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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 1 Problem set 9 Handout: 1/24 Due date: 1/31 Problem 1 Prove that the energy to first order for the lowest-energy state of a perturbed system is an upper bound for the exact energy of the lowest-energy state of the perturbed system, that is, that E () + E (1) E. The ground state energy is given by E =< ψ H ψ > (1) and the first correction to the ground state energy is given by E 1 =< ψ H 1 ψ > (2) where the total Hamiltonian is given by H = H + H 1. We know from the variational theorem that using the zeroth-order wavefunction as a trial function for the total Hamiltonian will give an upper bound to the energy < ψ H ψ > E (3) since ψ is not the wavefunction associated with the Hamiltonian. Therefore, we can prove that < ψ H ψ > E (4) < ψ H + H 1 ψ > E (5) < ψ H ψ > + < ψ H 1 ψ > E (6) E + E1 E (7) Problem 2 An electron moves in a harmonic potential, V = 1 2 x2. What is the effect, to first order, on the energies of a perturbing electric field, Ĥ 1 = Fx? Explain your reasoning. [hint: you do not need to evaluate any integrals to answer the question.]. (8) 1

2 Since the particle moves in a Harmonic potential the wavefunction squared is symmetric. An electric field perturbation V 1 = Ex which is antisymmetric will have no effect since < ψ V 1 ψ >= dx ψ 2 V 1 = dxsymmetric antisymmetric = (9) Problem 3 Calculate the energy to first order of He + in its lowest-energy state. Use the hydrogen atom in its ground state as your zeroth-order approximation. Use atomic units. The zeroth order Hamiltonian is for the hydrogen atom H = r (1) whereas the Hamiltonian for the He + is given by H = r (11) Thus, the total Hamiltonian can be split as The energy to first order is then H = R = H 1 r = H + H 1 (12) E = E + E 1 = < 1s 1 1s > (13) r Evaluating the first-order correction to the energy we get E 1 = < 1s 1 r 1s >= 4π π exp( 2r)rdr = 4 1 = 1 (14) 22 The energy to first-order is then E = 1/2 1 = 3/2. Problem 4 A hydrogen atom in its ground state is perturbed by applying a uniform electric field, Fz, along the z-direction. 2

3 1. What is the first-order change in the energy? 2. For the first-order correction to the ground-state wavefunction, we can consider a 2s,1s and a 2pz,1s to be the coefficients for mixing in 2s and 2p z character. Write down the expression for these two coefficients. 3. Predict the signs of these coefficients. Explain your reasoning. The ground state wavefunction for the hydrogen atom is 1s = 1 π exp( r). a) The first-order change in the energy due to the electric field is then given by = F 1 π = F 1 π b) The two cofficients are r 3 exp( 2r)dr < 1s H 1 1s >= F < 1s r cosθ 1s > (15) r 3 exp( 2r)dr π π cosθ sin θdθ sym. antisym. 2π 2π dφ (16) dφ = (17) c (1) 2s,1s = < 2s H1 1s > E 2s E 1s = (18) c (1) 2p z,1s = < 2p z H 1 1s > E 2pz E 1s = (19) The first coefficient are zero due to symmetry, i.e. it cannot improve the energy due to perturbation along the Z since it is symmetric in all directions. The second coefficient is positive since the 2p z function is oriented along the direction of the perturbation and will therefore improve the wavefunction. Problem 5 The Hellman-Feynman theorem establishes a connection between the change in the energy and the change in the Hamiltonian of a system experiences an electric field perturbation. The theorem is given by de df = Ĥ (2) where E is the energy, F is the electric field strength and Ĥ is the total hamiltonian for the system. If we apply an electric field perturbation along the z-direction we can write the first-order hamiltonian as Ĥ (1) = ˆµ z F (21) 3

