The Harmonic Oscillator: Zero Point Energy and Tunneling
|
|
- Lee Mitchell
- 6 years ago
- Views:
Transcription
1 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 for harmonic oscillator and describe the solutions. This module will not provide the method of solving the Schrodinger equation. 3. To express the energy levels of the harmonic oscillator in terms of the mass of the oscillator, the spring (force) constant and quantum numbers. 4. To plot the harmonic oscillator wave functions and their squares and identify the different sections of area under the square of the functions as probability of finding out the oscillator in that region. 5. To identify regions of the squares of wave functions beyond the classical potential energy (negative kinetic energy) region and associate with quantum mechanical tunneling. 6. To present an elementary summary of two dimensional and three dimensional isotropic harmonic oscillators using quantum mechanics. Lecture outcomes: 1. You will be able to solve for the vibrational frequencies of a simple diatomic molecule such as HCl, CO, NO using harmonic oscillator approximation.. You will be able to calculate bond force constants for different molecules using the harmonic oscillator approximation and arrange the molecule in terms of increasing stability. 3. You will be able to calculate the most probable vibrational coordinate values in which the molecule is likely to exist at very low temperatures. 4. By reading additional materials provided or pointed out in the last section, you will observe that harmonic model does not predict molecular dissociation at high temperatures or with the help of radiation, through vibrational motion, and therefore this model is only a preliminary account of vibrational spectroscopic concepts. Molecules do dissociate and through vibrational motion of the atoms around a bond. This is a deficiency which will hopefully prompt you to read more.
2 Lecture summary 1. The classical mechanical energy of the harmonic oscillator is given.. The Hamiltonian for the harmonic oscillator is given from the classical expression and the Schrödinger equation for the oscillator is written down. The equation is not solved in detail here. It is given in a supplement to this lecture. 3. The solutions, namely the eigenfunctions and eigenvalues of the Schrödinger equation, are described. 4. Plots of eigenvalues, eigenfunctions and the probabilities are included. A brief summary of why the study of harmonic oscillator is important for chemists and what new results that one gets from the analysis are indicated. 1. Classical Mechanics of the Oscillator: A simple harmonic oscillator is an example for small-amplitude motion of a mass point about an equilibrium point. Two flash movies of a grandfather clock and the diatomic molecule vibrating with small amplitudes should be included. They can be clipped from the adobe pdf file in the same folder. The restoring force F on the oscillating species is proportional to the magnitude of its displacement from equilibrium where the force is zero; the directions of the force and the displacement are opposite to each other. Put the diagram of the equilibrium position, the displacement and the restoring force be indicated. There are two pictures below the animations in the pdf file that can be put in here. The color background needs to be edited to blue as was done in earlier lectures. The restoring force on the oscillator undergoing harmonic motion is proportional to the displacement of the oscillator from its equilibrium position. For a diatomic molecule, the equilibrium distance is not zero but the bond distance at zero Kelvin in the limit of no vibration. The restoring force is therefore proportional to the difference between the instantaneous value for the bond distance (x) and the equilibrium bond distance x e. The proportionality constant k is known as the spring constant or the force constant. Formally
3 F x or F x( xx e ) F kx or F kx The restoring force of the harmonic oscillator for any given displacement is also equal to the negative derivative of the potential energy of the oscillator for that displacement, Hence it can be calculated immediately as F k x or F k x V x ; V(x) 1 kx constant or 1 kx constant The constant is the value of the potential energy in the absence of any displacement and in the case of molecule it is the potential energy of the molecule with no vibration. Clearly, these may not be zero, but for the study of harmonic oscillators in what follows, we ignore them as they contribute constant values to all our calculation. They will be set to zero. We shall use the following expression for the potential energy with the understanding that x stands for xalso in the case of molecular motion. 1 V( x) kx We can plot the potential energy as a function of the displacement and it is an even function and is a parabola centred at x 0A larger value of k means that the restoring force is high even for small displacement. It means, for a diatomic molecule, a stronger bond. A smaller value of k means for example, a weaker bond for an oscillating diatomic molecule. Illustrate the different values of k and the shape of the parabola with a figure. Please ask me. I have not included this in this directory. The kinetic energy of the oscillator is T 1 p mx m where x is the velocity of the oscillator with mass m and momentum p. For a diatomic molecule with atoms of mass m 1 and m, the mass of the oscillator is the reduced mass of the molecule,
