Computational Spectroscopy III. Spectroscopic Hamiltonians
|
|
- Frank Newton
- 5 years ago
- Views:
Transcription
1 Computational Spectroscopy III. Spectroscopic Hamiltonians (e) Elementary operators for the harmonic oscillator (f) Elementary operators for the asymmetric rotor (g) Implementation of complex Hamiltonians (h) Matrix diagonalization methods (i) Symmetry and wavefunction diagnostics (j) Selection rules Updated: April 10, 2008
2 (e) Elementary operators for the harmonic oscillator The harmonic oscillator problem The harmonic potential is Approximates a realistic potential for small amplitude vibrations. Is a starting point for solving the motion in a real vibrational potential. The Schrodinger equation is H" = E" or T" + V( x)" = E" # h2 $ 2 2µ $x " 2 n x Energy levels are where the harmonic frequency is the reduced mass is ( ) kx 2 " n ( x) = E n " n ( x) E n = ( n + 1 2)h" and the quantum number is n = 0, 1, 2, 3, V ( x) = 1 V ( x) kx 2 = 1 kx µ = m 1 m 2 m 1 + m 2 " = 2#$ = k µ φ 4 (x) φ 3 (x) φ 2 (x) φ 1 (x) Realistic Potential φ 0 (x)
3 Dimensionless coordinates x = ) x = x p = ) p = "ih # #x # Q = % µ" $ h & ( ' 1 2 x P = ( µh" ) 1 2 p H = H h" x - multiply by x operator for x coodinate. Dimension of length = [L]. p - operator for the momentum in the x direction. Dimension of momentum = [M][L]/[T] H has dimensions of energy = [M][L] 2 /[T] 2 P, Q, and H are dimensionless. The Shroedinger equation becomes H = 1 $ "# 2 2 #Q + ' & % 2 Q2 ) (
4 a " 1 2 a + " 1 2 Ladder operators ( Q + ip ) ( Q # ip ) a ( $ n ) = n $ n#1 a + ( $ n ) = n +1$ n +1 # h & 2 x = % ( a + + a $ 2µ" ' 1 ( ) # p = i hµ" & 2 % ( a + ) a $ 2 ' 1 ( ) Define the ladder operators a and a +. The ladder operators are NOT Hermitian. a is the lowering operator since it has the effect of transforming the harmonic oscillator wavefunction φ n (x) into a multiple of the next lower function φ n-1 (x). Likewise, a + is the raising operator. φ n+2 (x) φ n+1 (x) φ n (x) φ n-1 (x) φ n-2 (x) The x and p operators can be expressed in terms of the ladder operators a and a +. Therefore ANY Hamiltonian that can be writen in terms of the x and p operators can be expressed in terms of the ladder operators a and a +. In this context, a and a + will be our elementary operators. a + a + a + a + a a a a
5 Matrices for Ladder Operators To make the matrix for an operator, we first need to define our basis set. The eigenfunctions of a Hermitian operator are orthogonal, are (or can be) normalized, and form a complete set. Let s use the harmonic oscillator wave functions as our basis: {φ 0, φ 1, φ 2, φ 3, φ 4, } The matrix of a + + +$ is the set of elements a nm = % " * n ( x) a + (" #$ m ( x) )dx = m +1& n,m +1 This gives + a nm = m +1" n,m +1 n= m= " % $ ' $ ' a + = $ ' $ ' $ ' # $ : : : : &' a nm = m" n,m#1 n= m= " % $ ' $ ' a = $ ' $ ' $ ' # $ : : : : &'
6 Programing a + and a function aa=aplusm(n) % APLUSM calculates the matrix of dimension n for the raisning operator a+. aa=zeros(n,n); % start with an nxn matrix filled with zeros. for k=1:n-1 aa(k+1,k)=sqrt(k); % Fill the elements along the diagonal above the principal diagonal. end
7 Matrices for x, p, x 2, p 2, and H # h & 2 x = % ( a + + a $ 2µ" ' 1 ( ) # p = i hµ" & 2 % ( a + ) a $ 2 ' 1 ( ) Once the matices for a and a + have been calculated (aplus, a) and the constants h, µ, k, and ω are defined, just type in the algebraic formulae for the other operators! omega = sqrt(k/mu) x = sqrt(hbar/(2*mu*omega)*(aplus+a) p = i*sqrt(hbar*mu*omega/2)*(aplus-a) xsq = x*x psq = p*p H = (psq/(2*mu))+0.5*k*xsq For an anharmonic Hamiltonian, e.g., Hanh=H+cx 3 +dx 4 : Hanh = H + c*x^3 + d*x^4
8 (f) Elementary operators for the asymmetric rotor Rigid Body Rotation: Diatomic Molecules Center of mass transformation r m 2 Center of Mass r µ= m 1 m 2 m 1 +m 2 O Center of Mass m 1 Dick Zare s book, Angular Momentum - Spatial Aspects of Chemistry will be helpful for this part of the course.
