of the field strength to second order. 5) Finally, use this expression to identify the permanent dipole moment and polarizability of the system. [Hint: The final expression should be expression in terms of the electric dipole operator and the zerothorder wavefunctions de df = Ĥ = Ĥ µ z F = µ z (23) E = E + E F E + (24) 2 2F2 de df = E + 2 E + = µ 2F2 z α zzf + (25) Therefore, we can identify the dipole moment and polarizability as µ z = E (26) α zz = 2 E 2 (27) 4
5 4. The energy expression from perturbations theory reads: E = E + E 1 + E 2 + (28) = E + < ψ () H 1 ψ () > + < ψ m () H 1 ψ () >< ψ () H1 ψ () E E m m = E < ψ () µ z ψ () > F + < ψ m () µ z ψ () >< ψ () µ z ψ () E E m m 5. We can therefore identify the terms as m > m > + (29) + (3) and µ z = E =< ψ() µ z ψ () > (31) α zz = 2 E = 2 < ψ m () µ z ψ () >< ψ () µ z ψ () 2 E E m m m > (32) Problem 6 A given unperturbed system has a doubly generate energy level for which the perturbation integrals have values of H 1 11 = 6a, H1 22 = 8a, and H1 12 = 2a, where a is a positive constant. We also know that the unperturbed wavefunctions are orthonormal. 1. Find the firstorder correction to the energy in terms of the constant a. 2. Find the normalized correct zerothorder wave functions. 1. The secular equation is det(h EI) = H1 aa E H 1 ba H1 ab H 1 bb E = 6a E 2a 2a 8a E = (33) Expansion of the secular equation gives the following quadratic equation E 2 14Ea + 44a 2 = (34) 5
6 for which the solution is E = 7a ± 2a = 11.47a, 2.52a. The coefficients for the lowest eigenvalue is then given by Using normalization we find (H 1 aa E1 )c 1 + H ab c 2 = (35) (6a 2.52a)c 1 + 2ac 2 = (36) c 1 =.57c 2 (37) 1 =.57 2 c c2 2 ==> c 2 =.867 (38) Therefore, the normalized wavefunction is ψ 1 =.49ψ a +.867ψ b (39) 6
Quantum Mechanics Solutions
Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H
More informationFor a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as
VARIATIO METHOD For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as Variation Method Perturbation Theory Combination
More informationeigenvalues eigenfunctions
BornOppenheimer Approximation Atoms and molecules consist of heavy nuclei and light electrons. Consider (for simplicity) a diatomic molecule (e.g. HCl). Clamp/freeze the nuclei in space, a distance r
More informationIntroduction to multiconfigurational quantum chemistry. Emmanuel Fromager
Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg  Laboratoire de Chimie Quantique  Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Notations
More informationIntroduction to Molecular Vibrations and Infrared Spectroscopy
hemistry 362 Spring 2017 Dr. Jean M. Standard February 15, 2017 Introduction to Molecular Vibrations and Infrared Spectroscopy Vibrational Modes For a molecule with N atoms, the number of vibrational modes
More informationPhysics 505 Homework No. 1 Solutions S11
Physics 505 Homework No s S Some Preliminaries Assume A and B are Hermitian operators (a) Show that (AB) B A dx φ ABψ dx (A φ) Bψ dx (B (A φ)) ψ dx (B A φ) ψ End (b) Show that AB [A, B]/2+{A, B}/2 where
More information16.1. PROBLEM SET I 197
6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,
More informationProblem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:
Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m. 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m. 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More informationV( x) = V( 0) + dv. V( x) = 1 2
Spectroscopy 1: rotational and vibrational spectra The vibrations of diatomic molecules Molecular vibrations Consider a typical potential energy curve for a diatomic molecule. In regions close to R e (at
More informationProblem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1
Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2ad,g,h,j 2.6, 2.9; Chapter 3: 1ad,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate
More informationcharges q r p = q 2mc 2mc L (1.4) ptles µ e = g e
APAS 5110. Atomic and Molecular Processes. Fall 2013. 1. Magnetic Moment Classically, the magnetic moment µ of a system of charges q at positions r moving with velocities v is µ = 1 qr v. (1.1) 2c charges
More informationThe Einstein A and B Coefficients
The Einstein A and B Coefficients Austen Groener Department of Physics  Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics III December 10, 010 Abstract In this paper, the Einstein
More information2. The Schrödinger equation for oneparticle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for oneparticle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jean M Standard Problem Set 3 Solutions 1 Verify for the particle in a onedimensional box by explicit integration that the wavefunction ψ x) = π x ' sin ) is normalized To verify that
