Quantum Physics World

Tuesday, June 20, 2006

Physics, Math and Mental Gymnastics
While we enter physics to study the fascinating world of black holes, quarks and the quantum, the brutal truth is that mathematics is the central tool of the physicist. Gauss called mathematics the "Queen of the Sciences", and with good reason. If you don't have a solid grasp of mathematics, you aren't going to get very far.

One thing I noticed when getting my degrees in physics was that many of the students found math to be a painful "aside". In one case that really stands out in my memory, I was in a mechancs course and one of the homeworks required the calculation of a brutal integral. I worked very hard by myself over the weekend and managed to get the calculation out with a couple of pages of work. When I returned to class, I was surprised to find that the vast majority of the students had not even attempted to work out the integral. One student had obtained the answer-so he thought-using Mathematica. I looked at it carefully and saw that he had gotten the wrong answer. He argued with me-asserting that the computer cannot make a mistake-but we brought the TA over and it turned out he had entered the integral incorrectly. I had obtained the right answer by working it out by hand.

The student in question had thought he was interested in physics but didn't want to bother with the work of physics-which involves diving into the mathematics. But to become a good physicist-or a solid engineer-you need to bite the bullet and become a master of mathematics. It doesn't matter if you're going to be an astronomer, experimentalist, or engineer-in my view if you want to be the best at what you do in these fields, you should have a solid command of math. So if you are interested in physics but aren't a mathematical hot shot, how can you pull yourself to the top of the field? In my view, the answer is to view mathematics the way you would athletics. A friend of mine who shared this view coined the term "mental gymnastics" to characterize his outlook and study habits.

We all aren't Math Genuises
While for some students thinking mathematically comes natural, most of us aren't ready to master the intricacies of studying proofs when we're college freshmen. This article is written for those of us who aren't automatic math whiz kids. If you are a mere mortal who finds math a bit of work, don't be discouraged. It's my belief that average people can raise themselves up to become very good mathematicians with a little bit of hard work. What we need is some training--we need to train our minds to think mathematically. The best way to think about how you can get this done is to draw an analogy between math and athletics.

To master a sport you have to build your muscles and train your body to react in certain ways. For example, if you want to become a great basketball player, you could be lucky enough to be born Michael Jordan. But more likely, you'll have to work at building a basic skill set, and the truth is even players like Michael Jordan put extra work into their craft. Some activiities you might consider that could make you a better basketball player are

  • Lifting weights to build muscle mass
  • Run sprints to improve your ability to run up and down the court without getting tired
  • Spend a large amount of time shooting free throws, doing layups and practicing basic skills like passing

It turns out that becoming a successful physicist or engineer is in many ways similar to athletics. OK, so suppose you want to study Hawking radiation and string theory, but you are not a hot shot mathematician and weren't the best student. Instead of just reading a bunch of books or lamenting the fact we aren't an Einsteinian genius, what are the mathematical equivalents to lifting weights or running sprints we can do to improve our mathematical ability? In my view, we can begin by following two steps

  • learn the basic rules first
  • repeat, repeat, repeat

That is do tons of problems. In my view a student should start off simple. Don't try to understand the proofs. For example, in my recent book, "Calculus In Focus", I take the perspective that students need to learn math by following the formula: show, repeat, try it yourself. That is

  • Show the student a given rule, like the product rule for derivatives
  • Focus on mastering calculational skills first. Do this by showing the student how to apply the rule with multiple examples.
  • Repeat, repeat, repeat. Do a given type of problem multiple times so that it becomes second nature.

Once the "how" to solve problems is second nature, then go back for a deeper look at the material. Then learn the "why" and start learning the formality of mathematics through proofs and theorems. I use this approach to drill the central ideas of calculus in my book Calculus in Focus. More information can be found at http://www.quantumphysicshelp.com/calculus.htm.


In addition to the basic approach, a certain baseline has to be established if you want to build yourself up for a formal career in math, physics, or engineering. Let's build up a fundamental skill set that is going to build your fundamental math skills and help you master any subject. A few key areas I think students should focus on are outlined below.

The Importance of Algebra
If you study physics or engineering, algebra never goes away. So the first step on the road to becoming the next Stephen Hawking is to master this tedious yet fundamental subject. Do yourself a favor and pick up a decent algebra book and work through it. Do every problem so that by the end of the book, factoring equations, logarithms and other math basics are second nature for you. In the same way that lifting weights is going to make a football or basketball a better athlete when the games are actually played, mastering algebra will pay off later when you're doing your homework in dynamics or quantum theory.