4 where ˆµ z is the electric dipole moment operator in the z-direction. 1) Use the Hellman-Feynman theorem to express the change in the energy due to an electric field in the z-direction in terms of the dipole moment operator. 2) Expand the energy of a system in a Taylor expansion relative to the energy, E, in the absence of the field. The expectation value of the dipole moment operator can also be written as < ˆµ z >= µ z + α zzf + (22) where µ z is the permanent dipole moment of the system and α zz is the polarizability. 3) Use this expansion to express the permanent dipole moment and polarizability in terms of derivatives of the energy with respect to the electric field. 4) Use non-degenerate perturbation theory to obtain an expression for the energy of the system in terms of the field strength to second order. 5) Finally, use this expression to identify the permanent dipole moment and polarizability of the system. [Hint: The final expression should be expression in terms of the electric dipole operator and the zeroth-order wavefunctions de df = Ĥ = Ĥ µ z F = µ z (23) E = E + E F E + (24) 2 2F2 de df = E + 2 E + = µ 2F2 z α zzf + (25) Therefore, we can identify the dipole moment and polarizability as µ z = E (26) α zz = 2 E 2 (27) 4

5 4. The energy expression from perturbations theory reads: E = E + E 1 + E 2 + (28) = E + < ψ () H 1 ψ () > + < ψ m () H 1 ψ () >< ψ () H1 ψ () E E m m = E < ψ () µ z ψ () > F + < ψ m () µ z ψ () >< ψ () µ z ψ () E E m m 5. We can therefore identify the terms as m > m > + (29) + (3) and µ z = E =< ψ() µ z ψ () > (31) α zz = 2 E = 2 < ψ m () µ z ψ () >< ψ () µ z ψ () 2 E E m m m > (32) Problem 6 A given unperturbed system has a doubly generate energy level for which the perturbation integrals have values of H 1 11 = 6a, H1 22 = 8a, and H1 12 = 2a, where a is a positive constant. We also know that the unperturbed wavefunctions are orthonormal. 1. Find the first-order correction to the energy in terms of the constant a. 2. Find the normalized correct zeroth-order wave functions. 1. The secular equation is det(h EI) = H1 aa E H 1 ba H1 ab H 1 bb E = 6a E 2a 2a 8a E = (33) Expansion of the secular equation gives the following quadratic equation E 2 14Ea + 44a 2 = (34) 5

6 for which the solution is E = 7a ± 2a = 11.47a, 2.52a. The coefficients for the lowest eigenvalue is then given by Using normalization we find (H 1 aa E1 )c 1 + H ab c 2 = (35) (6a 2.52a)c 1 + 2ac 2 = (36) c 1 =.57c 2 (37) 1 =.57 2 c c2 2 ==> c 2 =.867 (38) Therefore, the normalized wavefunction is ψ 1 =.49ψ a +.867ψ b (39) 6

Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.

Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger

More information

Electric fields : Stark effect, dipole & quadrupole polarizability.

Electric fields : Stark effect, dipole & quadrupole polarizability. Electric fields : Stark effect, dipole & quadrupole polarizability. We are often interested in the effect of an external electric field on the energy levels and wavefunction of H and other one-electron

More information

Perturbation Theory 1

Perturbation Theory 1 Perturbation Theory 1 1 Expansion of Complete System Let s take a look of an expansion for the function in terms of the complete system : (1) In general, this expansion is possible for any complete set.

More information

PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8: Solutions. Topics covered: hydrogen fine structure

PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8: Solutions. Topics covered: hydrogen fine structure PHYS85 Quantum Mechanics II, Spring HOMEWORK ASSIGNMENT 8: Solutions Topics covered: hydrogen fine structure. [ pts] Let the Hamiltonian H depend on the parameter λ, so that H = H(λ). The eigenstates and

More information

Quantum 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. 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 information

Chemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):

Chemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron): April 6th, 24 Chemistry 2A 2nd Midterm. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (-electron): E n = m e Z 2 e 4 /2 2 n 2 = E Z 2 /n 2, n =, 2, 3,... where Ze is

More information

Time-Independent Perturbation Theory

Time-Independent Perturbation Theory 4 Phys46.nb Time-Independent Perturbation Theory.. Overview... General question Assuming that we have a Hamiltonian, H = H + λ H (.) where λ is a very small real number. The eigenstates of the Hamiltonian

More information

Time Independent Perturbation Theory Contd.