More informationLinear Algebra in Hilbert Space
Physics 342 Lecture 16 Linear Algebra in Hilbert Space Lecture 16 Physics 342 Quantum Mechanics I Monday, March 1st, 2010 We have seen the importance of the plane wave solutions to the potentialfree Schrödinger
More informationSummary: angular momentum derivation
Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. () () (3) Angular momentum commutation relations [L x, L y ] = i hl z (4) [L i, L j ] = i hɛ ijk L k (5) LeviCivita
More information22.02 Intro to Applied Nuclear Physics
22.02 Intro to Applied Nuclear Physics MidTerm Exam Solution Problem 1: Short Questions 24 points These short questions require only short answers (but even for yes/no questions give a brief explanation)
More informationAccelerator Physics Homework #7 P470 (Problems: 14)
Accelerator Physics Homework #7 P470 (Problems: 4) This exercise derives the linear transfer matrix for a skew quadrupole, where the magnetic field is B z = B 0 a z, B x = B 0 a x, B s = 0; with B 0 a
More informationChristophe De Beule. Graphical interpretation of the Schrödinger equation
Christophe De Beule Graphical interpretation of the Schrödinger equation Outline Schrödinger equation Relation between the kinetic energy and wave function curvature. Classically allowed and forbidden
More informationPostulates and Theorems of Quantum Mechanics
Postulates and Theorems of Quantum Mechanics Literally, a postulate is something taen as selfevident or assumed without proof as a basis for reasoning. It is simply is Postulate 1: State of a physical
More informationSolutions to chapter 4 problems
Chapter 9 Solutions to chapter 4 problems Solution to Exercise 47 For example, the x component of the angular momentum is defined as ˆL x ŷˆp z ẑ ˆp y The position and momentum observables are Hermitian;
More informationElectric Dipole Paradox: Question, Answer, and Interpretation
Electric Dipole Paradox: Question, Answer, and Interpretation Frank Wilczek January 16, 2014 Abstract Nonvanishing electric dipole moments for the electron, neutron, or other entities are classic signals
More informationLittle Orthogonality Theorem (LOT)
Little Orthogonality Theorem (LOT) Take diagonal elements of D matrices in RG * D R D R i j G ij mi N * D R D R N i j G G ij ij RG mi mi ( ) By definition, D j j j R TrD R ( R). Sum GOT over β: * * ( )
More informationLecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2
More informationr 2 dr h2 α = 8m2 q 4 Substituting we find that variational estimate for the energy is m e q 4 E G = 4
Variational calculations for Hydrogen and Helium Recall the variational principle See Chapter 16 of the textbook The variational theorem states that for a Hermitian operator H with the smallest eigenvalue
More informationBrief introduction to groups and group theory
Brief introduction to groups and group theory In physics, we often can learn a lot about a system based on its symmetries, even when we do not know how to make a quantitative calculation Relevant in particle
More informationSymmetries and Supersymmetries in Trapped Ion Hamiltonian Models
Proceedings of Institute of Mathematics of NAS of Ukraine 004, Vol. 50, Part, 569 57 Symmetries and Supersymmetries in Trapped Ion Hamiltonian Models Benedetto MILITELLO, Anatoly NIKITIN and Antonino MESSINA
More information1 Position Representation of Quantum State Function
C/CS/Phys C191 Quantum Mechanics in a Nutshell II 10/09/07 Fall 2007 Lecture 13 1 Position Representation of Quantum State Function We will motivate this using the framework of measurements. Consider first
More informationMolecular Simulation I
Molecular Simulation I Quantum Chemistry Classical Mechanics E = Ψ H Ψ ΨΨ U = E bond +E angle +E torsion +E nonbond Jeffry D. Madura Department of Chemistry & Biochemistry Center for Computational Sciences
More informationComputational Physics (6810): Session 8
Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential
More informationEngineering Jump Start  Summer 2012 Mathematics Worksheet #4. 1. What is a function? Is it just a fancy name for an algebraic expression, such as
Function Topics Concept questions: 1. What is a function? Is it just a fancy name for an algebraic expression, such as x 2 + sin x + 5? 2. Why is it technically wrong to refer to the function f(x), for
More informationTwolevel systems coupled to oscillators
Twolevel systems coupled to oscillators RLE Group Energy Production and Conversion Group Project Staff Peter L. Hagelstein and Irfan Chaudhary Introduction Basic physical mechanisms that are complicated
More informationSection 5.8. (i) ( 3 + i)(14 2i) = ( 3)(14 2i) + i(14 2i) = {( 3)14 ( 3)(2i)} + i(14) i(2i) = ( i) + (14i + 2) = i.