4 1 1 1 mm ; 1 m m m m m m 1 1 The Hamiltonian (in classical mechanics, the total energy) for the harmonic oscillator therefore, is p 1 H T V kx m This is a classical mechanical expression. Graphs are given below for different total energies of the oscillator. From the graphs, it is possible to obtain precise values for both the kinetic energy and the potential energy of the harmonic oscillator at any point during its motion and for any value of the total energy of oscillator. It is also important to note that the oscillator can have any energy depending upon the amplitude of vibration and it can be measured. All these possibilities are shown to be false in the realm of quantum mechanics.. The Schrödinger equation for a harmonic oscillator To describe the harmonic oscillator using quantum mechanics, use the operator form for p; this has been done earlier in the box problem; the quantum mechanical Hamiltonian (which is an operator) is, therefore, Hˆ d 1 kx mdx Solving the harmonics oscillator problem in quantum mechanics means that we obtain eigenvalues and eigenvectors for the oscillator by solving the Schrödinger equation d 1 mdx Hˆ kx E. We don't solve the equation here. We discuss only the solutions. A supplement describes the method of solving the equation. 3. Solutions of the Schrödinger equation for a harmonic oscillator: The solutions are constructed using Hermite polynomials and an exponential (Gaussian) function (In the process of solving the differential equation above, one gets the Hermite s
5 differential equation whose solutions are known as Hermite polynomials.) The polynomials are infinite in number and are classified as polynomial of first order, second order, third order etc. as given below: Denote Hermite polynomials (functions of a dimensionless coordinate x) by Hv ( x), v 1,, 3,... The functional forms for the polynomials are given below for the first few of them. H ( x) 1 0 H ( x) x 1 ( ) 4 3 3( ) ( ) ( ) ( ) ( ) H x x H x x x H x x x H x x x x H x x x x H x x x x x H ( x) 56x 3594x 13, 440x 13, 440x 1680 There is a recursion relation between these polynomials which can be used to generate the above ones as well as higher ones if necessary. The relation is given here, H ( x) xh ( x) vh ( x) v1 v v1 The harmonic oscillator energies (eigenvalues) and wave functions are given as E v k 1 v m 1 1 k h v ; ; v 0, 1,, m x / ( x) N H x e, v v v km and N v 1 v v! 1/4
6 Important: The harmonic oscillator energies are quantized and are equidistant. This statement means that the difference between any two adjacent energy levels is the same for all energy levels. (The energy levels are equally spaced). E 1 E 0 h E E 1 E 3 E E 4 E 3 E v1 E v h Also, the lowest value of quantum number allowed is 0, unlike the particle in a box. Plots of eigenvalues, eigenfunctions and probabilities Label all the plots with axis and quantum numbers.
7 The levels are equidistant as you will find by substituting 0,1,,..., in the formula for the eigenvalues given above. The harmonic oscillator model is quite important in molecular spectroscopy /infrared spectroscopy to be studied later. Important: The space available for the harmonic oscillator is all possible values of x, x to Therefore, if we want the wave functions to contain probability information in the same way we did with the particle in a box or the particle in a ring, we must ensure that the
8 wave function vanishes at the boundaries, namely at x and at x. The wave function will have to be finite in the entire region in between, and must have a unique value everywhere. Also the squares of the wave function must be such that when we add (x) dx for all values of x, i.e., we do the sum (x) dx, we must get the result of unity, since the probability of finding the oscillator anywhere in all its allowed region must be unity. The solutions for the eigenfunctions given above satisfy all of these conditions, namely, x / ( x) N H x e, where N v v v v 1 v v! 1/4 km and x x dx*( x) ( x) x ( Nv) dxexp( x ) Hv( x) x 1 Problem: Verify this by substituting the first few Hermite polynomials in the above equation. Significance of the solutions. (Why is harmonic oscillator important?) 1. The quantum mechanical harmonic oscillator has some energy even when the quantum number is zero. (i.e., it cannot have zero energy of vibration). Classically an oscillator having zero energy does not oscillate!! A state of absolute rest is possible, however, only in classical systems. Zero point energy (referring to the above) is a purely quantum mechanical concept and is fundamentally important for molecules and solids.