9 Rigid Body Rotation: Diatomic Molecules x z (x,y,z)! r y " x = rsin" cos# y = rsin" sin# z = rcos" Transformation to spherical polar coordinates The Schroedinger equation is H" = E" (1) For the rotation of a diatomic molecule, we have in Cartesian coordinates $ H = " h2 # 2 2µ #x + # 2 2 #y + # 2 ' & ) % 2 #z 2 ( Transform into spherical polar coordinates, we get for fixed r: & H = " h2 # 2 2µr 2 #$ + cot$ # 2 #$ + 1 # 2 ) ( + ' sin 2 $ #% 2 * Then solve eq (1) for " =" (#, $ ) (2) (3)
10 Angular Momentum The linear momentum of a mass m is given interms of the velocity: p = mv The angular momentum is given in terms of it position r = (x, y, z) and its linear momentum p = (p x, p y, p z ) as (4) l = r " p The operators for the components of the momentum are p x = "ih # #x p y = "ih # #y p z = "ih # #z (5) Using eq (5) in eq (4), we get the angular momentum operators $ l x = yp z " zp y = "i y # #z " z # ' & ) % #y ( $ l y = zp x " xp z = "i z # #x " x # ' & ) % #z ( $ l z = xp y " yp x = "i x # #y " y # ' & ) % #x ( (6)
11 Angular Momentum: Commutation Relationships The comutator of operators (matrices) A and B is defined [ A,B] " AB # BA = 0if theycommute $ 0 if they do not We already know that a coordinate does not commute with its corresponding linear momentum: [ x, p x ] = ih and x, p y [ ] = 0 etc. For the components of the angular momentum, we get [ l x,l y ] = ihl z [ l y,l z ] = ihl x l z,l x [ ] = ihl y These commutation relationships are considered to be so fundamental that we define a general angular momentum j as one whose components obey these rules: [ j x, j y ] = ihj z [ j y, j z ] = ihj x [ j z, j x ] = ihj y (7) (8)
12 Angular momentum eigenvectors Define A theorem in quantum mechanics says that we can find functions that are simultaneously eigenfunctions of two different Hermitian operators ONLY if those two operators commute. Let s choose j 2 and j z to be our two commuting operators, and we will denote the eigenfunctions as jm. We can show that the eigenvalue relationships are where j 2 = j x 2 + j y 2 + j z 2 We can show that j 2 commutes with its components [ j 2, j x ] = [ j 2, j y ] = j 2, j z j = 0, 1 2,1, 3 2,2,... [ ] = 0 j 2 jm = j( j +1)h 2 jm j z jm = mh jm (9) (10) (11) and m = ± j,± ( j "1),...± 1 or 0 2 (2j+1 values of m )
13 Ladder operators for angular momentum j mmax> Define the ladder operators: j + " j x + ij y and j - " j x # ij y (12) j m+1> j+ j- j m> m min = -j and m max = +j j m-1> j mmin> The action of the ladder operators on the angular momentum eigenfunctions is j + jm = h j( j +1) " m( m +1) j,m +1 j " jm = h j( j +1) " m( m "1) j,m "1 We can rearrange eq (12) to get j x = 1 2 ( j + + j " ) ( ) j y = 1 2i j + " j " (13) (14)
14 Matrix elements of angular momentum operators If we remember that the angular momentum eigenfunctions are normailized and orthognonal, the above results may be summarized as follows: jm j 2 j " m " = j( j +1)# j j " # m m " jm j z j " m " = m "# j j " # m m " [ ] 1 2 # j " [ m ( m " $1) ] 1 2 # j " [ m ( m " ±1) ] 1 2 # j " [ m ( m " ±1) ] 1 2 # j " jm j + j " m " = j( j +1) $ m "( m " +1) jm j $ j " " jm j x j " " jm j y j " " m = j( j +1) $ " m = 1 2 j( j +1) $ " m = mi 1 2 j( j +1) $ " j # m, m " +1 j # m, m " $1 j # m, m " ±1 j # m, m " ±1 (14) where we have used the bra - ket notation : * " n A " m = # " n A (" m )d$
