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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. Richard Feynmann, the 1965 Nobel Laureate in Physics, has said (in QED, The Strange Theory of Light and Matter),
"What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school--and you think I'm going to explain it to you so you can understand it? No, you're not going to be able to understand it. Why, then, am I going to bother you with all this? Why are you going to sit here all this time, when you won't be able to understand what I am going to say? It is my task to convince you not to turn away because you don't understand it. You see, my physics students don't understand it either. That is because I don't understand it. Nobody does."
In this course we will work our way through the recipes and try to reach an accomodation with 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.
I have posted answers to the exercises to number 119. 1318 14/xi/11 I now have control of the web page again. I have begun to post things that have happened since my loss of control. Will continue that as the days proceed. Let me know if any fail to work. 1635 7/xi/11
Robert Linck
Office, Ford Hall 311
Phone, X3836
Email, rlinck(at)email.smith.edu
Office Hours, CHREM, M, W, F, 8-8:30; Also Tu, 8-11; W 1-3; Th 10-12; or by appointment.
| Item | Link |
|---|---|
| Exercise Set 1 in a Mathematica notebook; exercises | 1-19 |
| Exercise Set 1 as a pdf file; exercises | 1-19 |
| Mathematica Notebook for | Mon, 12 Sept |
| Mathematica Notebook for | Wed, 14 Sept |
| Exercise Set 2; exercises as M. notebook | 20-55 |
| Exercise Set 2; exercises as pdf file | 20-55 |
| Notebook for Lecture on Monday | Sept 19 |
| Second Notebook for Lecture on Monday | Sept 19 |
| Answers to Exercise Set One | Probs 1-19 |
| Answers to Exercise Set 2, | complete |
| M. nb for numerical integration | Friday, 9/23 |
| M.nb showing the Manipulation | command |
| Notebook for Triangular Potential | Fri, 9/23 |
| Notebook for | numerical harmonic oscillator |
| Notebook for | step function |
| Notebook for | POP with barrier |
| Notebook for transcendental equations and | parve in a well |
| Exercise Set 3, Exercises | 56-74 |
| Answers to Exercise Set Three | complete |
| Notebook for Friday | Oct 14 |
| Exercise Set 5, | Exercises 104-110 |
| Exercise Set 4 | Exercises 75-103 |
| Notebook for First Order Perturbation Theory of a | Parve on a Pole |
| Notebook for Second Order Perturbation Theory of a | Parve on a Pole |
| Notebook for | Matrix Manipulation |
| Exercise Set 4 | Answers, to Exercises 75-103 |
| Notebook for | Variational Method |
| Notebook for | Huckel MO theory |
| Finite Matrix Method | Notebook |
| Exercise Set 5 | Answers, to Exercises 104-119 |
| Exercise Set 6 | Exercises 120-139 |
| Exercise Set 7 | Exercises 140-157 |
| Exercise Set 8 | Exercises 158-161 |
| Exercise Set 9 | Exercises 162-181 |
| Exercise Set 6 | Answers, to Exercises 120-139 |
| Exercise Set 7 | Answers, to Exercises 140-157 |
| Exercise Set 8 | Answers, to Exercises 158-161 |
| Some Answers for Exercise Set 9 | are available |
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 our heads 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, room to be determined; with CHREM sessions at 8 every morning.
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 I am bribing you: of the 1000 points in the course (see below, grading), 300 are available for doing problem sets and talking in class. The latter can be just asking questions, or at least trying to answer the ones that I ask. With respect to the former, I 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) and will compromise a reasonable portion of your grade. You are responsible for doing the assigned problems before each class. Because one purpose of the problems is to see where you are, the daily assigned problems must be handed in when you arrive in class. There are a lot of other problems that will be handed out; I would think as an intellectual exercise, you would work all of them. I would urge you to work every problem in the set if you really want to understand quantum mechanics. We will spend a reasonable portion of CHREM sesssions or 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 spend some time just sitting around in class, as a community of scholars, talking 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.
CHREM (CHemistry Reviews in the Early Morning) sessions are a chance for you to ask questions and for us to work out problems together. These meetings will occur each class day morning from 0800 to 0830, the start of class in Bass 210.
Text. I had a great deal of trouble finding a text for this course. When preparing and revising the material over the past four years, I used a set of twenty-six 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.
| Lecture date | Objective of lecture: In all cases insert the words "Be able to" before the expression. | Exercises DUE BEFORE CHREM this day |
Exercises covered to date. |
|---|
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.
| 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:
| 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.)
The schedule of examinations and quizzes is given below. Permission to postpone an examination or quiz will not be given except in exceptional circumstances.
| First Day of Class | 9 Sept |
| Conceptual Quiz One Due | 26 Sept |
| Practical Quiz One Due | 30 Sept |
| Fall Break | 8-11 Oct | Conceptual Exam One Due | 21 Oct |
| Practical Exam One Due | 24 Oct |
| Conceptual Exam Two Due | 17 Nov |
| Practical Exam Two Due | 21 Nov |
| Thanksgiving Break | 23-27 Nov |
| Last Class | 14 Dec |
| Takehome Final Exam Due | 22 Dec |
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.
<ψ|a+a-|ψ>
ψ(x,t) =
Σcnψn(x) e-iEt/hbar
[x, px] = i hbar
[x, py] = 0
p = - i hbar δ/δx
<x> = ∫ ψ* x ψ dx
H ψ = E ψ
ψ = A ei k x + B e-i k x