Quantum Final Project by Anustup Poddar and Cody Tripp 12/10/2013
|
|
- Bathsheba Shaw
- 5 years ago
- Views:
Transcription
1 Quantum Final Project by Anustup Poddar and Cody Tripp Introduction The Hamiltonian in the Schrӧdinger equation is the sum of a kinetic and potential energy operator. The Fourier grid Hamiltonian (FGH) method relies on the momentum representation of the kinetic energy operator and the coordinate representation for the potential energy operator. Kosloff 1 has shown that these ideas can be used to evaluate the expression ĤY. Kosloff represented both Y and ĤY as vectors whose components are the values of the function on a grid of points in coordinate space. In the paper we chose, Marston and Balint-Kurti 2 identify the matrix representation of Ĥ in this vector space. The calculation of the Ĥ matrix in this space is relatively simple. To calculate Ĥ, the potential is evaluated at the grid points and then a forward and reverse Fourier transform reduces the expression to a summation over cosine functions. This will be explained in the theory section of the paper. The FGH method is a special case of a discrete variable representation (DVR) originally introduced by Harris et al. in There have been numerous DVR techniques since Harris, but the FGH method has the advantage of simplicity over the other techniques. The simplicity of the process lies in the fact that the eigenfunctions of the Hamiltonian operator are generated directly as amplitudes of the wavefunctions on the grid points. This is in contrast to many other techniques that give the eigenfunctions as linear combinations of basis set functions. There are similarities to the variational methods we have done in class and the FGH method, which will become evident in the theory sections below. Theory The non-relativistic Hamiltonian is: th= t tp 2 T+V(tx)= +V(tx) 2m (1) We want to manipulate this Hamiltonian into a form that can be used in our matrix representation, but first some definitions are needed. Eqn. 2 is a standard orthogonality relationship in quantum mechanics which reduces to the Kronecker delta. Eqn. 2 is in terms of coordinates. < x x l>=d(x- x) l An identity relationship for momentum is also defined in eqn. 3. # ti k= k>< k dk (3) The transformation matrix elements between the coordinate and momentum representations can be derived as follows: (2)
2 tp=- i' 2 2x tpw k(x)='kw k(x) &- i' 2 2x W k(x)='kw k(x) &2W k(x)=ikw k(x)2x 2W k(x) & = ik2x W k(x) &W k(x)=e ikx (4) ow Ψ k (x) is the projection of the momentum vector, k, on the coordinate vector, x, represented as: W k(x)=< x k> The normalization constant,, can be calculated from the relation: # e ikx dx=d(k) Which gives =. Therefore, the forward and the reverse Fourier Transform can then be represented as: < x k>= 1 e ikx < k x>= 1 e -ikx The potential is the diagonal in the coordinate representation as shown in eqn x l V(tx) x2=v(x)d(x- x) l ow we can find out the expectation value of the Hamiltonian from eqn. 1 as: < x H t x l>=< x ( T+V(tx)) t x l> =< x T t x l>+<x V(tx) x l> &<x th l x >=< x t T l x >+V(x)d(x- l x) The identity relationship from eqn. 3 can now be inserted into eqn. 10. < x t H l x >=< x t T k>< k Since we have the eigenvalue equation as: G # J l x > dk+v(x)d(x- l x) (5) (6) (7 & 8) (9) (10) (11)