15 Angular momentum eigenfunctions: The spherical harmonics The eigenfunctions jm are called the spherical harmonics: ( ) " Y m j (#,$) jm " Y jm #,$ These are familiar to us as the angular parts of the hydrogen atom orbitals, and they are also the rotational wavefunctions for diatomic and linear molecules. They are orthogonal and normalized.
16 Rotation of Diatomic & Linear Molecules The rotational Hamiltonian is & H = " h2 # 2 2µr 2 #$ + cot$ # 2 #$ + 1 # 2 ) ( ' sin 2 $ #% 2 + * = J2 2µr 2 This is easier to deal with if we set h =1 in the definitions of the angular momentum operators and include all of the units in a single rotational constant: H = BJ 2 The rotational eigenfunctions are ten the spherical harmonics: HY jm = BJ 2 Y jm = Bj( j +1)Y jm The rotational energy levels are the eigenvalues, Bj(j+1) where j=0, 1, 2, 3,
17 The vector model of angular momentum The angular momentum vector has a fixed length and a fixed projection on the z-axis, which means that the angle θ is fixed. j 2 jm = j( j +1)h 2 jm j z jm = mh jm Figure from p 13 of Dick Zare s book. As is common in theoretical work, Dick has set h =1
18 Symmetric Tops Prolate Symmetric Top (stick-like) Oblate Symmetic Top (frisbee-like) The 3 principal moments of inertia are given by " i " i " i ( ) ( ) ( ) I x = m i y i 2 + z i 2 I y = m i x i 2 + z i 2 I z = m i x i 2 + y i 2 (18) I a = I b < I c("z) where i labels the atoms in the molecule. When two of the 3 principal monents of inertia are equal, the molecule is called a symmetric top. I a("z) < I b = I c
19 Symmetric Top Rotational Energy Levels The rotational Hamiltonian for a rigid body is H = J 2 a + J 2 b + J 2 c 2I a 2I b 2I c " AJ 2 a + BJ 2 2 b +CJ c For example consider a prolate top where B=C. We also use the relation J 2 = J 2 a + J 2 2 b + J c to rewrite the Hamiltonian H = AJ 2 a + BJ 2 b + BJ 2 c + BJ 2 2 a " BJ a = BJ ( A " B)J a Now we know that J 2 2 and J a commute so we can find eigenfunctions which are simultaneous eigenfunctions of both operators and that are therefore also eigenfunctions of H: H JK = J 2 JK = J(J +1)(h 2 ) JK ( ) JK J a JK = K h { J(J +1)(h 2 )B + K 2 (h 2 )( A " B) } JK (19) (20) (21) (22)
20 Symmetric Top Rotational Energy Levels It is convenient to set h =1 which means that we will include the units of h in the rotational constants to give (for the prolate symmetric rotor): J 2 JK = J(J +1) JK J a JK = K JK H JK = { J(J +1)B + K 2 ( A " B) } JK (23) (24) (25) We have still used a little slight of hand here since J a is the projection of J on an axis (a = z) fixed to the molecule, rather than a space-fixed axis Z as we did in the derivation of the angular momentum operators. However, we still have to consider the space-fixed axes and a full derivation shows that the symetric rotor wavefunctions JKM have 3 quantum numbers: J, K, and M. M is the quantum number for projection of the angular momentum on a spacefixed axis (Z) and K is the quantum number for the projection on a molecule-fixed axis (z). The rotational energy levels do not depend on M, so often we supress it. Dick Zare derives the explicit form of the symmetric rotor wavefunctions in terms of the three Euler angles, θ, φ, and χ.