Trigonometry
If you go on to become an electrical engineer and study circuit analysis or decide to master black hole physics, one fundamental area of business you'll have in common with your colleagues is trigonometry. Make sure you know your trig inside and out, learn what the trig functions really mean and master those pesky identities. Also don't over look this one crucial fact-trigonometry also provides a simple arena where you can learn how to prove and/or derive results. We all know that later, when you take advanced physics courses, you're going to see the words "show that" pop up frequently in your homework problems. This is sure to cause headaches among the mere mortals amongst us, but it turns out you can improve your skills in this area in a non-threatening way by deriving trig identities. Instead of viewing the derivation of trig identities as a tedious obstacle, start to look at this as an opportunity. All trig books have homework problems where you have to derive an identity so pick up a trig book and do it until your blue in the face. Take it seriously and write up each proof as if you were submitting a short paper to a major journal. This will teach you how to go from point A to point B mathematically and how to write up a derivation in a formal way that will allow someone else to understand what's going on. If you do, later it will be easier to get through homework in advanced classes, you'll get better grades, and you'll develop a good foundation for writing up theoretical derivations for research papers.

Graphing Functions
While any function can be graphed easily on the computer or on a graphing calculator, it is very important to be able to graph a function on the fly with nothing more than a pencil and paper. The key abilities you want to focus on are developing an intuitive sense for how functions behave and learning how to focus on how functions behave in various limits. That is, how does a function look when the argument is small? How does it behave as the argument goes to infinity? Dig out your calculus book and review techiques that use the first and second derivative to graph a function. I review these extensively in my recent book "Calculus in Focus".

Series and Complex Numbers
In my opinion, understanding the series expansion of functions and the behavior of complex numbers can't be underestimated. If you want to understand physics, you need to master the use of series. Start by learning how to expand a function in a series. Some series should be second nature ('oh yeah, that's cosine"). Learn about convergence. Get a copy of Arfken and review the solution of differential equations using series. Try to get an intuitive feel for cutting a series off at a given term while retaining the essential behavior of the function. These are tools that are important when studying theoretical physics or advanced engineering.

David McMahon is a physicist who consults at Sandia National Laboratories and is the author of Calculus In Focus. More information can be found at http://www.quantumphysicshelp.com/calculus.htm.

Monday, April 17, 2006

Quantum Mechanics Review: Schaum's Outline

Schaum's Outline of Quantum Mechanics
The Schaum's Outline of Quantum Mechanics promises the reader that this is the perfect supplement for the classroom. Whether or not this is true turns out to be a yes and no answer. The book has 16 chapters that cover everything from spin and the Schrodinger equation to scattering and the semi-classical treatment of radiation. Each chapter begins with a brief overview of the subject covered, with quick definitions and statements of equations and theorems. There is very little explanatory text. This is followed by solved examples, and each chapter concludes with a set of problems you can try on your own.



Strengths
Quantum mechanics is a tough course for anyone to get through, so supplements to the ridiculously expenseive and often impenetrable textbooks on the market are welcome. This book is helpful for two main reasons. The first is that it is very thorough. It covers every topic that you are going to see in either a senior level undergraduate course on quantum mechanics or a first year graduate course. Second, it has a good selection of solved problems, pretty much exposing you to the standard batch of problems you're likely to encounter in the standard textbooks.

Weaknesses
The book does suffer from several problems. The first is just the basic presentation style used in the Schaum's outlines. Granted these are not meant to replace a textbook, they are just supposed to be a classroom aid. Even so the extremely terse presentation leaves the reader hungry for more help. Difficult topics are presented in a mere page or two. There could be more explanatory text in the solved problems as well. Again the presentation seems to terse to be of use to students who are really struggling.

Conclusion
My advice is buy every quantum mechanics book you can since its a hard subject to master. The Schaum's Outlines are cheap, so you will only set yourself back maybe $17 picking up this title. So go ahead and add this one to your library. The best use I see for this book is in preparing for a preliminary or comprehensive exam in graduate school, after you've already gone a fair distance toward mastering quantum mechanics. If you are currently taking a course and find yourself hopelessly confused, then look elsewhere - in my opinion this book is not going to be that helpful.