1. Section 5.8 (i) ( 3 + i)(14 i) ( 3)(14 i) + i(14 i) {( 3)14 ( 3)(i)} + i(14) i(i) ( 4 + 6i) + (14i + ) 40 + 0i. (ii) + 3i 1 4i ( + 3i)(1 + 4i) (1 4i)(1 + 4i) (( + 3i) + ( + 3i)(4i) 1 + 4 10 + 11i 10
More informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/23//2017 Physics 5701 Lecture Outline 1 General Formulation of Quantum Mechanics 2 Measurement of physical quantities and observables 3 Representations
More informationCHAPTER 8: Matrices and Determinants
(Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means refer to Part A, (B) means refer to Part B, etc. Most of these exercises can be done without a
More informationRELATING CLASSICAL AND QUANTUM MECHANICS 1 I: Thinking about the wave function
RELATING CLASSICAL AND QUANTUM MECHANICS 1 I: Thinking about the wave function In quantum mechanics, the term wave function usually refers to a solution to the Schrödinger equation, Ψ(x, t) i = ĤΨ(x, t),
More information( 2 + k 2 )Ψ = 0 (1) c sinhα sinφ. x = y = c sinβ. z =
Power series Schrodinger eigenfunctions for a particle on the torus. Mario Encinosa and Babak Etemadi Department of Physics Florida A & M University Tallahassee, Florida 3307 Abstract The eigenvalues and
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More information6. Molecular structure and spectroscopy I
6. Molecular structure and spectroscopy I 1 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 lightmatter interaction 6.1 molecular spectroscopy introduction 2 Molecular
More informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1D case: 3D case Time Total energy = Hamiltonian To find out about
More informationVAN DER WAALS FORCES
VAN DER WAALS FORCES JANLOUIS MÖNNING The Van der Waals Force is an attracting force between neutral atoms with R the distance of the atoms. Our aim will be to show that such a force exists, which means
More informationElectric fields in matter
Electric fields in matter November 2, 25 Suppose we apply a constant electric field to a block of material. Then the charges that make up the matter are no longer in equilibrium: the electrons tend to
More informationDirectional States of SymmetricTop Molecules Produced by Combined Static and Radiative Electric Fields
Directional States of SymmetricTop Molecules Produced by Combined Static and Radiative Electric Fields Marko Härtelt and Bretislav Friedrich FritzHaberInstitut der MaxPlanckGesellschaft, Berlin, Germany
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More informationNewton s Method and Linear Approximations
Newton s Method and Linear Approximations Curves are tricky. Lines aren t. Newton s Method and Linear Approximations Newton s Method for finding roots Goal: Where is f (x) = 0? f (x) = x 7 + 3x 3 + 7x
More informationAtomic Physics: an exploration through problems and solutions (2nd edition)
Atomic Physics: an exploration through problems and solutions (2nd edition) We would be greatly indebted to our readers for informing us of errors and misprints in the book by sending an email to: budker@berkeley.edu.
More informationarxiv: v1 [grqc] 10 Jun 2009
MULTIPOLE CORRECTIONS TO PERIHELION AND NODE LINE PRECESSION arxiv:0906.1981v1 [grqc] 10 Jun 2009 L. FERNÁNDEZJAMBRINA ETSI Navales, Universidad Politécnica de Madrid, Arco de la Victoria s/n, E28040Madrid
More information2.5 The Fundamental Theorem of Algebra.