21 Symmetric Top Rotational Energy Levels Energy J=2 Prolate top A > B = C J=2 Oblate top A = B > C G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 25.
22 Asymmetric Rotor Rotational Energy Levels Asymmetric rotor molecules are between the prolate and oblate limits. The closeness of a given molecule to the prolate or oblate limts is measured by Ray s asymmetry parameter κ (kappa) defined as Zare, p 276 2B # A # C A#C The prolate limit is κ = -1 and in the oblate limit κ = 1. K is no longer a good quantum number, but! Kprolate = K-1 and Koblate = K+1 can be defined from the correlation diagram at right. The asymmetric rotor levels are labeled as "= J K "1 K +1 Most molecules are near the prolate limit, and so their spectra resemble prolate symmetric rotor spectra with additional asymmetry splittings. (Kappa)
23 Symmetric Top Selection Rules Example: a prolate symmetric rotor G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 417.
24 A Parallel Infrared Band of a Prolate Symmetric Rotor G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 418.
25 A Perpendicular Infrared Band of a Prolate Symmetric Rotor G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 425.
Ch 125a Problem Set 1
Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract
More informationRotations and vibrations of polyatomic molecules
Rotations and vibrations of polyatomic molecules When the potential energy surface V( R 1, R 2,..., R N ) is known we can compute the energy levels of the molecule. These levels can be an effect of: Rotation
More informationLecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor
Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that
More informationMolecular energy levels and spectroscopy
Molecular energy levels and spectroscopy 1. Translational energy levels The translational energy levels of a molecule are usually taken to be those of a particle in a three-dimensional box: n x E(n x,n
More informationChemistry 881 Lecture Topics Fall 2001
Chemistry 881 Lecture Topics Fall 2001 Texts PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon MATHEMATICS for PHYSICAL CHEMISTRY, Mortimer i. Mathematics Review (M, Chapters 1,2,3 & 4; M&S,
More informationSpectra of Atoms and Molecules. Peter F. Bernath
Spectra of Atoms and Molecules Peter F. Bernath New York Oxford OXFORD UNIVERSITY PRESS 1995 Contents 1 Introduction 3 Waves, Particles, and Units 3 The Electromagnetic Spectrum 6 Interaction of Radiation
More informationChemistry 483 Lecture Topics Fall 2009
Chemistry 483 Lecture Topics Fall 2009 Text PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon A. Background (M&S,Chapter 1) Blackbody Radiation Photoelectric effect DeBroglie Wavelength Atomic
More informationPHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions
PHYS851 Quantum Mechanics I, Fall 009 HOMEWORK ASSIGNMENT 10: Solutions Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum Some Key Concepts: Angular
More informationTime part of the equation can be separated by substituting independent equation
Lecture 9 Schrödinger Equation in 3D and Angular Momentum Operator In this section we will construct 3D Schrödinger equation and we give some simple examples. In this course we will consider problems where
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationCHAPTER 8 The Quantum Theory of Motion
I. Translational motion. CHAPTER 8 The Quantum Theory of Motion A. Single particle in free space, 1-D. 1. Schrodinger eqn H ψ = Eψ! 2 2m d 2 dx 2 ψ = Eψ ; no boundary conditions 2. General solution: ψ
More informationLecture 4: Polyatomic Spectra
Lecture 4: Polyatomic Spectra 1. From diatomic to polyatomic Ammonia molecule A-axis. Classification of polyatomic molecules 3. Rotational spectra of polyatomic molecules N 4. Vibrational bands, vibrational
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationQuantum mechanics (QM) deals with systems on atomic scale level, whose behaviours cannot be described by classical mechanics.
