Physics 302: Quantum Mechanics

Semester Two 2007-08

Peter Beckmann

Quantum mechanics is one of the most wonderful inventions of humankind. It gives us a window into the realm of molecules, atoms, and elementary particles undreamt of until its development in the period from the early 1900's to the 1960's. It also gives us a significant fraction of our modern technology. Quantum mechanics is a mathematical theory of measurement. It models physical systems in a manner that maximizes the knowledge that can be gained from experiment. It sometimes seems at odds with common sense but then its purpose is to model physical reality, not to fit into the cultural history of humankind. Whereas some fields (like General Relativity) were the brainchild of one or very few people, quantum mechanics involved a huge effort by dozens of scientists and mathematicians over many decades. No one name reigns supreme. I view quantum mechanics, like Darwinian evolution, to be aesthetically beautiful and culturally and personally liberating. Through quantum mechanics, my models of reality can transcend the notions I have had passed on to me for modeling my everyday realm. In that sense, quantum mechanics is like music and art.

Although we will certainly start at the beginning (don't all physics courses?), I will assume that you have made a first pass through one-dimensional quantum mechanics (infinite square well, harmonic oscillator, etc.) and that you have some foggy familiarity with the hydrogen atom. I will assume that you are comfortable with vector calculus and ordinary differential equations. If you have completed (1) Math 201, (2) Physics 206 or Math 203, and (3) Physics 306 (or Haverford equivalents) you will have the appropriate background. Math students who have not done second-year physics may be fine but must get permission from me first, please. Chemistry students are welcome so long as you have the appropriate mathematics background.

The software package Mathematica will be used extensively in this course and many assignments will involve using it. It will be available on many computers in the Physics Department and around campus. The text book is an "Introduction to Quantum Mechanics," second edition (David J Griffiths, Pearson/Prentice Hall, 2005, ISBN 0-13-111892-7). It is an excellent book and we will stick to it very closely.

We are scheduled to meet TTh 11:30 - 1:00. We will have recitation sessions and/or office hours as needed to work on basic concepts and assigned problems. I have an open-door policy. Weekly problem sets will be due every Thursday. During the semester we will have a series of 30-minute and 60-minute take-home exams at the end of each chapter and there will be a three-hour final examination. Part A of this final exam (one hour) will be the last "chapter exam" and Part B of this final exam (two hours) will be a comprehensive exam involving the material for the entire course. Two-thirds of the final grade will be based on the examinations and one-third will be based on the weekly problem sets. The rest of this resume involves the details of the materials covered in the course.

(1) Chapter One (one week) introduces us to the wave function and Schrödinger's Equation in one dimension. Schrödinger's Equation is to non-relativistic quantum mechanics as Newton's Equation is to non-relativistic classical mechanics. Right away, Griffiths tries to make us comfortable with the probabilistic interpretation of quantum mechanics. We are introduced to the momentum operator, the uncertainty principle, and the idea of expectation or average values. In addition to using Mathematica, there will be an Excel assignment here to make sure you can do mathematics using Excel.

(2A) It is convenient to split Chapter Two into two parts; A and B. Chapter 2A (two weeks) begins with the idea of the linear superposition of basis states and then uses the one-dimensional infinite square well, the one-dimensional harmonic oscillator, and the one-dimensional finite square well as examples of solving Schrödinger's equation and investigating the superposition of basis states. This is done in sections 2.1 to 2.3. Mathematica will be used extensively here both for bookkeeping and to do integrals. We learn that in one spatial dimension, solving Schrödinger's Equation leads to one quantum number, an integer that tells us the allowed energies a system can have. We will not do all parts of Griffith's presentation of the harmonic oscillator, it's just too much.

(2B) Two models that will be new to most of you are the notion of a wave packet to describe a free particle and the Dirac-delta function potential in Sections 2.4 and 2.5 (one week). These are very important topics. The former introduces us to the use of Fourier transforms in quantum mechanics and the equivalency of representing the state of a system in either the position or momentum representations. Some of the ideas presented here are mathematically confusing and we shall tread carefully.

(3) Chapter Three (two weeks) introduces the formal mathematical structure of quantum mechanics. We learn about Hilbert spaces, operators, eigenoperators, eigenstates, and eigenvalues. You learned much of the basic mathematics here in your second-year algebra courses (physics 206 or math 203). The new formalism allows us to return to the ideas behind the statistical interpretation of quantum mechanics and the uncertainty principle introduced in chapter one and deepen our understanding of these potentially confusing concepts. Finally, in Chapter Three, Griffiths introduces Dirac notation (bras, kets, and all that) which lets us do everything all over again using different notation. This matters because whereas the Schrödinger equation and its wave function describe physical reality in spacetime, much of quantum mechanics deals with attributes that have nothing to do with spacetime and require a whole new way of thinking. Dirac notation is a first step on this path to freedom from the intellectual confinements of spacetime.