3 tt k>= T k k> eqn. 11 reduces to: < x th x l>= # < x k> T k< k x l> dk+v(x)d(x- x) l (12) (13) We now use a forward and reverse Fourier transform (eqns. 7 & 8) to further simplify the equation to exponential from: Discretization < x H t x l>= 1 e ik(x- x) l # T kdk+v(x)d(x- l The Fourier grid Hamiltonian method transforms the continuous range of coordinate values to a grid of discrete values, x i, where, x i =iδx. Here Δx is the uniform spacing between the grid points in coordinate space and i is the index number of the grid points. We use grid values for x. The total length of the coordinate space is Δx. As a result of this discretization, we have to re-define of the normalization condition of the wave function: } * # (x)}(x)dx=1 } * (x i) }(x i)dx=1 Dx } i 2 =1 (14) becomes i=1, or, i=1 (15) since Ψ i = Ψ(x i ) is the value of the wave function at the grid point i. The total length of the coordinate space represents the maximum wavelength and therefore the smallest frequency in the reciprocal momentum space. Therefore, we may say that the spacing between the grid points in momentum space, Δk, can be represented as x) = = (16) ow, quantum particles can either move to the right or to the left, i.e., positive or negative k. So here we consider that the momentum space is centered at k=0, with n grid values in both the positive and the negative directions, where 2n=-1. Therefore, we have a relation similar to our coordinate space, k=lδk, where l is the index number of the grid points. In the paper by Marston and Balint-Kurti the FGH method must employ an odd number of grid points,. In our new coordinate space we can use bra-ket notation to show the values of the wave function at specific grid points: < x i }>=}(x i)=}i (17)
4 Then the next task is to redefine orthogonality and identity relations for our new coordinate space: i=1 ti x= x i>dx<x i Dx<x i x j>=d (19) Armed with the needed machinery we now proceed to find the discretized analog of our Hamiltonian in eqn. 14. < x th x l>= 1 e ik(x-x) l # T kdk+v(x)d(x- l x) We use the relations k=lδk, eqn. 19, and the relation for kinetic energy, T k : (18) (20) Tl= & 2 k 2 = & 2 2m 2m (ltk) (21) We represent the kinetic energy as T l because we now use the l-index in our equation. We then convert the integration from negative infinity to positive infinity into a sum from n to +n: H =< x i H t x j> n = 1 e il Dk(x i - x j ) ' 2 G 2m (ldk) 2 JDk+ l=-n V(x i)d Dx We then use the relation between Δk and Δx (eqn. 16), to further simplify our equation to: (22) H = 1 or,h = 1 Dx S G n Tl l=-n +n e il2 r(i-j) Tl l=-n Dx X e [il(2 r Dx)(i-j)Dx] F I+ F I+ V(x i)d J V(x i)d Dx (23) We now use Euler s formula, = +, for both the positive and negative values of l. The sine terms being both positive and negative cancel out and with T 0 being zero (eqn. 21), we are left with a sum of cosines. H = 1 Dx or,h = 1 Dx n G 2cos(l(i-j)) T l+ V(x i)d J l=1 n G2 2cos(l(i-j))Tl+ V(x i)d J l=1 (24)
5 Using the above equation we can now find out the eigenvalues and amplitudes of the wavefunctions at the grid points utilizing the variational method. We use the identity relation (eqn. 18) and the fact that < > = (which is the amplitude of the wave function at the particular grid point i) to find a discrete analog of our wave function. i=1 }>=t I x }>= x i>dx<x i }> = x i>dx} i i=1 (25) The expectation value of the energy for the arbitrary wave function can now be written using the orthogonality relation (eqn. 19): } * i Dx<x i H x j>dx} j <} H }> E= = <} }> } * i Dx<x i x j>dx} j } * i DxH Dx} j = Dx } * i d } j (26) Then, a renormalized Hamiltonian matrix is defined for the wave function: H 0 = G2 n 2cos(l(i-j))Tl+ V(x i)d J= H Dx l=1 (27) Therefore, the expression of the expectation value of the energy changes: } * i DxH Dx} j E= Dx } * i d } j = } * i DxH } j } i * } jd = } * i H 0 } j } i * } jd (28) According to the variational method we minimize this energy, taking the derivatives with respect to the Ψ j (which here acts as our coefficients, unlike the actual coefficients of the wave functions in the normal variational principle) and as a result we get a set of secular equations: " H 0 - Emd % } m j=0 j (29) In the above equation, λ represents the quantum number of the energy level for the particular