A 10-MINUTE RATHER QUICK INTRODUCTION TO QUANTUM MECHANICS 1. What is quantum mechanics (as opposed to classical mechanics)? Quantum mechanics (QM) deals with systems on atomic scale level, whose behaviours
More informationPAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 16 (CLASSIFICATION OF MOLECULES)
Subject Chemistry Paper No and Title Module No and Title Module Tag 8: Physical Spectroscopy 16: Classification of Molecules CHE_P8_M16 TABLE OF CONTENTS 1. Learning Outcomes. Introduction 3. Classification
More informationPhysics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I
Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular
More informationFun With Carbon Monoxide. p. 1/2
Fun With Carbon Monoxide p. 1/2 p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results C V (J/K-mole) 35 30 25
More informationPhysics 70007, Fall 2009 Answers to Final Exam
Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,
More informationLecture 10. Central potential
Lecture 10 Central potential 89 90 LECTURE 10. CENTRAL POTENTIAL 10.1 Introduction We are now ready to study a generic class of three-dimensional physical systems. They are the systems that have a central
More informationeigenvalues eigenfunctions
Born-Oppenheimer 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 informationReading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom.
Chemistry 356 017: Problem set No. 6; Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom. The H atom involves spherical coordinates and angular momentum, which leads to the shapes
More informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationAngular Momentum Algebra
Angular Momentum Algebra Chris Clark August 1, 2006 1 Input We will be going through the derivation of the angular momentum operator algebra. The only inputs to this mathematical formalism are the basic
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationRotations in Quantum Mechanics
Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific
More informationAngular momentum. Quantum mechanics. Orbital angular momentum
Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular
More informationRotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.
Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect
More informationChemistry 795T. NC State University. Lecture 4. Vibrational and Rotational Spectroscopy
Chemistry 795T Lecture 4 Vibrational and Rotational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule
More informationAddition of Angular Momenta
Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed
More informationChapter 12. Linear Molecules
Chapter 1. Linear Molecules Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (1998), Chap. 17. 1.1 Rotational Degrees of Freedom For a linear molecule, it is customary
More informationPhysics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I
Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from
More informationExercises 16.3a, 16.5a, 16.13a, 16.14a, 16.21a, 16.25a.
SPECTROSCOPY Readings in Atkins: Justification 13.1, Figure 16.1, Chapter 16: Sections 16.4 (diatomics only), 16.5 (omit a, b, d, e), 16.6, 16.9, 16.10, 16.11 (omit b), 16.14 (omit c). Exercises 16.3a,
More informationAngular momentum & spin
Angular momentum & spin January 8, 2002 1 Angular momentum Angular momentum appears as a very important aspect of almost any quantum mechanical system, so we need to briefly review some basic properties
More informationMore On Carbon Monoxide
More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations of CO 1 / 26 More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations
More informationLecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators
Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform
More informationQuantum Mechanics of Molecular Structures
Quantum Mechanics of Molecular Structures Bearbeitet von Kaoru Yamanouchi 1. Auflage 2013. Buch. xiv, 267 S. Hardcover ISBN 978 3 642 32380 5 Format (B x L): 15,5 x 23,5 cm Gewicht: 590 g Weitere Fachgebiete
More information26 Group Theory Basics
26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.