9 . The boundary for an oscillator is endless. What does the potential look like? It is a parabola and the classical oscillator can ONLY oscillate between the two ends of the parabola. The total energy of the oscillator is all potential energy at the turning points and all kinetic energy midway between the two points. However, in quantum mechanics the wave function outside of this classical parabola can be nonzero for all values of x, though very small. Thus, the square of the wave function which represents probability density is nonzero everywhere, except for a finite number of points. It goes to zero as x tends towards infinity. This is often stated as that quantum mechanical systems can be found in regions where they are strictly forbidden if classical mechanics were applicable. This is also known as quantum mechanical tunneling. Though tunneling can often be explained by solution of the Schrödinger equation for particles in one dimensional finite potential barrier models in physics, chemistry students are introduced to tunneling from harmonic oscillator model. It is a finite potential barrier model anyway. Experimentally tunneling phenomena have been amply demonstrated. The invention of scanning tunneling electron microscope which was credited with a Nobel Prize in physics amply demonstrates the idea. which are forbidden by classical mechanics.. Squares of Harmonic oscillator wave functions. The plots of squares of wave functions above are worth analyzing. The fraction of the area under the square of the wave function in the middle region of the potential curve decreases with increasing total energy, so the harmonic oscillator behaves more and more classically with increasing energy of the oscillator. Also the square of the wavefunction tends towards a maximum at the turning points for larger and larger values of total energy meaning that the likelihood of finding the oscillator near the turning point region increases with energy. Thus the oscillator is more likely to be found where its motion is arrested as would one expect classically. For small values of energy you notice that the oscillator is more likely to be found in the middle of the potential well. (The wave functions associated with lower energies have peaks in the centre of the parabolic curve. This is against classical intuition as it is in these regions that the oscillator has the maximum kinetic energy, classically speaking, and is least likely to be spotted!!
10 3. Neither the kinetic energy alone nor the potential energy alone can be determined exactly for the harmonic oscillator. Only average values can be obtained through many measurements for which a theoretical limiting value can be provided. Mathematically this can be stated as follows: the kinetic energy operator does not commute with the potential energy operator and therefore eigenvalues for both cannot be obtained simultaneously. 4. The diatomic molecular vibrational energy is quantized and the simplest model above explains the basic features of the vibrational spectra of most stable molecules. Replace the mass of the oscillator by the reduced mass of the diatomic molecule and the connection between the two systems is established immediately.
Lecture 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 informationHarmonic Oscillator Eigenvalues and Eigenfunctions
Chemistry 46 Fall 217 Dr. Jean M. Standard October 4, 217 Harmonic Oscillator Eigenvalues and Eigenfunctions The Quantum Mechanical Harmonic Oscillator The quantum mechanical harmonic oscillator in one
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 informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Schrödinger Wave Equation Expectation Values Infinite Square-Well Potential Finite Square-Well Potential Three-Dimensional Infinite-Potential
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
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 informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 9, February 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 9, February 8, 2006 The Harmonic Oscillator Consider a diatomic molecule. Such a molecule
More informationAe ikx Be ikx. Quantum theory: techniques and applications
Quantum theory: techniques and applications There exist three basic modes of motion: translation, vibration, and rotation. All three play an important role in chemistry because they are ways in which molecules
More information1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2
15 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2 2mdx + 1 2 2 kx2 (15.1) where k is the force
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 information(Refer Slide Time: 1:20) (Refer Slide Time: 1:24 min)
Engineering Chemistry - 1 Prof. K. Mangala Sunder Department of Chemistry Indian Institute of Technology, Madras Lecture - 5 Module 1: Atoms and Molecules Harmonic Oscillator (Continued) (Refer Slide Time:
More informationLecture 6 Quantum Mechanical Systems and Measurements
Lecture 6 Quantum Mechanical Systems and Measurements Today s Program: 1. Simple Harmonic Oscillator (SHO). Principle of spectral decomposition. 3. Predicting the results of measurements, fourth postulate
