Chm331_09 page

The Front Range of CO from Chief Mountain.

Introduction

Quantum mechanics is absurd! It offers a recipe (a highly mathematical recipe) for finding incredibly accurate answers to problems, yet there is nothing in our common experience that allows us to grasp what is "really" happening. In this course we will work our way through the recipes and try to find meaning in the weirdness. We will have to do some math, but we will make extensive use of computer based mathematical systems to keep as much drudgery out of our effort as possible. I am completely convinced that the only way you learn quantum mechanics is by doing quantum mechanics. Therefore we will do it. We will do lots of problems. To be able to get a grip on this subject is very rewarding. Obtaining that grip is the aim of the course.

Messages

I have posted the notebook for Huckel MO treatment 1149 17/xi/09
I have posted answers to Daily Exercises up to 54 and to Other Exercises up to 47. See the answers table below. 1520 4/xi/09
I have managed to catch up on some grading. The papers are on the table outside the Ford Hall office, 311. 1152 17/xi/09

Instructor

Robert Linck
Office, Sabin-Reed 429
Phone, X3836
Email, rlinck(at)email.smith.edu
Office Hours, Tu, 8-11; Th 10-12; 1-3; F, 1-3; or by appointment.

Problem Sets, Answers, and Exam Answers

Fall, 2009
Item Link

Problem Set 1 DP 1-12; OP 1-5.