More informationSimple Harmonic Oscillator
Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in
More informationThe one and three-dimensional particle in a box are prototypes of bound systems. As we
6 Lecture 10 The one and three-dimensional particle in a box are prototypes of bound systems. As we move on in our study of quantum chemistry, we'll be considering bound systems that are more and more
More informationVibrations and Rotations of Diatomic Molecules
Chapter 6 Vibrations and Rotations of Diatomic Molecules With the electronic part of the problem treated in the previous chapter, the nuclear motion shall occupy our attention in this one. In many ways
More informationQM and Angular Momentum
Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that
More informationLecture 7. More dimensions
Lecture 7 More dimensions 67 68 LECTURE 7. MORE DIMENSIONS 7.1 Introduction In this lecture we generalize the concepts introduced so far to systems that evolve in more than one spatial dimension. While
More informationHarmonic Oscillator with raising and lowering operators. We write the Schrödinger equation for the harmonic oscillator in one dimension as follows:
We write the Schrödinger equation for the harmonic oscillator in one dimension as follows: H ˆ! = "!2 d 2! + 1 2µ dx 2 2 kx 2! = E! T ˆ = "! 2 2µ d 2 dx 2 V ˆ = 1 2 kx 2 H ˆ = ˆ T + ˆ V (1) where µ is
More informationSt Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial:
St Hugh s 2 nd Year: Quantum Mechanics II Reading The following sources are recommended for this tutorial: The key text (especially here in Oxford) is Molecular Quantum Mechanics, P. W. Atkins and R. S.
More informationChapter 4. Q. A hydrogen atom starts out in the following linear combination of the stationary. (ψ ψ 21 1 ). (1)
Tor Kjellsson Stockholm University Chapter 4 4.5 Q. A hydrogen atom starts out in the following linear combination of the stationary states n, l, m =,, and n, l, m =,, : Ψr, 0 = ψ + ψ. a Q. Construct Ψr,
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationNuclear models: Collective Nuclear Models (part 2)
Lecture 4 Nuclear models: Collective Nuclear Models (part 2) WS2012/13: Introduction to Nuclear and Particle Physics,, Part I 1 Reminder : cf. Lecture 3 Collective excitations of nuclei The single-particle
More information20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R
20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationNPTEL/IITM. Molecular Spectroscopy Lecture 2. Prof.K. Mangala Sunder Page 1 of 14. Lecture 2 : Elementary Microwave Spectroscopy. Topics.
Lecture 2 : Elementary Microwave Spectroscopy Topics Introduction Rotational energy levels of a diatomic molecule Spectra of a diatomic molecule Moments of inertia for polyatomic molecules Polyatomic molecular
More information(3.1) Module 1 : Atomic Structure Lecture 3 : Angular Momentum. Objectives In this Lecture you will learn the following
Module 1 : Atomic Structure Lecture 3 : Angular Momentum Objectives In this Lecture you will learn the following Define angular momentum and obtain the operators for angular momentum. Solve the problem
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: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture
More informationLecture 10 Diatomic Vibration Spectra Harmonic Model
Chemistry II: Introduction to Molecular Spectroscopy Prof. Mangala Sunder Department of Chemistry and Biochemistry Indian Institute of Technology, Madras Lecture 10 Diatomic Vibration Spectra Harmonic
More informationVibrational states of molecules. Diatomic molecules Polyatomic molecules
Vibrational states of molecules Diatomic molecules Polyatomic molecules Diatomic molecules V v 1 v 0 Re Q Birge-Sponer plot The solution of the Schrödinger equation can be solved analytically for the
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationLecture 19 (Nov. 15, 2017)
Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Chemistry 3502/4502 Final Exam Part I May 14, 2005 1. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle (e) The
More informationVibrational and Rotational Analysis of Hydrogen Halides
Vibrational and Rotational Analysis of Hydrogen Halides Goals Quantitative assessments of HBr molecular characteristics such as bond length, bond energy, etc CHEM 164A Huma n eyes Near-Infrared Infrared