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 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 informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More information5.1 Classical Harmonic Oscillator
Chapter 5 Harmonic Oscillator 5.1 Classical Harmonic Oscillator m l o l Hooke s Law give the force exerting on the mass as: f = k(l l o ) where l o is the equilibrium length of the spring and k is the
More information1.3 Harmonic Oscillator
1.3 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: H = h2 d 2 2mdx + 1 2 2 kx2 (1.3.1) where k is the force
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 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 informationVibrational motion. Harmonic oscillator ( 諧諧諧 ) - A particle undergoes harmonic motion. Parabolic ( 拋物線 ) (8.21) d 2 (8.23)
Vibrational motion Harmonic oscillator ( 諧諧諧 ) - A particle undergoes harmonic motion F == dv where k Parabolic V = 1 f k / dx = is Schrodinge h m d dx ψ f k f x the force constant x r + ( 拋物線 ) 1 equation
More informationQuantum Mechanics: Vibration and Rotation of Molecules
Quantum Mechanics: Vibration and Rotation of Molecules 8th April 2008 I. 1-Dimensional Classical Harmonic Oscillator The classical picture for motion under a harmonic potential (mass attached to spring
More informationIntroduction to Vibrational Spectroscopy
Introduction to Vibrational Spectroscopy Harmonic oscillators The classical harmonic oscillator The uantum mechanical harmonic oscillator Harmonic approximations in molecular vibrations Vibrational spectroscopy
More informationChemistry 2. Lecture 1 Quantum Mechanics in Chemistry
Chemistry 2 Lecture 1 Quantum Mechanics in Chemistry Your lecturers 8am Assoc. Prof Timothy Schmidt Room 315 timothy.schmidt@sydney.edu.au 93512781 12pm Assoc. Prof. Adam J Bridgeman Room 222 adam.bridgeman@sydney.edu.au
More information* = 2 = Probability distribution function. probability of finding a particle near a given point x,y,z at a time t
Quantum Mechanics Wave functions and the Schrodinger equation Particles behave like waves, so they can be described with a wave function (x,y,z,t) A stationary state has a definite energy, and can be written
More informationThe Quantum Harmonic Oscillator
The Classical Analysis Recall the mass-spring system where we first introduced unforced harmonic motion. The DE that describes the system is: where: Note that throughout this discussion the variables =
More information2m dx 2. The particle in a one dimensional box (of size L) energy levels are
Name: Chem 3322 test #1 solutions, out of 40 marks I want complete, detailed answers to the questions. Show all your work to get full credit. indefinite integral : sin 2 (ax)dx = x 2 sin(2ax) 4a (1) with
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 informationChem 452 Mega Practice Exam 1
Last Name: First Name: PSU ID #: Chem 45 Mega Practice Exam 1 Cover Sheet Closed Book, Notes, and NO Calculator The exam will consist of approximately 5 similar questions worth 4 points each. This mega-exam
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 informationAppendix A. The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System
Appendix A The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System Real quantum mechanical systems have the tendency to become mathematically
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 10, February 10, / 4
Chem 350/450 Physical Chemistry II (Quantum Mechanics 3 Credits Spring Semester 006 Christopher J. Cramer Lecture 10, February 10, 006 Solved Homework We are asked to find and for the first two
More informationThe Schrödinger Equation in One Dimension
The Schrödinger Equation in One Dimension Introduction We have defined a comple wave function Ψ(, t) for a particle and interpreted it such that Ψ ( r, t d gives the probability that the particle is at
More informationModel for vibrational motion of a diatomic molecule. To solve the Schrödinger Eq. for molecules, make the Born- Oppenheimer Approximation:
THE HARMONIC OSCILLATOR Features Example of a problem in which V depends on coordinates Power series solution Energy is quantized because of the boundary conditions Model for vibrational motion of a diatomic
More informationLecture 12: Particle in 1D boxes & Simple Harmonic Oscillator
Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator U(x) E Dx y(x) x Dx Lecture 12, p 1 Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound
More informationTHEORY OF MOLECULE. A molecule consists of two or more atoms with certain distances between them
THEORY OF MOLECULE A molecule consists of two or more atoms with certain distances between them through interaction of outer electrons. Distances are determined by sum of all forces between the atoms.
More informationLecture 3 Dynamics 29
Lecture 3 Dynamics 29 30 LECTURE 3. DYNAMICS 3.1 Introduction Having described the states and the observables of a quantum system, we shall now introduce the rules that determine their time evolution.