6 system. The energies calculated are below the dissociation energy, V(x), of the system and that gives us the bound state energies. The eigenvector, Ψ l λ, gives us the amplitudes of the wave function at particular grid points. umerical Implementation The potential of the Morse curve is modeled by an equation of the form V= D[1- e ( - b (x-x e) ] 2 (30) where D has units of energy and is a scaling factor that dictates the horizontal asymptote the potential approaches as x. If VD is plotted, the Morse potential approaches 1 as x. This is the form of Figures 1-3 below. β is in units of inverse distance and sets the width of the Morse potential. The variable x e is in units of distance and is the equilibrium bond distance for the molecule of interest. These three parameters will change the shape of the Morse potential depending on the molecule of interest. The parameters for H 2 are given in Table 1. Table 1. Morse parameters for H 2 in atomic units (a.u.) D = a.u. = ev β = a.u. = x10 10 m -1 x e = a.u. = x10-10 m The Morse parameters, along with the grid size and maximum x-value, are the only inputs needed to get approximate values for the energies and wavefunction amplitudes. We have chosen a grid size of =129, which is what Marston uses in the paper. Only energies below the Morse dissociation energy are valid, so an estimate of the number of valid energy levels is needed: ymax= 2Dnb=17.42 (31) The maximum vibrational quantum number was taken to be 16 (the 17 th state). The maximum x-value was taken to be 1.5 times larger than the outer turning point of the 16 th eigenstate which ensures that all of the relevant features are captured. This value comes out to x The maximum value of x is then split into uniform steps: x i=i3x, 0# i# -1 (32) The energy levels for vibrational quantum numbers 0-16 and the deviation from the analytical values is shown in Table 2. When taken to 8 digits, the energy values calculated by Marston deviate from the analytical values by an average of 0.3 ppm. Our calculations had higher deviations. The smallest deviation was 9 ppm, while the largest was 102 ppm. The average of the deviations was 65 ppm. The deviation becomes progressively smaller as ν 16.
7 Although the calculated deviations were a factor of 200 larger than Marston s, the overall accuracy is still high, on the order of ppm. Table 2. Eigenvalue energies (in Hartree) calculated with a 129x129 grid comparing results with both Marston and the analytical Morse energies Poddar and Tripp Marston Vibrational Quantum umber Energy (a.u.) Deviation (ppm) Energy (a.u) Deviation (ppm) Exact Avg. Deviation (ppm) References (1) Kosloff, R.; Tal-Exer, H. A Direct Relaxation Method for Calculating Eigenfunctions and Eigenvalues of the Schrodinger Equation on a Grid. Chem. Phys. Lett. 1986, 127, (2) Marston, C. C.; Balint-Kurti, G. G. The Fourier Grid Hamiltonian Method for Bound State Eigenvalues and Eigenfunctions. J. Chem. Phys. 1989, 91, 3571.
8 (a) (b) Figure 1. H 2 wavefunctions for ν=0 with Morse potential and energy level: (a) from Marston 2 ;(b) reproduced using Marston s parameters. (a) (b) Figure 2. H 2 wavefunctions for ν=5 with Morse potential and energy level: (a) from Marston 2 ; (b) reproduced using Marston s parameters. (a) (b) Figure 3. H 2 wavefunctions for ν=15 with Morse potential and energy level: (a) from Marston 2 ; (b) reproduced using Marston s parameters
9 Supplemental Figures Deviation from Exact Energy Values for 129x129 Grid Error (ppm) Vibrational Quantum umber 600 Processing Time for Grid Size Time (s) Grid Length