Thursday, April 06, 2006

Learning physics should be easy

While its true that not all of us are Einsteins, should it be so difficult to learn math, science, and engineering that only a small handful of people can get degrees in these fields?

Part of the problem is the way that math, physics, and engineering are taught. I haven't decided if there is a conscious conspiracy or not-but the truth is these subjects are generally taught in a way that is not helpful to most people. Maybe its because the vast majority of people that become professors are simply quite a bit smarter than the rest of us, and they don't realize what they're doing because they just "get it" and figure if you don't "get it" you are'nt cut out to be a physicist or mathematician.

In a typical college experience, I took "Electrodynamics" in graduate school. The professor was a great lecturer, but his lectures were really a complete waste of time. Basically, we spent our days in class listening to him spit out the book. He would recite the theorems and prove them. Had the book not been available his lectures would have been gold, but since we could buy a book, in fact since we were required to buy a book that had all this exact material in it, the lectures turned out to be no help at all.

It's important to reinforce concepts to be sure, but physics, math and engineering are about doing things. These are active fields where problems must be solved. Its not about memorizing a theorem, its about applying it or being able to derive a new one.

Rather than "lecture", I would prefer that professors assign a book they are going to follow and then use the class time to help students solve problems. They should have the students read a given chapter before coming to class, and then spend class showing students how to solve some problems. Homework can then be assigned allowing students to build on what they did in class to learn how to solve the problems on their own.

To make matters worse, physics professors of late seem to want to avoid sticking to a book at all (yet that is what they really end up doing when all is said and done). I don't know how many times in graduate school a professor would announce he wasn't sticking to a particular book, but you may want to buy these 10 different texts. Please.

Maybe physics professors like having a realm of mystery surround them. They like to feel smarter than everyone else and often aren't interested in helping people learn. So they keep problem solving tricks to themselves, and then tell the students who don't "get it" that they should become experimentalists or engineers.

In Quantum Mechanics Demystified I have attempted to provide readers with a format that makes learning physics straightforward. I show you how to solve quantum mechanics problems, and then you can try to do similar problems on your own.

Wednesday, April 05, 2006

Mathematica

In addition to Quantum Mechanics Demystified, with Dan Topa I am the co-author of A Beginner's Guide to Mathematica, a book we hope will help introduce new users to this powerful computational tool. To help readers decide if they are interested in the book, they can download sample chapters at

http://www.davidmcmahonbooks.com/work5.htm

Saturday, April 01, 2006

From the fringes of quantum physics and relativity theory comes Bob Lazaar, an interesting man who claims to have worked on UFO's at Area 51. Tonight on coast to coast AM, the show will be guest hosted by George Knapp, a TV reporter who broke the Bob Lazaar story several years ago.
Lazaar is a very well spoken and charismatic man, but his story doesn't add up. I am not even talking about his wild claim to have worked on UFO's and "antigravity" propulsion at Area 51. Let's just start with his basic storyline.
He claims to be a physicist that worked at several places where a clearance is required, including Los Alamos and Area 51. I don't recall where he claimed to have gone to school, it might have been Cal Tech or MIT, but something that stood out for me was he claimed not to recall any professors names.
Anyone who has gotten a technical degree will recognize this claim as absurd. Students in math, physics and engineering run into hard professors and nutty professors, and good professors that just downright torture you during the semester. At least some of the names of these professors stick to you like glue throughout your life and its something that binds you to your fellow students at the institution where you got your degree. So when I heard Lazaar make this statement it struck me as odd to say the least.

Then there is the problem of his academic record. As I recall he claimed to have an advanced degree in physics yet there was no record of him having attended any of these prestigious instituions. This was explained by the claim that the CIA wiped out his record or something like that.

It has also been difficult to verify his work record. The sole evidence he worked at Los Alamos is a single paystub for a 2 week period where he worked as a technician. Again, I believe this is explained away by the vast powers in the CIA wiping out his record.

Anyone who believes the CIA or any government entity is that powerful or that places like Los Alamos are that secretive has been spending too much time watching television! The fact is its no secret who works at Los Alamos or any other government lab. The only things that are secret are the details of the project they work on. Its easy to find out that Joe Schmo works at X national lab in department Y. To me, the fact that no such record exists for Lazaar indicates that at best he worked as a temp here and there doing contract work. He was not some top scientist that would be called upon in the extremely unlikely event that they needed someone to "reverse engineer" a UFO. In a nutshell, I basically don't buy into Lazaar or his crazy story.