More informationQuantum Harmonic Oscillator
Quantum Harmonic Oscillator Chapter 13 P. J. Grandinetti Chem. 4300 Oct 20, 2017 P. J. Grandinetti (Chem. 4300) Quantum Harmonic Oscillator Oct 20, 2017 1 / 26 Kinetic and Potential Energy Operators Harmonic
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 informationA Brief Introduction to the Quantum Harmonic Oscillator
A Brief Introduction to the Quantum Harmonic Oscillator Salvish Goomanee King s College London, UK Email address: salvish.goomanee@kcl.ac.uk Abstract In this short paper, a very brief introduction of the
More informationPHYS 3313 Section 001 Lecture #20
PHYS 3313 Section 001 ecture #0 Monday, April 10, 017 Dr. Amir Farbin Infinite Square-well Potential Finite Square Well Potential Penetration Depth Degeneracy Simple Harmonic Oscillator 1 Announcements
More informationNPTEL/IITM. Molecular Spectroscopy Lectures 1 & 2. Prof.K. Mangala Sunder Page 1 of 15. Topics. Part I : Introductory concepts Topics
Molecular Spectroscopy Lectures 1 & 2 Part I : Introductory concepts Topics Why spectroscopy? Introduction to electromagnetic radiation Interaction of radiation with matter What are spectra? Beer-Lambert
More informationPhysical Chemistry II Exam 2 Solutions
Chemistry 362 Spring 2017 Dr Jean M Standard March 10, 2017 Name KEY Physical Chemistry II Exam 2 Solutions 1) (14 points) Use the potential energy and momentum operators for the harmonic oscillator to
More informationChemistry 3502/4502. Exam I Key. September 19, ) This is a multiple choice exam. Circle the correct answer.
D Chemistry 350/450 Exam I Key September 19, 003 1) This is a multiple choice exam. Circle the correct answer. ) There is one correct answer to every problem. There is no partial credit. 3) A table of
More information6. Qualitative Solutions of the TISE
6. Qualitative Solutions of the TISE Copyright c 2015 2016, Daniel V. Schroeder Our goal for the next few lessons is to solve the time-independent Schrödinger equation (TISE) for a variety of one-dimensional
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 informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 7, February 1, 2006
Chem 350/450 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 006 Christopher J. Cramer ecture 7, February 1, 006 Solved Homework We are given that A is a Hermitian operator such that
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationPage 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19
Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page
More informationThe Simple Harmonic Oscillator
The Simple Harmonic Oscillator Your introductory physics textbook probably had a chapter or two discussing properties of Simple Harmonic Motion (SHM for short). Your modern physics textbook mentions SHM,
More informationProblems and Multiple Choice Questions
Problems and Multiple Choice Questions 1. A momentum operator in one dimension is 2. A position operator in 3 dimensions is 3. A kinetic energy operator in 1 dimension is 4. If two operator commute, a)
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 informationPhysical Chemistry Laboratory II (CHEM 337) EXPT 9 3: Vibronic Spectrum of Iodine (I2)
Physical Chemistry Laboratory II (CHEM 337) EXPT 9 3: Vibronic Spectrum of Iodine (I2) Obtaining fundamental information about the nature of molecular structure is one of the interesting aspects of molecular
More informationApplications of Quantum Theory to Some Simple Systems
Applications of Quantum Theory to Some Simple Systems Arbitrariness in the value of total energy. We will use classical mechanics, and for simplicity of the discussion, consider a particle of mass m moving
More informationQuantum Mechanics. The Schrödinger equation. Erwin Schrödinger
Quantum Mechanics The Schrödinger equation Erwin Schrödinger The Nobel Prize in Physics 1933 "for the discovery of new productive forms of atomic theory" The Schrödinger Equation in One Dimension Time-Independent
More informationQUANTUM CHEMISTRY PROJECT 2: THE FRANCK CONDON PRINCIPLE
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 4, 2017 OUTLINE QUANTUM CHEMISTRY PROJECT 2: THE FRANCK CONDON PRINCIPLE This project deals with the Franck-Condon Principle, electronic transitions
More informationREVIEW: The Matching Method Algorithm
Lecture 26: Numerov Algorithm for Solving the Time-Independent Schrödinger Equation 1 REVIEW: The Matching Method Algorithm Need for a more general method The shooting method for solving the time-independent
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 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 informationChapter 6 Vibrational Spectroscopy
Chapter 6 Vibrational Spectroscopy As with other applications of symmetry and group theory, these techniques reach their greatest utility when applied to the analysis of relatively small molecules in either
More informationChemistry 3502/4502. Exam I. September 19, ) This is a multiple choice exam. Circle the correct answer.