Mathematica Notebook for Wed, 9 Sept

Problem Set 2 DP 11-30; OP 6-22

Answers to Problem Set One DP 1-12

Answers to Other Problems OP 1-5

M.nb for POP and superposition Mon, 9/20

M.nb for numerical integration Mon, 9/20

Notebook for Expansion of η

Notebook for V potential

Answers to Daily Problems DP 13-30

Answers to Other Problems OP 8-20

Notebook for numerical harmonic oscillator

Notebook for barriers in POP

Notebook for tunneling in POP

Answers to Conceptual Quiz

REAL derivation of parve in well

Notebook for Transcendental Equations

Answers to Practical Quiz Number One

Notebook for Harmonic Osillator Fri, Oct 2

Notebook on Eigenfunction, Eigenvalues, 2/x/09

Problem Set 3 DP 31-45; OP23-33

Dr. Robinson's Analysis of Basis of Numerical Integration

Practice Conceptual Exam

Answers to Practice Conceptual Exam

Answers to Daily Problems 30-45

Answers to Other Problems 20-33

Practice Practical Exam One

Answers to Practical Exam One

Problem Set 4 DP 46-59; OP 34-49

Notebook on Spherical Harmonics 28/x/09

Answers to Practical Exam Number One

Notebook for Laguerre Polynomical Solutions for radial part of H atom

Notebook for Perturbation Theory of a Parve on a Pole

Notebook for Matrix Manipulation

Answers to Problem Set Four 46-59

Answers to Other Problems, Set Four 34-49

Notebook for Two Dimensional Parve on Sheet and Degenerate Perturbation Theory

Problem Set 5 DP 60-72, OP 50-69

Notebook for Variational Method

Notebook for Huckel MO theory

**Fall, 2009**
Item	Link
Problem Set 1	DP 1-12; OP 1-5.
Mathematica Notebook for	Wed, 9 Sept
Problem Set 2	DP 11-30; OP 6-22
Answers to Problem Set One	DP 1-12
Answers to Other Problems	OP 1-5
M.nb for POP and superposition	Mon, 9/20
M.nb for numerical integration	Mon, 9/20
Notebook for	Expansion of η
Notebook for	V potential
Answers to Daily Problems	DP 13-30
Answers to Other Problems	OP 8-20
Notebook for	numerical harmonic oscillator
Notebook for	barriers in POP
Notebook for	tunneling in POP
Answers to Conceptual	Quiz
REAL derivation of	parve in well
Notebook for Transcendental	Equations
Answers to Practical Quiz	Number One
Notebook for Harmonic Osillator	Fri, Oct 2
Notebook on Eigenfunction,	Eigenvalues, 2/x/09
Problem Set 3	DP 31-45; OP23-33
Dr. Robinson's Analysis of Basis of	Numerical Integration
Practice Conceptual	Exam
Answers to Practice Conceptual	Exam
Answers to Daily Problems	30-45
Answers to Other Problems	20-33
Practice Practical	Exam One
Answers to Practical	Exam One
Problem Set 4	DP 46-59; OP 34-49
Notebook on Spherical	Harmonics 28/x/09
Answers to Practical Exam	Number One
Notebook for Laguerre Polynomical Solutions	for radial part of H atom
Notebook for Perturbation Theory of a	Parve on a Pole
Notebook for	Matrix Manipulation
Answers to Problem Set Four	46-59
Answers to Other Problems, Set Four	34-49
Notebook for	Two Dimensional Parve on Sheet and Degenerate Perturbation Theory
Problem Set 5	DP 60-72, OP 50-69
Notebook for	Variational Method
Notebook for	Huckel MO theory

Syllabus

Purpose. This course develops the methods to use quantum mechanics to derive the answers to problems involving small things--like electrons and atoms--the things with which chemists are always dealing. We will start with some simple (unreal but useful) one dimensional problems, work on the harmonic oscillator, learn the postulates that govern quantum mechanics, move on to an actual atom, then deal with the subjects necessary to handle multi-electron atoms. This course is of necessity highly mathematical. We will swim in equations throughout the course, trying at all times to keep head above water, to keep our sense of what we really have buried in all that math. We will do some limited amount of math "by hand," but a large part of the exercises are much more easily done by computer based mathematical techniques. This is good because we are interested in the answers to the problems, not the math getting us there.

Meetings. The course is scheduled to meet three times a week, MWF, 0830-0950 in Bass 210.

Learning. I have been to a number of workshops over the years that stress that learning is most effective when students are actively engaged. I believe this. To facilitate student participation we will use a radio frequency response system, affectionately called "clicker" technology by most. This will allow me to ask questions of the class and have you answer without having to admit to others that you know (or don't know) the answer--and heaven forbid that we would reveal this dark truth to others! You should go to the CMP and check out a clicker. Instructions for use will be discussed in class. I also believe, based on my own experience, that the only way you learn anything about quantum mechanics is by doing it. Accordingly, this course will have assigned problems nearly every class session. These will be graded (at the least in an effort expended sense), either by a peer grading system, or by the instructor, and will compromise a reasonable portion of your grade. You are responsible for doing the assigned problems before each class. We will spend a reasonable portion of class time on problems. We will discuss them together, talk about the merits of a given answer, accept or reject it, improve upon it, refine it, learn from the effort. That is, we will sit around in class as a community of scholars and talk about methods in quantum chemistry. I encourage you to establish some groups which meet regularly to work on problems--and you will need to meet near a computer for most of those problems. This activity will require you to spend time before each class doing problems; I would estimate between one and three hours will be required. That is a reasonable fraction of the 12 hours per week that you are expected (and will probably need) to spend on this class. I expect this class will be intensive. But by the same token, you will learn a tremendous amount: by the end of the first month you will understand how small particles can tunnel through barriers and be able to predict the probability that they will do so--which is all that quantum mechanics can ever say about anything. After six weeks, you will see how by using the nonsensical operators called a₊ and a_- you will be able to find the allowed energy levels of an oscillator, and so on.

Text. I had a great deal of trouble finding a text for this course. When preparing the material, I used a set of seventeen books, copyrighted between 1930 and 2006, consulting almost all of them on each subject. No one of those books had everything that I wanted. No one of those books explained everything clearly. But from the collection, some sort of sense emerged. As a pure compromise, I chose for a text Quantum Mechanics, by Robert Scherrer, which is available in the bookstore. It treats most subjects we will do, just not always as clearly (or completely) as some other book might treat them.