Time 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 information

Problem Set 5 Solutions

Problem 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 information

Quantum Mechanics Solutions

Quantum Mechanics Solutions Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H

More information

( ) in the interaction picture arises only

( ) in the interaction picture arises only Physics 606, Quantum Mechanics, Final Exam NAME 1 Atomic transitions due to time-dependent electric field Consider a hydrogen atom which is in its ground state for t < 0 For t > 0 it is subjected to a

More information

Approximation Methods in QM

Approximation Methods in QM Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden

More information

0 belonging to the unperturbed Hamiltonian H 0 are known

0 belonging to the unperturbed Hamiltonian H 0 are known Time Independent Perturbation Theory D Perturbation theory is used in two qualitatively different contexts in quantum chemistry. It allows one to estimate (because perturbation theory is usually employed

More information

Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split

Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split Electron Spin Electron spin hypothesis Solution to H atom problem gave three quantum numbers, n,, m. These apply to all atoms. Experiments show not complete description. Something missing. Alkali metals

More information

For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as

For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as VARIATIO METHOD For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as Variation Method Perturbation Theory Combination

More information

Quantum Physics III (8.06) Spring 2008 Final Exam Solutions

Quantum Physics III (8.06) Spring 2008 Final Exam Solutions Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,

More information

10 Time-Independent Perturbation Theory

10 Time-Independent Perturbation Theory S.K. Saiin Oct. 6, 009 Lecture 0 0 Time-Independent Perturbation Theory Content: Non-degenerate case. Degenerate case. Only a few quantum mechanical problems can be solved exactly. However, if the system

More information

Final Examination. Tuesday December 15, :30 am 12:30 pm. particles that are in the same spin state 1 2, + 1 2

Final Examination. Tuesday December 15, :30 am 12:30 pm. particles that are in the same spin state 1 2, + 1 2 Department of Physics Quantum Mechanics I, Physics 57 Temple University Instructor: Z.-E. Meziani Final Examination Tuesday December 5, 5 :3 am :3 pm Problem. pts) Consider a system of three non interacting,

More information

Chemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer.

Chemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer. B Chemistry 3502/4502 Exam III All Hallows Eve/Samhain, 2003 1) This is a multiple choice exam. Circle the correct answer. 2) There is one correct answer to every problem. There is no partial credit. 3)

More information

Introduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti

Introduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti Introduction to Quantum Mechanics PVK - Solutions Nicolas Lanzetti lnicolas@student.ethz.ch 1 Contents 1 The Wave Function and the Schrödinger Equation 3 1.1 Quick Checks......................................

More information

Electron States of Diatomic Molecules

Electron States of Diatomic Molecules IISER Pune March 2018 Hamiltonian for a Diatomic Molecule The hamiltonian for a diatomic molecule can be considered to be made up of three terms Ĥ = ˆT N + ˆT el + ˆV where ˆT N is the kinetic energy operator

More information

Helium and two electron atoms

Helium and two electron atoms 1 Helium and two electron atoms e 2 r 12 e 1 r 2 r 1 +Ze Autumn 2013 Version: 04.12.2013 2 (1) Coordinate system, Schrödinger Equation 3 slides Evaluation of repulsion term 2 slides Radial Integral - details

More information

Problem 1: A 3-D Spherical Well(10 Points)

Problem 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 information

Chemistry 3502/4502. Exam III. March 28, ) Circle the correct answer on multiple-choice problems.