D Chemistry 350/450 Exam I September 9, 003 ) This is a multiple choice exam. Circle the correct answer. ) There is one correct answer to every problem. There is no partial credit. 3) A table of useful
More informationEigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator.
PHYS208 spring 2008 Eigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator. 07.02.2008 Adapted from the text Light - Atom Interaction PHYS261 autumn 2007 Go to list of topics
More informationHarmonic oscillator - Vibration energy of molecules
Harmonic oscillator - Vibration energy of molecules The energy of a molecule is approximately the sum of the energies of translation of the electrons (kinetic energy), of inter-atomic vibration, of rotation
More informationQuantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :
Student Selected Module 2005/2006 (SSM-0032) 17 th November 2005 Quantum Mechanics Outline : Review of Previous Lecture. Single Particle Wavefunctions. Time-Independent Schrödinger equation. Particle in
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules CHEM3006P or similar background knowledge is required for this course. This course has two parts: Part 1: Quantum Chemistry techniques for simulations of molecular
More informationMolecular orbitals, potential energy surfaces and symmetry
Molecular orbitals, potential energy surfaces and symmetry mathematical presentation of molecular symmetry group theory spectroscopy valence theory molecular orbitals Wave functions Hamiltonian: electronic,
More informationFrom The Picture Book of Quantum Mechanics, S. Brandt and H.D. Dahmen, 4th ed., c 2012 by Springer-Verlag New York.
1 Fig. 6.1. Bound states in an infinitely deep square well. The long-dash line indicates the potential energy V (x). It vanishes for d/2 < x < d/2 and is infinite elsewhere. Points x = ±d/2 are indicated
More informationLecture 6. Four postulates of quantum mechanics. The eigenvalue equation. Momentum and energy operators. Dirac delta function. Expectation values
Lecture 6 Four postulates of quantum mechanics The eigenvalue equation Momentum and energy operators Dirac delta function Expectation values Objectives Learn about eigenvalue equations and operators. Learn
More informationThe Sommerfeld Polynomial Method: Harmonic Oscillator Example
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic
More informationHarmonic Oscillator I
Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering
More informationIllustrating the Bohr Correspondence Principle
Illustrating the Bohr Correspondence Principle Glenn V. Lo Department of Physical Sciences Nicholls State University Thibodaux, LA 70310 phsc-gl@nicholls.edu Copyright 2002 by the Division of Chemical
More informationBrief review of Quantum Mechanics (QM)
Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt
More informationLecture-XXVI. Time-Independent Schrodinger Equation
Lecture-XXVI Time-Independent Schrodinger Equation Time Independent Schrodinger Equation: The time-dependent Schrodinger equation: Assume that V is independent of time t. In that case the Schrodinger equation
More informationECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline:
ECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline: Linearly Varying Potential Triangular Potential Well Time-Dependent Schrödinger Equation Things you should know when you leave Key
More informationvan Quantum tot Molecuul
10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based
More information8 Wavefunctions - Schrödinger s Equation
8 Wavefunctions - Schrödinger s Equation So far we have considered only free particles - i.e. particles whose energy consists entirely of its kinetic energy. In general, however, a particle moves under
More informationExperiment 6: Vibronic Absorption Spectrum of Molecular Iodine
Experiment 6: Vibronic Absorption Spectrum of Molecular Iodine We have already seen that molecules can rotate and bonds can vibrate with characteristic energies, each energy being associated with a particular
More informationLecture 2: simple QM problems
Reminder: http://www.star.le.ac.uk/nrt3/qm/ Lecture : simple QM problems Quantum mechanics describes physical particles as waves of probability. We shall see how this works in some simple applications,
More informationIntroduction. Quantum Mechanical Computation. By Sasha Payne N. Diaz. CHEMISTRY 475-L03 1 January 2035
Quantum Mechanical Computation By Sasha Payne N. Diaz CHEMISTRY 47-L 1 January Introduction Backgrounds for computational chemistry Understanding the behavior of materials at the atomic scale is fundamental
More informationRotation and vibration of Molecules
Rotation and vibration of Molecules Overview of the two lectures... 2 General remarks on spectroscopy... 2 Beer-Lambert law for photoabsorption... 3 Einstein s coefficients... 4 Limits of resolution...