Exams. We will have a quiz, two exams, and a final during this course (see the schedule below for dates). Each of these will be in two parts, a conceptual and a practical. The conceptual part will be the usual type of examination in which you will be asked to answer questions without the benefit of notes or texts. This part will not stress the memorization of equations, but rather will try to test your knowledge of the general principles and consequences of quantum mechanics. The practical part will be an examination in which you will be allowed to use any material you wish (notes, texts, etc), including a computer to do fancy math (such as integrations). In this second part you will be expected to produce the right equation for the right situation, either from your memory or by looking it up. (Of course, having to look everything up will be quite time consuming if you are not prepared for the exam!) I will arrange so that all of these assessments are available to you during a period of time so that you may take them when you desire. Both kinds of exams will have time limits imposed.

This table subject to frequent change! Check frequently.
Entries of darker color have been completed.

Fall, 2009
Lecture date Objective of lecture: In all cases insert the words "Be able to" before the expression. DP OP

Part I Preliminaries

Sept 9 Use Mathematica to solve 0simple problems.
Outline the weird behavior for which qm has to account.
Know the historical failures of classical mechanics. 1-4 None-

Sept 11 Formulate the probability of an occurrence.
Show what normalization means.
Calculate the expectation value of a value.
Calculate the expectation value of a function.
Show that Sin[x/λ - t/T] shows wave behavior.
Deduce that the sign of k measures direction of wave motion.
Appreciate the formation of a standing wave.
State why we use an exponential representation of a wave.
Use a phase shift to express a Sin wave as a Cos wave.
Express a wave as the real or imaginary part of Exp[iφ]. 5-8 1-2

Sept 14 State what two quantum rules go into Schrödinger's formulation.
See how Schrödinger might have been led to his equation.
Write the general form of Schrödinger's equation.Appreciate the statistical interpretation of ψ.
State the realist, orthodox, and alternative positions on qm. 9-12 3-5

Part II SE and Consequences

Sept 16
Follow, but not necessarily reproduce, the derivation of the <p>, especially where the operator is placed in the expression.
Use the rule: <Q> = $∫ ψ* Q$ _op ψ dx .
Use the separation of variables concept to write solutions to an equation.
Apply separation of variables to the time dependent S.E.
Write SE for a parve on a pole (with infinite potential outside the pole.
Express the wave function solutions to SE for a POP. 13-16 6-9

Sept 18 Give the energy levels in terms of parameters for a POP.
Show that the uncertainty principle holds for quantum systems.
Write an arbitrary function in terms of eigenfunctions of the Hamiltonian.
Evaluate the coefficients of that expansion.
Predict the frequency with which a given eigenvalue is obtained in this expansion. 17-19 10-13

Sept 21 Show how qm and classical interpretations of the POP differ.
Apply time dependency to a system.
Prove that SE requires ψ curves toward the ψ = 0 axis when (E-V) is positive.
Prove that ψ curves away from the ψ = 0 axis when (E-V) is negative (the non-classical situation).
Understand that the only acceptable option in the (E-V) case is that ψ and δψ/δx both approach zero. 20-22 None

Poor timing space

Sept 25 Solve POP problem numerically.
Solve problems with various potentials numerically.Numerically solve the well with a barrier problem.
Write the general solution for a free particle. 23 14

Sept 28
Follow the argument that shows reflection from a barrier even when E > V_barrier.
Set up boundary conditions for a parve in a finite well.
Use M to solve the resulting transcendental equation. 24-25 15-17

Sept 30 Set up the Hamiltonian for a harmonic oscillator.
Use the variable q to get a solution at large q.
Introduce waviness with a power series.
Evaluate what SE requires of the power series. 26- 18-

Oct 2 Apply the boundary condition to get a recursive equation.
Find the coefficients from the recursive equation. 27 28 19-