(4A) I think Chapter Four is actually two chapters and I will refer to them as Chapter 4A (Sections 1-3) and Chapter 4B (Section 4). Chapter 4A (two weeks) takes us into three spatial dimensions and now we have three quantum numbers, one for each spatial dimension. If the potential energy function is independent of orientation, then the solutions to Schrödinger's Equation are the spherical harmonics. These functions characterize the isotropy of space and have use far beyond quantum mechanics. (They are the funky lobes encountered in chemistry.) They give rise to two orientational quantum numbers, one characterizing the square of the angular momentum and the other characterizing the projection of the angular momentum on one (and only one) axis. For the Coulomb interaction, the one-dimensional radial wave function gives rise to the hydrogen atom and its quantized energies. We will use Mathematica extensively here since we want to repeat the same calculations (mainly integrals and plots) several times for different states of the hydrogen atom.

(4B) Chapter 4B (one week) is about spin and we deal almost exclusively with a spin ½ system; the simplest physical system in the universe. Although the basic linear algebra is straightforward there seems to be a huge conceptual, almost psychological hurdle here for most students. This is the first time we meet an identity property that is not single valued but has nothing to do with spacetime. For example, charge is a single-valued property (a scalar) that describes particles in a way that does not involve spacetime. Yet we have never met a fundamental non-spatial property that is not single valued (i.e., spin up and/or spin down). (Another example of this is the three colors of quarks.) Since there are no cultural crutches available to us here, we have no choice but to rely on pure mathematics. It is a hugely liberating experience. At this point, students might want to

(a) go to the Philadelphia Museum of Art (or go on line) and look at Duchamp's "Nude Descending a Staircase," (b) listen to Shostakovich's Fourth Symphony, and (c) write an essay on how these two works are similar to quantum mechanics.

(5) In the humble opinion of the instructor, the notion of identical particles and wave function symmetry is the essence of quantum mechanics. Griffiths deals with this beautifully in Chapter Five (one week). This notion of particle symmetry is the cornerstone of chemistry. In Section 1 we learn about distinguishable particles and the two types of indistinguishable particles (fermions and bosons). Section 5.2 leads us into the periodic table and the many interactions in real atoms. I will begin this section bravely and be reduced to mumbling and drooling by the time we get to about neon (Z = 10). Beyond that, you're on your own. (Take a course from Michelle Francl or Sharon Burgmayer.) We will not do Sections 5.3 (solids) or 5.4 (quantum statistical mechanics) which your humble servant teaches as physics 322 and physics 303 respectively.

(12) After Chapter 5, we will look at Chapter 12 (one week) where we consider some of the conceptual, almost epistemological, issues in quantum mechanics. As a society, we seem to still be dealing with the fact that Einstein didn't like quantum mechanics. Griffiths has a beautiful discussion of the Einstein-Podolsky-Rosen paradox ("action at a distance"), Bell's 1964 Theorem (suggesting that hidden variables are inconsistent with quantum mechanics), Schrödinger's Cat (a truly stupid example of the measurement problem - I agree with Griffiths here), and other "issues" concerning the interpretation of quantum mechanics that trouble some folks. These ideas are the cornerstone of quantum encryption, the application of which is becoming quite widespread.

(6) Finally, having learned a lot, we actually do something in Chapter Six (three weeks). There are three problems with Schrödinger's Equation. First, when you put in real potential energy functions, it's just too hard to solve. So, we do perturbation theory to introduce small additional interactions to see how our well known unperturbed energies change. This is especially important for the hydrogen atom where we can learn about several interactions (in addition to the coulomb interaction) between the electron and the proton. Second, Schrödinger's equation is non-relativistic, so we have to introduce relativistic effects as a perturbation - as if it were a new interaction (which it isn't). Third, spin doesn't appear anywhere in Schrödinger's equation (because it has nothing to do with space) and again, it has to be put in by hand - as an perturbative interaction. The latter two problems can be rectified in one fell swoop by using Dirac's Equation. Here relativity is part of the formulation and spin pops out - literally. (So does the idea of antiparticles.) Griffiths does not deal with this but I will provide a handout for students to look at since I think it's important to have at least an introduction to the Dirac Equation. So, chapter Six deals with relativity and the spin-orbit interaction as perturbations which is very instructive. Fittingly, we end with the purely quantum mechanical interaction between the spin of the proton and the spin of the electron which provides the source of the famous 21-centimeter line. Which is why we know that the universe is made mostly of hydrogen and so it would be good to understand it. Applying quantum mechanics to a spacetime-independent interaction in the hydrogen atom leads to a basic understanding of the spacetime of the cosmos as a whole. Go figure.