Chemistry 3502/4502. Exam III. March 28, ) Circle the correct answer on multiple-choice problems. A Chemistry 352/452 Exam III March 28, 25 1) Circle the correct answer on multiple-choice problems. 2) There is one correct answer to every multiple-choice problem. There is no partial credit. On the short-answer

More information

Intermission: Let s review the essentials of the Helium Atom

Intermission: Let s review the essentials of the Helium Atom PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the

More information

1 Time-Dependent Two-State Systems: Rabi Oscillations

1 Time-Dependent Two-State Systems: Rabi Oscillations Advanced kinetics Solution 7 April, 16 1 Time-Dependent Two-State Systems: Rabi Oscillations a In order to show how Ĥintt affects a bound state system in first-order time-dependent perturbation theory

More information

MATH325 - QUANTUM MECHANICS - SOLUTION SHEET 11

MATH325 - QUANTUM MECHANICS - SOLUTION SHEET 11 MATH35 - QUANTUM MECHANICS - SOLUTION SHEET. The Hamiltonian for a particle of mass m moving in three dimensions under the influence of a three-dimensional harmonic oscillator potential is Ĥ = h m + mω

More information

Advanced Spectroscopy. Dr. P. Hunt Rm 167 (Chemistry) web-site:

Advanced Spectroscopy. Dr. P. Hunt Rm 167 (Chemistry) web-site: Advanced Spectroscopy Dr. P. Hunt p.hunt@imperial.ac.uk Rm 167 (Chemistry) web-site: http://www.ch.ic.ac.uk/hunt Maths! Coordinate transformations rotations! example 18.1 p501 whole chapter on Matrices

More information

International Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-2016 ISSN

International Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-2016 ISSN 1574 International Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-216 Energy and Wave-function correction for a quantum system after a small perturbation Samuel Mulugeta Bantikum

More information

Physics 115C Homework 2

Physics 115C Homework 2 Physics 5C Homework Problem Our full Hamiltonian is H = p m + mω x +βx 4 = H +H where the unperturbed Hamiltonian is our usual and the perturbation is H = p m + mω x H = βx 4 Assuming β is small, the perturbation

More information

5.1 Classical Harmonic Oscillator

5.1 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 information

Quantum Mechanics Solutions. λ i λ j v j v j v i v i.

Quantum 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 information

Chapter 4. Some Important Tools of Theory

Chapter 4. Some Important Tools of Theory Chapter 4. Some Important Tools of Theory For all but the most elementary problems, many of which serve as fundamental approximations to the real behavior of molecules (e.g., the Hydrogenic atom, the harmonic

More information

Chemistry 532 Problem Set 7 Spring 2012 Solutions

Chemistry 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 information

Joel Broida UCSD Fall 2009 Phys 130B QM II. Homework Set 4

Joel Broida UCSD Fall 2009 Phys 130B QM II. Homework Set 4 Joel Broida UCSD Fall 009 Phys 130B QM II Homework Set 4 1. Consider the particle-in-a-box problem but with a delta function potential H (x) = αδ(x l/) at the center (with α = const): H = αδ(x l/) 0 l/

More information

$ +! j. % i PERTURBATION THEORY AND SUBGROUPS (REVISED 11/15/08)

$ +! j. % i PERTURBATION THEORY AND SUBGROUPS (REVISED 11/15/08) PERTURBATION THEORY AND SUBGROUPS REVISED 11/15/08) The use of groups and their subgroups is of much importance when perturbation theory is employed in understanding molecular orbital theory and spectroscopy

More information

Consider Hamiltonian. Since n form complete set, the states ψ (1) and ψ (2) may be expanded. Now substitute (1-5) into Hψ = Eψ,

Consider Hamiltonian. Since n form complete set, the states ψ (1) and ψ (2) may be expanded. Now substitute (1-5) into Hψ = Eψ, 3 Time-independent Perturbation Theory I 3.1 Small perturbations of a quantum system Consider Hamiltonian H 0 + ˆV, (1) where H 0 and ˆV both time-ind., and ˆV represents small perturbation to Hamiltonian

More information

Chapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)

Chapter 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 information

3: Many electrons. Orbital symmetries. l =2 1. m l

3: Many electrons. Orbital symmetries. l =2 1. m l 3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience

More information

Non-stationary States and Electric Dipole Transitions

Non-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 information

eigenvalues eigenfunctions

eigenvalues 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 information

Non-degenerate Perturbation Theory. and where one knows the eigenfunctions and eigenvalues of

Non-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 information

Sample Quantum Chemistry Exam 2 Solutions

Sample 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 information

Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom.