Oct 5 Mountain Day

Part III Postulates and Operators

Oct 7 Use the harmonic oscillator results for practical purposes.
Know the postulates of QM.
Understand communtation of operators.
Use Dirac notation. 29 30 20-21

Oct 9 Appreciate Hermitian operators and how to establish it.
State the consequences of Hermitian operators. 31 34 22-24

Oct 14 Understand how communtation of operators leads to the Uncertainty Principle.
Use the common eigenfunctions of commuting operators. 35 36 25-

Oct 16 See the relationship between a₊ and a_- and the Hamiltonian for a harmonic oscillator.
Obtain communtation relationships between the raising and lowering operators.
Show that a_-|n> produces c|n-1>.
Evaluate c.
Appreciate that you can't lower |0>. 37 None-

Part IV Two and Three Dimensional Problems

Oct 19 Express the coordinates of two connected balls in terms of the center of mass coordinates. 38 39 26-30

Oct 21 State the definition of angular momentum.
See how the two connected masses becomes a reduced mass around a point.
Use DEL in spherical coordinates; pay attention to the integrating factor.
Seperate variables in the rigid rotor problem.
Understand the generation of the Phi and Theta solutions. 40 31-

Oct 23 Show that l_z on Y gives the z projection of the angular momentum.
Deduce the commutation properties of l_x, l_y, and l_z.
Show l² commutes with l_x, l_y, and l_z. 41 44 32-33

Oct 26 Follow the operator proof that | l' m' > is the same as Y_l,m. 45 46 34-

Oct 28 Apply the 'vector model' to an | l m >.
See a method of localizing Y, and appreciating the cost.
Understand rotational spectra.
Appreciate that the H atom problem can be treated by separation of variables.
See that the Φ solution has the quantum number m. 47 49 35-36

Oct 30 Follow the finish of the separation of R(r) from Θ(θ).
See the similarities in the H atom solution to that of the harmonic oscillator by the power series method. 50 51 37-40

Nov 2 Plot the radial wave functions of a hydrogen atom.
Understand the set up for perturbation theory.
Express the first order energy correction. 52 54 42-47

Part V Perturbation and Variational Theory

Nov 4 Find the first order energy correction for excited states.
Find the first order wave function corrections.
Use perturbation theory to get energy and wave function corrections. 55 57 48-49

Nov 6 Identify why degenerate perturbation theory is different.
Write the degenerate perturbation equations.
Use matrix techniques to solve them.
Understand the matrix/eigenvector/eigenvalue/eigenvector equation. 59 None-

Nov 9 Solve the parve on a sheet perturbation problem.
Use Mathematica to solve degenerate perturbation problems.
Prove the the variational energy is greater than the real energy.
Use the Variational Theorem. 60 61 50-54

Nov 11 Set up the Roothaan-Hall equations.
Apply Huckel MO theory to appropriate molecules. 62 55-58

Nov 13 Use the properties of quantum mechanical operators to develop an intrinsic angular momentum.
Evaluate c in the expression j₊ | j m_j > = c | j m_j+1. 63 59-

Poor Timing Space

Part VI Spin

Nov 18 Construct matricies to represent spin operators.
Build a superposition wave function.
Analyze the Stern-Gerloch experiment. 64-65 60-

Nov 20 Evaluate for two (or more) parves, the conjugation properties of angular momentum.
Specify a state in coupled or uncoupled form. 66-67 61-62

Nov 23 Build a coupled state from uncoupled ones.
Evaluate j² in terms of j_i components.
Operate and find singlet and triplet states. 68-69 63-

**Fall, 2009**
Lecture date	Objective of lecture: In all cases insert the words "Be able to" before the expression.	DP	OP