More informationLecture #8: Quantum Mechanical Harmonic Oscillator
5.61 Fall, 013 Lecture #8 Page 1 Last time Lecture #8: Quantum Mechanical Harmonic Oscillator Classical Mechanical Harmonic Oscillator * V(x) = 1 kx (leading term in power series expansion of most V(x)
More information221A Lecture Notes Convergence of Perturbation Theory
A Lecture Notes Convergence of Perturbation Theory Asymptotic Series An asymptotic series in a parameter ɛ of a function is given in a power series f(ɛ) = f n ɛ n () n=0 where the series actually does
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 informationSIMPLE QUANTUM SYSTEMS
SIMPLE QUANTUM SYSTEMS Chapters 14, 18 "ceiiinosssttuu" (anagram in Latin which Hooke published in 1676 in his "Description of Helioscopes") and deciphered as "ut tensio sic vis" (elongation of any spring
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 informationApplied Nuclear Physics (Fall 2006) Lecture 3 (9/13/06) Bound States in One Dimensional Systems Particle in a Square Well
22.101 Applied Nuclear Physics (Fall 2006) Lecture 3 (9/13/06) Bound States in One Dimensional Systems Particle in a Square Well References - R. L. Liboff, Introductory Quantum Mechanics (Holden Day, New
More informationSample Quantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)
More informationChem 344 Final Exam Tuesday, Dec. 11, 2007, 3-?? PM
Chem 344 Final Exam Tuesday, Dec. 11, 2007, 3-?? PM Closed book exam, only pencils and calculators permitted. You may bring and use one 8 1/2 x 11" paper with anything on it. No Computers. Put all of your
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 #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box
561 Fall 017 Lecture #5 page 1 Last time: Lecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box 1-D Wave equation u x = 1 u v t * u(x,t): displacements as function of x,t * nd -order:
More informationQuantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 17 Dr. Jean M. Standard November 8, 17 Name KEY Quantum Chemistry Exam Solutions 1.) ( points) Answer the following questions by selecting the correct answer from the choices provided.
More informationProbability and Normalization
Probability and Normalization Although we don t know exactly where the particle might be inside the box, we know that it has to be in the box. This means that, ψ ( x) dx = 1 (normalization condition) L
More informationPHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101
PHY 114 A General Physics II 11 AM-1:15 PM TR Olin 101 Plan for Lecture 3 (Chapter 40-4): Some topics in Quantum Theory 1. Particle behaviors of electromagnetic waves. Wave behaviors of particles 3. Quantized
More informationPARABOLIC POTENTIAL WELL
APPENDIX E PARABOLIC POTENTIAL WELL An example of an extremely important class of one-dimensional bound state in quantum mechanics is the simple harmonic oscillator whose potential can be written as V(x)=
More informationAdvanced Quantum Mechanics, Notes based on online course given by Leonard Susskind - Lecture 8
Advanced Quantum Mechanics, Notes based on online course given by Leonard Susskind - Lecture 8 If neutrinos have different masses how do you mix and conserve energy Mass is energy. The eigenstates of energy
More informationAtkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 8: Quantum Theory: Techniques and Applications
Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas Chapter 8: Quantum Theory: Techniques and Applications TRANSLATIONAL MOTION wavefunction of free particle, ψ k = Ae ikx + Be ikx,
More informationSimple Harmonic Motion Practice Problems PSI AP Physics 1
Simple Harmonic Motion Practice Problems PSI AP Physics 1 Name Multiple Choice Questions 1. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the
More informationBorn-Oppenheimer Approximation
Born-Oppenheimer Approximation Adiabatic Assumption: Nuclei move so much more slowly than electron that the electrons that the electrons are assumed to be obtained if the nuclear kinetic energy is ignored,
More information