2.5. THE FUNDAMENTAL THEOREM OF ALGEBRA. 79 2.5 The Fundamental Theorem of Algebra. We ve seen formulas for the (complex) roots of quadratic, cubic and quartic polynomials. It is then reasonable to ask:
More informationTaylor Series. richard/math230 These notes are taken from Calculus Vol I, by Tom M. Apostol,
Taylor Series Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math230 These notes are taken from Calculus Vol
More informationThe A, B, C and D are determined by these 4 BCs to obtain
Solution:. Floquet transformation: (a) Defining a new coordinate η = y/ β and φ = (/ν) s 0 ds/β, we find ds/dφ = νβ, and dη dφ = ds dη dφ d 2 η dφ 2 = ν2 β ( β y ) ( 2 β 3/2 β y = ν β /2 y ) 2 β /2 β y,
More informationAbout the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a threeyear curriculum development project
About the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a threeyear curriculum development project undertaken by a consortium of five English universities led by
More informationExam TMA4120 MATHEMATICS 4K. Monday , Time:
Exam TMA4 MATHEMATICS 4K Monday 9.., Time: 9 3 English Hjelpemidler (Kode C): Bestemt kalkulator (HP 3S eller Citizen SR7X), Rottmann: Matematisk formelsamling Problem. a. Determine the value ( + i) 6
More informationVibrations of Carbon Dioxide and Carbon Disulfide
Vibrations of Carbon Dioxide and Carbon Disulfide Purpose Vibration frequencies of CO 2 and CS 2 will be measured by Raman and Infrared spectroscopy. The spectra show effects of normal mode symmetries
More informationEfficiency of genetic algorithm and determination of ground state energy of impurity in a spherical quantum dot
Efficiency of genetic algorithm and determination of ground state energy of impurity in a spherical quantum dot +DOXNùDIDN 1* 0HKPHWùDKLQ 1, Berna Gülveren 1, Mehmet Tomak 1 Selcuk University, Faculty
More informationQuantum chemistry and vibrational spectra
Chapter 3 Quantum chemistry and vibrational spectra This chapter presents the quantum chemical results for the systems studied in this work, FHF (Section 3.) and OHF (Section 3.3). These triatomic anions
More informationBornOppenheimer Corrections Near a RennerTeller Crossing
BornOppenheimer Corrections Near a RennerTeller Crossing Mark S. Herman Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the
More informationNewton s Method and Linear Approximations
Newton s Method and Linear Approximations Newton s Method for finding roots Goal: Where is f (x) =0? f (x) =x 7 +3x 3 +7x 2 1 21 0.5 0.52 Newton s Method for finding roots Goal: Where is f (x) =0? f
More informationGood Vibes: Introduction to Oscillations
Good Vibes: Introduction to Oscillations Description: Several conceptual and qualitative questions related to main characteristics of simple harmonic motion: amplitude, displacement, period, frequency,
More informationStochastic Processes. Monday, November 14, 11
Stochastic Processes 1 Definition and Classification X(, t): stochastic process: X : T! R (, t) X(, t) where is a sample space and T is time. {X(, t) is a family of r.v. defined on {, A, P and indexed
More informationONEELECTRON AND TWOELECTRON SPECTRA
ONEELECTRON AND TWOELECTRON SPECTRA (A) FINE STRUCTURE AND ONEELECTRON SPECTRUM PRINCIPLE AND TASK The wellknown spectral lines of He are used for calibrating the diffraction spectrometer. The wavelengths
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationActionAngle Variables and KAMTheory in General Relativity
ActionAngle Variables and KAMTheory in General Relativity Daniela Kunst, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany Workshop in Oldenburg
More informationReview of Linear Algebra
Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources
More informationLikewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H
Finite Dimensional systems/ilbert space Finite dimensional systems form an important subclass of degrees of freedom in the physical world To begin with, they describe angular momenta with fixed modulus
More informationQuiz 5 R = litatm/molk 1 (25) R = J/molK 2 (25) 3 (25) c = X 10 8 m/s 4 (25)
ADVANCED INORGANIC CHEMISTRY QUIZ 5 and FINAL December 18, 2012 INSTRUCTIONS: PRINT YOUR NAME > NAME. QUIZ 5 : Work 4 of 15 (The lowest problem will be dropped) FINAL: #6 (10 points ) Work 6 of 7 to 14
More informationNewton s Method and Linear Approximations 10/19/2011
Newton s Method and Linear Approximations 10/19/2011 Curves are tricky. Lines aren t. Newton s Method and Linear Approximations 10/19/2011 Newton s Method Goal: Where is f (x) =0? f (x) =x 7 +3x 3 +7x