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationPhysical Chemistry - Problem Drill 15: Vibrational and Rotational Spectroscopy
Physical Chemistry - Problem Drill 15: Vibrational and Rotational Spectroscopy No. 1 of 10 1. Internal vibration modes of a molecule containing N atoms is made up of the superposition of 3N-(5 or 6) simple
More informationStatistical Interpretation
Physics 342 Lecture 15 Statistical Interpretation Lecture 15 Physics 342 Quantum Mechanics I Friday, February 29th, 2008 Quantum mechanics is a theory of probability densities given that we now have an
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
More informationCHM Physical Chemistry II Chapter 12 - Supplementary Material. 1. Einstein A and B coefficients
CHM 3411 - Physical Chemistry II Chapter 12 - Supplementary Material 1. Einstein A and B coefficients Consider two singly degenerate states in an atom, molecule, or ion, with wavefunctions 1 (for the lower
More informationQuantum Mechanics: The Hydrogen Atom
Quantum Mechanics: The Hydrogen Atom 4th April 9 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen
More informationChemistry 532 Practice Final Exam Fall 2012 Solutions
Chemistry 53 Practice Final Exam Fall Solutions x e ax dx π a 3/ ; π sin 3 xdx 4 3 π cos nx dx π; sin θ cos θ + K x n e ax dx n! a n+ ; r r r r ˆL h r ˆL z h i φ ˆL x i hsin φ + cot θ cos φ θ φ ) ˆLy i
More informationCHAPTER 13 LECTURE NOTES
CHAPTER 13 LECTURE NOTES Spectroscopy is concerned with the measurement of (a) the wavelengths (or frequencies) at which molecules absorb/emit energy, and (b) the amount of radiation absorbed at these
More informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationRotational motion of a rigid body spinning around a rotational axis ˆn;
Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with
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 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 informationStructure of diatomic molecules
Structure of diatomic molecules January 8, 00 1 Nature of molecules; energies of molecular motions Molecules are of course atoms that are held together by shared valence electrons. That is, most of each
More informationMatrices of Dirac Characters within an irrep
Matrices of Dirac Characters within an irrep irrep E : 1 0 c s 2 c s D( E) D( C ) D( C ) 3 3 0 1 s c s c 1 0 c s c s D( ) D( ) D( ) a c b 0 1 s c s c 2 1 2 3 c cos( ), s sin( ) 3 2 3 2 E C C 2 3 3 2 3
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem
More information1 The postulates of quantum mechanics
1 The postulates of quantum mechanics The postulates of quantum mechanics were derived after a long process of trial and error. These postulates provide a connection between the physical world and the
More informationAngular Momentum. Classical. J r p. radius vector from origin. linear momentum. determinant form of cross product iˆ xˆ J J J J J J
Angular Momentum Classical r p p radius vector from origin linear momentum r iˆ ˆj kˆ x y p p p x y determinant form of cross product iˆ xˆ ˆj yˆ kˆ ˆ y p p x y p x p y x x p y p y x x y Copyright Michael
More informationThe Schrodinger Wave Equation (Engel 2.4) In QM, the behavior of a particle is described by its wave function Ψ(x,t) which we get by solving:
When do we use Quantum Mechanics? (Engel 2.1) Basically, when λ is close in magnitude to the dimensions of the problem, and to the degree that the system has a discrete energy spectrum The Schrodinger
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More informationobtained in Chapter 14 to this case requires that the E1 approximation
Chapter 15 The tools of time-dependent perturbation theory can be applied to transitions among electronic, vibrational, and rotational states of molecules. I. Rotational Transitions Within the approximation
More informationP. W. Atkins and R. S. Friedman. Molecular Quantum Mechanics THIRD EDITION
P. W. Atkins and R. S. Friedman Molecular Quantum Mechanics THIRD EDITION Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1997 Introduction and orientation 1 Black-body radiation 1 Heat capacities 2 The
More information2. Griffith s 4.19 a. The commutator of the z-component of the angular momentum with the coordinates are: [L z. + xyp x. x xxp y.