10 Supplemental Figures 60 Average Percent Error in Calculated Energies 50 Average Percent Error Grid Length
11 Division of Labor Both Anustup and I worked closely on all aspects of the project.
PHYS-454 The position and momentum representations
PHYS-454 The position and momentum representations 1 Τhe continuous spectrum-a n So far we have seen problems where the involved operators have a discrete spectrum of eigenfunctions and eigenvalues.! n
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 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 informationPhysics 342 Lecture 17. Midterm I Recap. Lecture 17. Physics 342 Quantum Mechanics I
Physics 342 Lecture 17 Midterm I Recap Lecture 17 Physics 342 Quantum Mechanics I Monday, March 1th, 28 17.1 Introduction In the context of the first midterm, there are a few points I d like to make about
More information1 Infinite-Dimensional Vector Spaces
Theoretical Physics Notes 4: Linear Operators In this installment of the notes, we move from linear operators in a finitedimensional vector space (which can be represented as matrices) to linear operators
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 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 informationto the potential V to get V + V 0 0Ψ. Let Ψ ( x,t ) =ψ x dx 2
Physics 0 Homework # Spring 017 Due Wednesday, 4/1/17 1. Griffith s 1.8 We start with by adding V 0 to the potential V to get V + V 0. The Schrödinger equation reads: i! dψ dt =! d Ψ m dx + VΨ + V 0Ψ.
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 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 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 informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationdf(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation
Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations
More informationLecture 8. 1 Uncovering momentum space 1. 2 Expectation Values of Operators 4. 3 Time dependence of expectation values 6
Lecture 8 B. Zwiebach February 29, 206 Contents Uncovering momentum space 2 Expectation Values of Operators 4 Time dependence of expectation values 6 Uncovering momentum space We now begin a series of
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 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 informationWaves and the Schroedinger Equation
Waves and the Schroedinger Equation 5 april 010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form
More informationHomework Problem Set 3 Solutions
Chemistry 460 Dr. Jean M. Standard Homework Problem Set 3 Solutions 1. See Section 2-5 of Lowe and Peterson for more information about this example. When a particle experiences zero potential energy over
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about
More informationVibrationally Mediated Bond Selective Dissociative Chemisorption of HOD on Cu(111) Supporting Information
Submitted to Chem. Sci. 8/30/202 Vibrationally Mediated Bond Selective Dissociative Chemisorption of HOD on Cu() Supporting Information Bin Jiang,,2 Daiqian Xie,,a) and Hua Guo 2,a) Institute of Theoretical
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 informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
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 Electronic Structure Theory
Introduction to Electronic Structure Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2002 Last Revised: June 2003 1 Introduction The purpose of these
More information5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More informationCHEM 301: Homework assignment #5
CHEM 30: Homework assignment #5 Solutions. A point mass rotates in a circle with l =. Calculate the magnitude of its angular momentum and all possible projections of the angular momentum on the z-axis.
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 information( ) = 9φ 1, ( ) = 4φ 2.
Chemistry 46 Dr Jean M Standard Homework Problem Set 6 Solutions The Hermitian operator A ˆ is associated with the physical observable A Two of the eigenfunctions of A ˆ are and These eigenfunctions are
More informationPhysics 217 Problem Set 1 Due: Friday, Aug 29th, 2008
Problem Set 1 Due: Friday, Aug 29th, 2008 Course page: http://www.physics.wustl.edu/~alford/p217/ Review of complex numbers. See appendix K of the textbook. 1. Consider complex numbers z = 1.5 + 0.5i and
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 informationQuantum Mechanical Tunneling
Chemistry 460 all 07 Dr Jean M Standard September 8, 07 Quantum Mechanical Tunneling Definition of Tunneling Tunneling is defined to be penetration of the wavefunction into a classically forbidden region
More informationContinuous quantum states, Particle on a line and Uncertainty relations