Reading: 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 information

Appendix 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 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 information

IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance

IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.

More information

Phys 622 Problems Chapter 5

Phys 622 Problems Chapter 5 1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit

More information

ψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.

ψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions. 1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product

More information

QUARK EFFECT ON H-ATOM SPECTRA(1S)

QUARK EFFECT ON H-ATOM SPECTRA(1S) QUARK EFFECT ON H-ATOM SPECTRA1S March 8, 18 1 A QUARK MODEL OF THE PROTON The purpose of this paper is to test a theory of how the quarks in the proton might affect the H-atom spectra. There are three

More information

Physics 139B Solutions to Homework Set 4 Fall 2009

Physics 139B Solutions to Homework Set 4 Fall 2009 Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates

More information

B7 Symmetry : Questions

B7 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 information

Chemistry 431. NC State University. Lecture 17. Vibrational Spectroscopy

Chemistry 431. NC State University. Lecture 17. Vibrational Spectroscopy Chemistry 43 Lecture 7 Vibrational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule vibrates.

More information

UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2

UNIVERSITY 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 information

Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager

Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Notations

More information

Physics 828 Problem Set 7 Due Wednesday 02/24/2010

Physics 828 Problem Set 7 Due Wednesday 02/24/2010 Physics 88 Problem Set 7 Due Wednesday /4/ 7)a)Consider the proton to be a uniformly charged sphere of radius f m Determine the correction to the s ground state energy 4 points) This is a standard problem

More information

2m 2 Ze2. , where δ. ) 2 l,n is the quantum defect (of order one but larger

2m 2 Ze2. , where δ. ) 2 l,n is the quantum defect (of order one but larger PHYS 402, Atomic and Molecular Physics Spring 2017, final exam, solutions 1. Hydrogenic atom energies: Consider a hydrogenic atom or ion with nuclear charge Z and the usual quantum states φ nlm. (a) (2

More information

Solution of Second Midterm Examination Thursday November 09, 2017

Solution of Second Midterm Examination Thursday November 09, 2017 Department of Physics Quantum Mechanics II, Physics 570 Temple University Instructor: Z.-E. Meziani Solution of Second Midterm Examination Thursday November 09, 017 Problem 1. (10pts Consider a system

More information

CHAPTER 11 MOLECULAR ORBITAL THEORY

CHAPTER 11 MOLECULAR ORBITAL THEORY CHAPTER 11 MOLECULAR ORBITAL THEORY Molecular orbital theory is a conceptual extension of the orbital model, which was so successfully applied to atomic structure. As was once playfuly remarked, a molecue

More information

We now turn to our first quantum mechanical problems that represent real, as

We now turn to our first quantum mechanical problems that represent real, as 84 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like

More information

PH 451/551 Quantum Mechanics Capstone Winter 201x

PH 451/551 Quantum Mechanics Capstone Winter 201x These are the questions from the W7 exam presented as practice problems. The equation sheet is PH 45/55 Quantum Mechanics Capstone Winter x TOTAL POINTS: xx Weniger 6, time There are xx questions, for

More information

Perturbation Theory. Andreas Wacker Mathematical Physics Lund University

Perturbation Theory. Andreas Wacker Mathematical Physics Lund University Perturbation Theory Andreas Wacker Mathematical Physics Lund University General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates

More information

I. Perturbation Theory and the Problem of Degeneracy[?,?,?]

I. Perturbation Theory and the Problem of Degeneracy[?,?,?] MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 19 THE VAN VLECK TRANSFORMATION IN PERTURBATION THEORY 1 Although frequently it is desirable to carry a perturbation treatment to second or third

More information

6.1 Nondegenerate Perturbation Theory

6.1 Nondegenerate Perturbation Theory 6.1 Nondegenerate Perturbation Theory Analytic solutions to the Schrödinger equation have not been found for many interesting systems. Fortunately, it is often possible to find expressions which are analytic