Physics Homework #8 Spring 6 Due Friday 5/7/6. Download the Mathematica notebook from the notes page of the website (http://minerva.union.edu/labrakes/physics_notes_s5.htm) or the homework page of the
More informationDiatomic Molecules. 7th May Hydrogen Molecule: Born-Oppenheimer Approximation
Diatomic Molecules 7th May 2009 1 Hydrogen Molecule: Born-Oppenheimer Approximation In this discussion, we consider the formulation of the Schrodinger equation for diatomic molecules; this can be extended
More informationThe Harmonic Oscillator: Zero Point Energy and Tunneling
The Harmonic Oscillator: Zero Point Energy and Tunneling Lecture Objectives: 1. To introduce simple harmonic oscillator model using elementary classical mechanics.. To write down the Schrodinger equation
More informationΨ t = ih Ψ t t. Time Dependent Wave Equation Quantum Mechanical Description. Hamiltonian Static/Time-dependent. Time-dependent Energy operator
Time Dependent Wave Equation Quantum Mechanical Description Hamiltonian Static/Time-dependent Time-dependent Energy operator H 0 + H t Ψ t = ih Ψ t t The Hamiltonian and wavefunction are time-dependent
More informationThe Simple Harmonic Oscillator
The Simple Harmonic Oscillator Asaf Pe er 1 November 4, 215 This part of the course is based on Refs [1] [3] 1 Introduction We return now to the study of a 1-d stationary problem: that of the simple harmonic
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More information( ). Expanding the square and keeping in mind that
One-electron atom in a Magnetic Field When the atom is in a magnetic field the magnetic moment of the electron due to its orbital motion and its spin interacts with the field and the Schrodinger Hamiltonian
More informationSymmetry and degeneracy
Symmetry and degeneracy Let m= degeneracy (=number of basis functions) of irrep i: From ( irrep) 1 one can obtain all the m irrep j by acting with off-diagonal R and orthogonalization. For instance in
More informationB7 Symmetry : Questions
B7 Symmetry 009-10: Questions 1. Using the definition of a group, prove the Rearrangement Theorem, that the set of h products RS obtained for a fixed element S, when R ranges over the h elements of the
More informationQuantum Mechanics for Scientists and Engineers
Quantum Mechanics for Scientists and Engineers Syllabus and Textbook references All the main lessons (e.g., 1.1) and units (e.g., 1.1.1) for this class are listed below. Mostly, there are three lessons
More informationP3317 HW from Lecture and Recitation 7
P3317 HW from Lecture 1+13 and Recitation 7 Due Oct 16, 018 Problem 1. Separation of variables Suppose we have two masses that can move in 1D. They are attached by a spring, yielding a Hamiltonian where
More informationThe energy of the emitted light (photons) is given by the difference in energy between the initial and final states of hydrogen atom.
Lecture 20-21 Page 1 Lectures 20-21 Transitions between hydrogen stationary states The energy of the emitted light (photons) is given by the difference in energy between the initial and final states of
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More informationThe Hydrogen Atom. Chapter 18. P. J. Grandinetti. Nov 6, Chem P. J. Grandinetti (Chem. 4300) The Hydrogen Atom Nov 6, / 41
The Hydrogen Atom Chapter 18 P. J. Grandinetti Chem. 4300 Nov 6, 2017 P. J. Grandinetti (Chem. 4300) The Hydrogen Atom Nov 6, 2017 1 / 41 The Hydrogen Atom Hydrogen atom is simplest atomic system where
More informationAssignment 11 (C + C ) = (C + C ) = (C + C) i(c C ) ] = i(c C) (AB) = (AB) = B A = BA 0 = [A, B] = [A, B] = (AB BA) = (AB) AB
Arfken 3.4.6 Matrix C is not Hermition. But which is Hermitian. Likewise, Assignment 11 (C + C ) = (C + C ) = (C + C) [ i(c C ) ] = i(c C ) = i(c C) = i ( C C ) Arfken 3.4.9 The matrices A and B are both
More informationUGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM PH 05 PHYSICAL SCIENCE TEST SERIES # 1. Quantum, Statistical & Thermal Physics
UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM BOOKLET CODE SUBJECT CODE PH 05 PHYSICAL SCIENCE TEST SERIES # Quantum, Statistical & Thermal Physics Timing: 3: H M.M: 00 Instructions. This test
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More informationIntro/Review of Quantum
Intro/Review of Quantum QM-1 So you might be thinking I thought I could avoid Quantum Mechanics?!? Well we will focus on thermodynamics and kinetics, but we will consider this topic with reference to the
More information