Continuous quantum states, Particle on a line and Uncertainty relations So far we have considered k-level (discrete) quantum systems. Now we turn our attention to continuous quantum systems, such as a
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 informationSample Quantum Chemistry Exam 1 Solutions
Chemistry 46 Fall 217 Dr Jean M Standard September 27, 217 Name SAMPE EXAM Sample Quantum Chemistry Exam 1 Solutions 1 (24 points Answer the following questions by selecting the correct answer from the
More informationQuantum Physics Lecture 8
Quantum Physics ecture 8 Steady state Schroedinger Equation (SSSE): eigenvalue & eigenfunction particle in a box re-visited Wavefunctions and energy states normalisation probability density Expectation
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationdf(x) dx = h(x) Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation
Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations
More informationThe Schrödinger Equation
Chapter 13 The Schrödinger Equation 13.1 Where we are so far We have focused primarily on electron spin so far because it s a simple quantum system (there are only two basis states!), and yet it still
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 informationChapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)
Tor Kjellsson Stockholm University Chapter 6 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: where α is a constant. H = αδ(x a/ ( a Find the first-order correction
More informationBASICS OF QUANTUM MECHANICS. Reading: QM Course packet Ch 5
BASICS OF QUANTUM MECHANICS 1 Reading: QM Course packet Ch 5 Interesting things happen when electrons are confined to small regions of space (few nm). For one thing, they can behave as if they are in an
More informationBasic Quantum Mechanics
Frederick Lanni 10feb'12 Basic Quantum Mechanics Part I. Where Schrodinger's equation comes from. A. Planck's quantum hypothesis, formulated in 1900, was that exchange of energy between an electromagnetic
More informationPhysics 137A Quantum Mechanics Fall 2012 Midterm II - Solutions
Physics 37A Quantum Mechanics Fall 0 Midterm II - Solutions These are the solutions to the exam given to Lecture Problem [5 points] Consider a particle with mass m charge q in a simple harmonic oscillator
More informationNumerical solution of the time-independent 1-D Schrödinger equation. Matthias E. Möbius. September 24, 2010
Numerical solution of the time-independent 1-D Schrödinger equation Matthias E. Möbius September 24, 2010 1 Aim of the computational lab Numerical solution of the one-dimensional stationary Schrödinger
More information1 Planck-Einstein Relation E = hν
C/CS/Phys C191 Representations and Wavefunctions 09/30/08 Fall 2008 Lecture 8 1 Planck-Einstein Relation E = hν This is the equation relating energy to frequency. It was the earliest equation of quantum
More informationProblem 1: A 3-D Spherical Well(10 Points)
Problem : A 3-D Spherical Well( Points) For this problem, consider a particle of mass m in a three-dimensional spherical potential well, V (r), given as, V = r a/2 V = W r > a/2. with W >. All of the following
More informationThe Birth of Quantum Mechanics. New Wave Rock Stars
The Birth of Quantum Mechanics Louis de Broglie 1892-1987 Erwin Schrödinger 1887-1961 Paul Dirac 1902-1984 Werner Heisenberg 1901-1976 New Wave Rock Stars Blackbody radiation: Light energy is quantized.
More informationUNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2
Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL,
More informationProject: Vibration of Diatomic Molecules
Project: Vibration of Diatomic Molecules Objective: Find the vibration energies and the corresponding vibrational wavefunctions of diatomic molecules H 2 and I 2 using the Morse potential. equired Numerical
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 information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
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/CS/Phys 191 Quantum Mechanics in a Nutshell I 10/04/05 Fall 2005 Lecture 11
C/CS/Phys 191 Quantum Mechanics in a Nutshell I 10/04/05 Fall 2005 Lecture 11 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 8, February 3, 2006 & L "
Chem 352/452 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 26 Christopher J. Cramer Lecture 8, February 3, 26 Solved Homework (Homework for grading is also due today) Evaluate
More informationProblem Set 5 Solutions
Chemistry 362 Dr Jean M Standard Problem Set 5 Solutions ow many vibrational modes do the following molecules or ions possess? [int: Drawing Lewis structures may be useful in some cases] In all of the
More informationDiscrete Variable Representation