More information

( ). Expanding the square and keeping in mind that

( ). 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 information

opposed to idealized, systems. These problems are the structures of atoms. We will begin

opposed to idealized, systems. These problems are the structures of atoms. We will begin 88 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like

More information

Introduction to Molecular Vibrations and Infrared Spectroscopy

Introduction to Molecular Vibrations and Infrared Spectroscopy hemistry 362 Spring 2017 Dr. Jean M. Standard February 15, 2017 Introduction to Molecular Vibrations and Infrared Spectroscopy Vibrational Modes For a molecule with N atoms, the number of vibrational modes

More information

Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid

Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures

More information

PHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep

PHY 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 information

Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!

Qualifying 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 information

Problem 1: Spin 1 2. particles (10 points)

Problem 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 information

Theoretical Photochemistry WiSe 2017/18

Theoretical Photochemistry WiSe 2017/18 Theoretical Photochemistry WiSe 2017/18 Lecture 7 Irene Burghardt (burghardt@chemie.uni-frankfurt.de) http://www.theochem.uni-frankfurt.de/teaching/ Theoretical Photochemistry 1 Topics 1. Photophysical

More information

11 Perturbation Theory

11 Perturbation Theory S.K. Saikin Oct. 8, 009 11 Perturbation Theory Content: Variational Principle. Time-Dependent Perturbation Theory. 11.1 Variational Principle Lecture 11 If we need to compute the ground state energy of

More information

σ u * 1s g - gerade u - ungerade * - antibonding σ g 1s

σ u * 1s g - gerade u - ungerade * - antibonding σ g 1s One of these two states is a repulsive (dissociative) state. Other excited states can be constructed using linear combinations of other orbitals. Some will be binding and others will be repulsive. Thus

More information

Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14

Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Because the wavelength of light (400-700nm) is much greater than the diameter of an atom (0.07-0.35

More information

one-dimensional box with harmonic interaction

one-dimensional box with harmonic interaction On the symmetry of four particles in a arxiv:1607.00977v [quant-ph] 8 Jul 016 one-dimensional box with harmonic interaction Francisco M. Fernández INIFTA (CONICET, UNLP), División Química Teórica Blvd.

More information

Electric properties of molecules

Electric properties of molecules Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the

More information

Last Name or Student ID

Last Name or Student ID 12/05/18, Chem433 Final Exam answers Last Name or Student ID 1. (2 pts) 12. (3 pts) 2. (6 pts) 13. (3 pts) 3. (3 pts) 14. (2 pts) 4. (3 pts) 15. (3 pts) 5. (4 pts) 16. (3 pts) 6. (2 pts) 17. (15 pts) 7.

More information

Problems and Multiple Choice Questions

Problems 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 information

α β β α β β 0 β α 0 0 0

α β β α β β 0 β α 0 0 0 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry 5.6 Quantum Mechanics Fall 03 Problem Set #7 Reading Assignment: McQuarrie 9.-9.5, 0.-0.5,Matlab and Linear Algebra Handouts ( = Easier = More

More information

I. CSFs Are Used to Express the Full N-Electron Wavefunction

I. CSFs Are Used to Express the Full N-Electron Wavefunction Chapter 11 One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N- Electron Configuration Functions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon

More information

Homework assignment 3: due Thursday, 10/26/2017

Homework assignment 3: due Thursday, 10/26/2017 Homework assignment 3: due Thursday, 10/6/017 Physics 6315: Quantum Mechanics 1, Fall 017 Problem 1 (0 points The spin Hilbert space is defined by three non-commuting observables, S x, S y, S z. These

More information

Lecture 3 Dynamics 29

Lecture 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 information

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II January 22, 2016 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.

More information

The general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation

The general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent

More information

1.6. Quantum mechanical description of the hydrogen atom

1.6. Quantum mechanical description of the hydrogen atom 29.6. Quantum mechanical description of the hydrogen atom.6.. Hamiltonian for the hydrogen atom Atomic units To avoid dealing with very small numbers, let us introduce the so called atomic units : Quantity

More information

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II August 23, 208 9:00 a.m. :00 p.m. Do any four problems. Each problem is worth 25 points.