	Part I Preliminaries
Sept 9	Use Mathematica to solve 0simple problems. Outline the weird behavior for which qm has to account. Know the historical failures of classical mechanics.	1-4	None-
Sept 11	Formulate the probability of an occurrence. Show what normalization means. Calculate the expectation value of a value. Calculate the expectation value of a function. Show that Sin[x/λ - t/T] shows wave behavior. Deduce that the sign of k measures direction of wave motion. Appreciate the formation of a standing wave. State why we use an exponential representation of a wave. Use a phase shift to express a Sin wave as a Cos wave. Express a wave as the real or imaginary part of Exp[iφ].	5-8	1-2
Sept 14	State what two quantum rules go into Schrödinger's formulation. See how Schrödinger might have been led to his equation. Write the general form of Schrödinger's equation.Appreciate the statistical interpretation of ψ. State the realist, orthodox, and alternative positions on qm.	9-12	3-5
	Part II SE and Consequences
Sept 16	Follow, but not necessarily reproduce, the derivation of the <p>, especially where the operator is placed in the expression. Use the rule: <Q> = $∫ ψ* Q$ _op ψ dx . Use the separation of variables concept to write solutions to an equation. Apply separation of variables to the time dependent S.E. Write SE for a parve on a pole (with infinite potential outside the pole. Express the wave function solutions to SE for a POP.	13-16	6-9
Sept 18	Give the energy levels in terms of parameters for a POP. Show that the uncertainty principle holds for quantum systems. Write an arbitrary function in terms of eigenfunctions of the Hamiltonian. Evaluate the coefficients of that expansion. Predict the frequency with which a given eigenvalue is obtained in this expansion.	17-19	10-13
Sept 21	Show how qm and classical interpretations of the POP differ. Apply time dependency to a system. Prove that SE requires ψ curves toward the ψ = 0 axis when (E-V) is positive. Prove that ψ curves away from the ψ = 0 axis when (E-V) is negative (the non-classical situation). Understand that the only acceptable option in the (E-V) case is that ψ and δψ/δx both approach zero.	20-22	None
Poor timing space
Sept 25	Solve POP problem numerically. Solve problems with various potentials numerically.Numerically solve the well with a barrier problem. Write the general solution for a free particle.	23	14
Sept 28	Follow the argument that shows reflection from a barrier even when E > V_barrier. Set up boundary conditions for a parve in a finite well. Use M to solve the resulting transcendental equation.	24-25	15-17
Sept 30	Set up the Hamiltonian for a harmonic oscillator. Use the variable q to get a solution at large q. Introduce waviness with a power series. Evaluate what SE requires of the power series.	26-	18-
Oct 2	Apply the boundary condition to get a recursive equation. Find the coefficients from the recursive equation.	27 28	19-
Oct 5	Mountain Day
	Part III Postulates and Operators
Oct 7	Use the harmonic oscillator results for practical purposes. Know the postulates of QM. Understand communtation of operators. Use Dirac notation.	29 30	20-21
Oct 9	Appreciate Hermitian operators and how to establish it. State the consequences of Hermitian operators.	31 34	22-24
Oct 14	Understand how communtation of operators leads to the Uncertainty Principle. Use the common eigenfunctions of commuting operators.	35 36	25-
Oct 16	See the relationship between a₊ and a_- and the Hamiltonian for a harmonic oscillator. Obtain communtation relationships between the raising and lowering operators. Show that a_-\|n> produces c\|n-1>. Evaluate c. Appreciate that you can't lower \|0>.	37	None-
	Part IV Two and Three Dimensional Problems
Oct 19	Express the coordinates of two connected balls in terms of the center of mass coordinates.	38 39	26-30
Oct 21	State the definition of angular momentum. See how the two connected masses becomes a reduced mass around a point. Use DEL in spherical coordinates; pay attention to the integrating factor. Seperate variables in the rigid rotor problem. Understand the generation of the Phi and Theta solutions.	40	31-
Oct 23	Show that l_z on Y gives the z projection of the angular momentum. Deduce the commutation properties of l_x, l_y, and l_z. Show l² commutes with l_x, l_y, and l_z.	41 44	32-33
Oct 26	Follow the operator proof that \| l' m' > is the same as Y_l,m.	45 46	34-
Oct 28	Apply the 'vector model' to an \| l m >. See a method of localizing Y, and appreciating the cost. Understand rotational spectra. Appreciate that the H atom problem can be treated by separation of variables. See that the Φ solution has the quantum number m.	47 49	35-36
Oct 30	Follow the finish of the separation of R(r) from Θ(θ). See the similarities in the H atom solution to that of the harmonic oscillator by the power series method.	50 51	37-40
Nov 2	Plot the radial wave functions of a hydrogen atom. Understand the set up for perturbation theory. Express the first order energy correction.	52 54	42-47
	Part V Perturbation and Variational Theory
Nov 4	Find the first order energy correction for excited states. Find the first order wave function corrections. Use perturbation theory to get energy and wave function corrections.	55 57	48-49
Nov 6	Identify why degenerate perturbation theory is different. Write the degenerate perturbation equations. Use matrix techniques to solve them. Understand the matrix/eigenvector/eigenvalue/eigenvector equation.	59	None-
Nov 9	Solve the parve on a sheet perturbation problem. Use Mathematica to solve degenerate perturbation problems. Prove the the variational energy is greater than the real energy. Use the Variational Theorem.	60 61	50-54
Nov 11	Set up the Roothaan-Hall equations. Apply Huckel MO theory to appropriate molecules.	62	55-58
Nov 13	Use the properties of quantum mechanical operators to develop an intrinsic angular momentum. Evaluate c in the expression j₊ \| j m_j > = c \| j m_j+1.	63	59-
Poor Timing Space
	Part VI Spin
Nov 18	Construct matricies to represent spin operators. Build a superposition wave function. Analyze the Stern-Gerloch experiment.	64-65	60-
Nov 20	Evaluate for two (or more) parves, the conjugation properties of angular momentum. Specify a state in coupled or uncoupled form.	66-67	61-62
Nov 23	Build a coupled state from uncoupled ones. Evaluate j² in terms of j_i components. Operate and find singlet and triplet states.	68-69	63-