Discrete Variable Representation Rocco Martinazzo E-mail: rocco.martinazzo@unimi.it Contents Denition and properties 2 Finite Basis Representations 3 3 Simple examples 4 Introduction Discrete Variable
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 informationSpectral Broadening Mechanisms
Spectral Broadening Mechanisms Lorentzian broadening (Homogeneous) Gaussian broadening (Inhomogeneous, Inertial) Doppler broadening (special case for gas phase) The Fourier Transform NC State University
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 informationQuantum Mechanics Final Exam Solutions Fall 2015
171.303 Quantum Mechanics Final Exam Solutions Fall 015 Problem 1 (a) For convenience, let θ be a real number between 0 and π so that a sinθ and b cosθ, which is possible since a +b 1. Then the operator
More informationPhysics 218 Quantum Mechanics I Assignment 6
Physics 218 Quantum Mechanics I Assignment 6 Logan A. Morrison February 17, 2016 Problem 1 A non-relativistic beam of particles each with mass, m, and energy, E, which you can treat as a plane wave, is
More informationThe Postulates of Quantum Mechanics Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: 3-D case
The Postulates of Quantum Mechanics Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about the
More informationThe quantum state as a vector
The quantum state as a vector February 6, 27 Wave mechanics In our review of the development of wave mechanics, we have established several basic properties of the quantum description of nature:. A particle
More informationThe Schroedinger equation
The Schroedinger equation After Planck, Einstein, Bohr, de Broglie, and many others (but before Born), the time was ripe for a complete theory that could be applied to any problem involving nano-scale
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 informationEnergy Level Sets for the Morse Potential
Energy Level Sets for the Morse Potential Fariel Shafee Department of Physics Princeton University Princeton, NJ 08540 Abstract: In continuation of our previous work investigating the possibility of the
More informationWe start with some important background material in classical and quantum mechanics.
Chapter Basics We start with some important background material in classical and quantum mechanics.. Classical mechanics Lagrangian mechanics Compared to Newtonian mechanics, Lagrangian mechanics has the
More informationSpectral Broadening Mechanisms. Broadening mechanisms. Lineshape functions. Spectral lifetime broadening
Spectral Broadening echanisms Lorentzian broadening (Homogeneous) Gaussian broadening (Inhomogeneous, Inertial) Doppler broadening (special case for gas phase) The Fourier Transform NC State University
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 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 informationNon-stationary States and Electric Dipole Transitions
Pre-Lab Lecture II Non-stationary States and Electric Dipole Transitions You will recall that the wavefunction for any system is calculated in general from the time-dependent Schrödinger equation ĤΨ(x,t)=i
More informationChemistry 532 Problem Set 7 Spring 2012 Solutions
Chemistry 53 Problem Set 7 Spring 01 Solutions 1. The study of the time-independent Schrödinger equation for a one-dimensional particle subject to the potential function leads to the differential equation
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 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationPhysics 215 Quantum Mechanics 1 Assignment 5
Physics 15 Quantum Mechanics 1 Assignment 5 Logan A. Morrison February 10, 016 Problem 1 A particle of mass m is confined to a one-dimensional region 0 x a. At t 0 its normalized wave function is 8 πx
More informationQuantum Mechanics I - Session 9
Quantum Mechanics I - Session 9 May 5, 15 1 Infinite potential well In class, you discussed the infinite potential well, i.e. { if < x < V (x) = else (1) You found the permitted energies are discrete:
More informationNon-degenerate Perturbation Theory. and where one knows the eigenfunctions and eigenvalues of
on-degenerate Perturbation Theory Suppose one wants to solve the eigenvalue problem ĤΦ = Φ where µ =,1,2,, E µ µ µ and where Ĥ can be written as the sum of two terms, ˆ ˆ ˆ ˆ ˆ ˆ H = H + ( H H ) = H +
More information2 Feynman rules, decay widths and cross sections
2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In non-relativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in
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 informationAssignment #1 Chemistry 314 Summer 2008
Assignment #1 Due Thursday, July 17. Hand in for grading, including especially the graphs and tables of values for question 2. 1. This problem develops the classical treatment of the harmonic oscillator.