More information

Chemistry 543--Final Exam--Keiderling May 5, pm SES

Chemistry 543--Final Exam--Keiderling May 5, pm SES Chemistry 543--Final Exam--Keiderling May 5,1992 -- 1-5pm -- 174 SES Please answer all questions in the answer book provided. Make sure your name is clearly indicated and that the answers are clearly numbered,

More information

( ) = 9φ 1, ( ) = 4φ 2.

( ) = 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 information

Lecture #9 Redfield theory of NMR relaxation

Lecture #9 Redfield theory of NMR relaxation Lecture 9 Redfield theory of NMR relaxation Topics Redfield theory recap Relaxation supermatrix Dipolar coupling revisited Scalar relaxation of the st kind Handouts and Reading assignments van de Ven,

More information

( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1

( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1 Physics 624, Quantum II -- Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is

More information

Chm 331 Fall 2015, Exercise Set 4 NMR Review Problems

Chm 331 Fall 2015, Exercise Set 4 NMR Review Problems Chm 331 Fall 015, Exercise Set 4 NMR Review Problems Mr. Linck Version.0. Compiled December 1, 015 at 11:04:44 4.1 Diagonal Matrix Elements for the nmr H 0 Find the diagonal matrix elements for H 0 (the

More information

Consequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators

Consequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators VI. SPIN-ADAPTED CONFIGURATIONS A. Preliminary Considerations We have described the spin of a single electron by the two spin functions α(ω) α and β(ω) β. In this Sect. we will discuss spin in more detail

More information

Particle in one-dimensional box

Particle in one-dimensional box Particle in the box Particle in one-dimensional box V(x) -a 0 a +~ An example of a situation in which only bound states exist in a quantum system. We consider the stationary states of a particle confined

More information

Solution Exercise 12

Solution Exercise 12 Solution Exercise 12 Problem 1: The Stark effect in the hydrogen atom a) Since n = 2, the quantum numbers l can take the values, 1 and m = -1,, 1.We obtain the following basis: n, l, m = 2,,, 2, 1, 1,

More information

University of Michigan Physics Department Graduate Qualifying Examination

University of Michigan Physics Department Graduate Qualifying Examination Name: University of Michigan Physics Department Graduate Qualifying Examination Part II: Modern Physics Saturday 17 May 2014 9:30 am 2:30 pm Exam Number: This is a closed book exam, but a number of useful

More information

Physics 216 Spring The Variational Computation of the Ground State Energy of Helium

Physics 216 Spring The Variational Computation of the Ground State Energy of Helium Physics 26 Spring 22 The Variational Computation of the Ground State Energy of Helium I. Introduction to the variational computation where The Hamiltonian for the two-electron system of the helium atom

More information

TP computing lab - Integrating the 1D stationary Schrödinger eq

TP computing lab - Integrating the 1D stationary Schrödinger eq TP computing lab - Integrating the 1D stationary Schrödinger equation September 21, 2010 The stationary 1D Schrödinger equation The time-independent (stationary) Schrödinger equation is given by Eψ(x)

More information

16.1. PROBLEM SET I 197

16.1. PROBLEM SET I 197 6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,

More information

Physical Chemistry I Fall 2016 Second Hour Exam (100 points) Name:

Physical Chemistry I Fall 2016 Second Hour Exam (100 points) Name: Physical Chemistry I Fall 2016 Second Hour Exam (100 points) Name: (20 points) 1. Quantum calculations suggest that the molecule U 2 H 2 is planar and has symmetry D 2h. D 2h E C 2 (z) C 2 (y) C 2 (x)

More information

Ψ t = ih Ψ t t. Time Dependent Wave Equation Quantum Mechanical Description. Hamiltonian Static/Time-dependent. Time-dependent Energy operator

Ψ 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 information