Grading

Grading in this course reflects the importance I place on collaborative learning, working on the problems before class, and contributing to our classroom discussion. Grading will be based on 1000 points, distributed as indicated in the table.

Distribution of Points Among Various Assessment Tools
Quiz 100 points

Exams 300 points

Final Exam 300 points

Discussion and Problem Sets 300 points

**Distribution of Points Among Various Assessment Tools**
Quiz	100 points
Exams	300 points
Final Exam	300 points
Discussion and Problem Sets	300 points

Your letter grade in the course will depend on the number of points out of the thousand available that you obtain as follows:

**Letter Grades Assigned to Points in Course**
A	>80% of points
B	>65% of points
C	>45% of points
D	>40% of points

(Plus and minus attachments to these letters will occur near the respective limits of the ranges.)

Schedule

The schedule of examinations and quizzes is given below. Permission to postpone an examination or quiz will not be given except in exceptional circumstances.

**Important Dates in CHM 331, Fall, 2009**
First Day of Class	9 Sept
Conceptual Quiz One Due	25 Sept
Practical Quiz One Due	30 Sept
Fall Break	12-13 Oct
Conceptual Exam One Due	21 Oct
Practical Exam One Due	28 Oct
Conceptual Exam Two Due	18 Nov
Practical Exam Two Due	23 Nov
Thanksgiving Break	25-29 Nov
Last Class	14 Dec
Takehome Final Exam Due	22 Dec

Honor Code

Chemistry 331 is a Smith College course: the Honor Code applies. Any work that you submit for a grade must be your own work unless specifically indicated otherwise by the instructor. The quiz, the two examinations, and the final examinations are work that you are expected to do on your own. Preparation of the daily problems may be the work of a group. You are STRONGLY encouraged to do problems with others.