More informationAn operator is a transformation that takes a function as an input and produces another function (usually).
Formalism of Quantum Mechanics Operators Engel 3.2 An operator is a transformation that takes a function as an input and produces another function (usually). Example: In QM, most operators are linear:
More informationCHM320 EXAM #2 USEFUL INFORMATION
CHM30 EXAM # USEFUL INFORMATION Constants mass of electron: m e = 9.11 10 31 kg. Rydberg constant: R H = 109737.35 cm 1 =.1798 10 18 J. speed of light: c = 3.00 10 8 m/s Planck constant: 6.66 10 34 Js
More informationPhysics 443, Solutions to PS 2
. Griffiths.. Physics 443, Solutions to PS The raising and lowering operators are a ± mω ( iˆp + mωˆx) where ˆp and ˆx are momentum and position operators. Then ˆx mω (a + + a ) mω ˆp i (a + a ) The expectation
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 informationSection 9 Variational Method. Page 492
Section 9 Variational Method Page 492 Page 493 Lecture 27: The Variational Method Date Given: 2008/12/03 Date Revised: 2008/12/03 Derivation Section 9.1 Variational Method: Derivation Page 494 Motivation
More informationNov : Lecture 18: The Fourier Transform and its Interpretations
3 Nov. 04 2005: Lecture 8: The Fourier Transform and its Interpretations Reading: Kreyszig Sections: 0.5 (pp:547 49), 0.8 (pp:557 63), 0.9 (pp:564 68), 0.0 (pp:569 75) Fourier Transforms Expansion of a
More information-state problems and an application to the free particle
-state problems and an application to the free particle Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 2013 3 September, 2013 Outline 1 Outline 2 The Hilbert space 3 A free particle 4 Keywords
More informationAyan Chattopadhyay Mainak Mustafi 3 rd yr Undergraduates Integrated MSc Chemistry IIT Kharagpur
Ayan Chattopadhyay Mainak Mustafi 3 rd yr Undergraduates Integrated MSc Chemistry IIT Kharagpur Under the supervision of: Dr. Marcel Nooijen Associate Professor Department of Chemistry University of Waterloo
More informationReview of paradigms QM. Read McIntyre Ch. 1, 2, 3.1, , , 7, 8
Review of paradigms QM Read McIntyre Ch. 1, 2, 3.1, 5.1-5.7, 6.1-6.5, 7, 8 QM Postulates 1 The state of a quantum mechanical system, including all the informaion you can know about it, is represented mathemaically
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 informationarxiv:quant-ph/ v1 29 Mar 2003
Finite-Dimensional PT -Symmetric Hamiltonians arxiv:quant-ph/0303174v1 29 Mar 2003 Carl M. Bender, Peter N. Meisinger, and Qinghai Wang Department of Physics, Washington University, St. Louis, MO 63130,
More informationComparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation Student: Yael Elmatad
Comparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation Student: Yael Elmatad Overview The goal of this research was to examine the
More informationSupplementary information I Hilbert Space, Dirac Notation, and Matrix Mechanics. EE270 Fall 2017
Supplementary information I Hilbert Space, Dirac Notation, and Matrix Mechanics Properties of Vector Spaces Unit vectors ~xi form a basis which spans the space and which are orthonormal ( if i = j ~xi
More informationQM1 - Tutorial 2 Schrodinger Equation, Hamiltonian and Free Particle
QM - Tutorial Schrodinger Equation, Hamiltonian and Free Particle Yaakov Yuin November 07 Contents Hamiltonian. Denition...................................................... Example: Hamiltonian of a
More informationTime Independent Perturbation Theory Contd.
Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n
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 informationQuantum mechanics is a physical science dealing with the behavior of matter and energy on the scale of atoms and subatomic particles / waves.
Quantum mechanics is a physical science dealing with the behavior of matter and energy on the scale of atoms and subatomic particles / waves. It also forms the basis for the contemporary understanding
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 information