More informationGroup representation theory and quantum physics
Group representation theory and quantum physics Olivier Pfister April 29, 2003 Abstract This is a basic tutorial on the use of group representation theory in quantum physics, in particular for such systems
More informationPoles, Residues, and All That
hapter Ten Poles, Residues, and All That 0.. Residues. A point z 0 is a singular point of a function f if f not analytic at z 0, but is analytic at some point of each neighborhood of z 0. A singular point
More informationELECTRIC DIPOLE MOMENT OF THE ELECTRON AND ITS COSMOLOGICAL IMPLICATIONS
ELECTRIC DIPOLE MOMENT OF THE ELECTRON AND ITS COSMOLOGICAL IMPLICATIONS H S NATARAJ Under the Supervision of Prof. B P DAS NonAccelerator Particle Physics Group Indian Institute of Astrophysics Bangalore
More informationPhysicsAndMathsTutor.com. GCE Edexcel GCE. Core Mathematics C2 (6664) January Mark Scheme (Results) Core Mathematics C2 (6664) Edexcel GCE
GCE Edexcel GCE Core Mathematics C (666) January 006 Mark Scheme (Results) Edexcel GCE Core Mathematics C (666) January 006 666 Core Mathematics C Mark Scheme. (a) +5 + c = 0 or  + c = 0 c = A () (b)
More informationwave mechanics applied to semiconductor heterostructures
wave mechanics applied to semiconductor heterostructures GERALD BASTARD les editions de physique Avenue du Hoggar, Zone Industrielle de Courtaboeuf, B.P. 112, 91944 Les Ulis Cedex, France Contents PREFACE
More informationClick here for answers. f x dy dx sl 2 x 2. (b) Determine the function y f x. f x y. Find the minimum value of f. I n. I k 1
CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER 6 A Click here for answers. S Click here for solutions. ;. Three mathematics students have ordered a 4inch pizza. Instead of slicing it in the traditional
More informationMath 311, Partial Differential Equations, Winter 2015, Midterm
Score: Name: Math 3, Partial Differential Equations, Winter 205, Midterm Instructions. Write all solutions in the space provided, and use the back pages if you have to. 2. The test is out of 60. There
More informationBianchi I Spacetimes and Loop Quantum Cosmology
Bianchi I Spacetimes and Loop Quantum Cosmology Edward WilsonEwing Institute for Gravitation and the Cosmos The Pennsylvania State University Work with Abhay Ashtekar October 23, 2008 E. WilsonEwing
More information7. Arrange the molecular orbitals in order of increasing energy and add the electrons.
Molecular Orbital Theory I. Introduction. A. Ideas. 1. Start with nuclei at their equilibrium positions. 2. onstruct a set of orbitals that cover the complete nuclear framework, called molecular orbitals
More information9 Particleinabox (PIB)
9 Particleinabox (PIB) 1. Consider a linear polyene.. The electrons are completely delocalized inside the polyene, but cannot leave the molecular framework. 3. Let us approximate this system by a
More information2 Resolvents and Green s Functions
Course Notes Solving the Schrödinger Equation: Resolvents 057 F. Porter Revision 09 F. Porter Introduction Once a system is wellspecified, the problem posed in nonrelativistic quantum mechanics is to
More informationTransformed E&M I homework. Multipole Expansion (Griffiths Chapter 3)
Transformed E&M I homework Multipole Expansion (Griffiths Chapter 3) Multipole Expansion Question 1. Multipole moment of charged wire CALCULATION; EXPANSION (U. Nauenberg, HW3, solutions available) A charge
More informationSolution to the exam in TFY4275/FY8907 CLASSICAL TRANSPORT THEORY May 20, 2011
NTNU Page 1 of 5 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk Solution to the exam in TFY4275/FY8907 CLASSICAL TRANSPORT THEORY May 20, 2011 This solution consists of 5 pages. Problem
More informationTutorial for PhET Sim Quantum Bound States
Tutorial for PhET Sim Quantum Bound States J. Mathias Weber * JILA and Department of Chemistry & Biochemistry, University of Colorado at Boulder With this PhET Sim, we will explore the properties of some
More informationProve proposition 68. It states: Let R be a ring. We have the following
Theorem HW7.1. properties: Prove proposition 68. It states: Let R be a ring. We have the following 1. The ring R only has one additive identity. That is, if 0 R with 0 +b = b+0 = b for every b R, then
More informationNucleon Pairing in Atomic Nuclei
ISSN 739, Moscow University Physics Bulletin,, Vol. 69, No., pp.. Allerton Press, Inc.,. Original Russian Text B.S. Ishkhanov, M.E. Stepanov, T.Yu. Tretyakova,, published in Vestnik Moskovskogo Universiteta.
More informationSection 3 Electronic Configurations, Term Symbols, and States
Section 3 Electronic Configurations, Term Symbols, and States Introductory Remarks The Orbital, Configuration, and State Pictures of Electronic Structure One of the goals of quantum chemistry is to allow
More informationDynamics of spinning particles in Schwarzschild spacetime
Dynamics of spinning particles in Schwarzschild spacetime, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany 08.05.2014 RTG Workshop, Bielefeld
More informationPOEM: Physics of Emergent Materials
POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Reference: Bernevig Topological Insulators and Topological Superconductors Tutorials:
More informationThe Hydrogen Atom Chapter 20
4/4/17 Quantum mechanical treatment of the H atom: Model; The Hydrogen Atom Chapter 1 r 1 Electron moving aroundpositively charged nucleus in a Coulombic field from the nucleus. Potential energy term
More information5. Atoms and the periodic table of chemical elements. Definition of the geometrical structure of a molecule
Historical introduction The Schrödinger equation for oneparticle problems Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical elements
More informationMOLECULAR SPECTROSCOPY
MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective
More informationExploring the energy landscape
Exploring the energy landscape ChE210D Today's lecture: what are general features of the potential energy surface and how can we locate and characterize minima on it Derivatives of the potential energy
More informationIR absorption spectroscopy
IR absorption spectroscopy IR spectroscopy  an analytical technique which helps determine molecules structure When a molecule absorbs IR radiation, the vibrational energy of the molecule increase! The
More informationPhysics 221A Fall 2017 Notes 25 The Zeeman Effect in Hydrogen and Alkali Atoms
Copyright c 2017 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 25 The Zeeman Effect in Hydrogen and Alkali Atoms 1. Introduction The Zeeman effect concerns the interaction of atomic systems with
More informationChapter 9 Global Nonlinear Techniques
Chapter 9 Global Nonlinear Techniques Consider nonlinear dynamical system 0 Nullcline X 0 = F (X) = B @ f 1 (X) f 2 (X). f n (X) x j nullcline = fx : f j (X) = 0g equilibrium solutions = intersection of
More informationThe Multivariate Gaussian Distribution
The Multivariate Gaussian Distribution Chuong B. Do October, 8 A vectorvalued random variable X = T X X n is said to have a multivariate normal or Gaussian) distribution with mean µ R n and covariance
More informationEducreator Research Journal (ERJ) ISSN :
ESTIMATION OF GROUNDSTATE ENERGY OF PARTICLE IN 1D POTENTIAL WELLVARIATION METHOD R.B. Ahirrao Uttamrao Patil College of Art s and Science, Dahivel (M. S.), Abstract: Quantum mechanics provides approximate
More informationThe Spherical Harmonics
Physics 6C Fall 202 The Spherical Harmonics. Solution to Laplace s equation in spherical coordinates In spherical coordinates, the Laplacian is given by 2 = ( r 2 ) ( + r 2 r r r 2 sin 2 sinθ ) + θ θ θ
More informationVariational Method Applied to the Harmonic Oscillator
Variational Method Applied to the Harmonic Oscillator Department of Chemistry Centre College Danville, Kentucky 44 dunnk@centre.edu Copyright by the Division of Chemical Education, Inc., American Chemical
More informationSection 10: Many Particle Quantum Mechanics Solutions
Physics 143a: Quantum Mechanics I Section 10: Many Particle Quantum Mechanics Solutions Spring 015, Harvard Here is a summary of the most important points from this week (with a few of my own tidbits),
More information