If anything, the Lazaar phenomenon is an interesting study in human behavior and our wish to live in a universe inhabited by aliens. Despite my skepticism I find the story interesting and intend to tune in at least part of the evening to George Knapp on Coast to Coast AM tonight.

Wednesday, March 29, 2006

Algebra and Richard Cohen

In a recent column in the Washington Post, Richard Cohen laments the demise of a high school student named Gabriella who couldn't pass algebra after 7 or so tries. Cohen tells poor Gabriella that he hates math and has never used algebra.

Cohen misses the point. Whether or not you use something later in life is not always the basis of what is important to take in school. It is important to have a broadly educated citizenry who have been exposed to the wide range of human knowledge. This includes among other things history, the arts, science, and yes mathematics.

Furthermore, it shouldn't be lost on people that we live in a highly technical society. The importance of science and technology is increasing, yet the populace as a whole is woefully ignorant in these topics. People are being asked to make decisions about nuclear power, global warming, biotechnology, stem cells, space travel and the like. These are all scientific issues and having a reasonable understanding of them is important. The population should have their own understanding of the issues to a certain degree rather than having to rely on experts for everything. This can only be done through the education system, and while these are conceptually based topics ultimately there is a mathematical underpinning. I am not suggesting that people should be going out and doing their own calculations of say uranium half-lives, but doing some calculations like that in school will allow someone to make more reasoned judgements on many issues-like storing nuclear wastes at Yucca Mountain.

Cohen says that algebra isn't a high or the highest form of human reasoning, and that writing is. Frankly Mr. Cohen I beg to differ. Mathematics is the highest form of human reasoning and is the basic underpinning of our modern society. It transcends the sciences, being at the root of the human genome project, the design of lasers, electric power, radio and cell phones and the internet. In short the entire modern world is fundamentally mathematically based. Writing was already highly developed long before calculus came into existence.

Cohen also claims that a computer or calculator can do math while they can't write. Cohen's understanding of how computers are used in mathematics is naive. In higher mathematics, doing a solution like the quadratic equation is trivial. Its the understanding of the solutions and properties of equations that require higher reasoning. The computer is used to churn out solutions to equations that are too hard for the human mind to solve directly (no analytic or closed form solution). In the end a human being has to analyze and interpret the results-something a computer can't do.

I am sorry that Gabriella couldn't pass the course, but that's too bad. We do need higher standards in this country. Otherwise China and India will come to dominate world affairs in the coming century.

Tuesday, March 28, 2006

One of the themes touched on in "Equations of Eternity" by David Darling is the unreasonable effectiveness of mathematics in describing the physical world. Time and again, as Darling points out, mathematicians have worked on some obscure theoretical idea or area that seems to have nothing to do with reality. Then years later physicists stumble on it and discover that it describes some physical process in absolute detail, down to the last dotted i and crossed t.
A great example of the connection between mathematics and the physical world is the discovery by Maxwell of well, Maxwell's equations. During the 19th century the frontiers of science were being pushed by people studying electromagnetic phenomena. Years earlier Coloumb had figured out how to describe the electric force between two charges. In the early to mid-1800's physics had moved quite a bit beyond that to consider electric currents and magnetic fields. It was here that mathematical insight would prove to be an unusually effective tool-revealing properties of nature hidden to the senses.
In about the 1830's Ampere worked out a "law" that relates the magnetic field to a flow of current. Ampere's law has a very precise mathematical form which was worked out from careful experimental observation. In short Ampere figured out that the curl of the magnetic field B is related to the current density J via



While this "law" was worked out by careful measurement and experimental observation-it violates a tenet of vector calculus. We know from vector calculus that the divergence of the curl of any vector is zero:






Maxwell basically applied this fact to Ampere's law and discovered that Ampere's law could not be true-based strictly on mathematical reasoning. This in turn led to the discovery of radio. Let's get a basic idea of what he did. And so, taking the divergence of the left hand side of Ampere's law, we should get zero



But on the right side, we have






But this isn't zero. In fact, using the conservation of current density together with Gauss's law, it can be shown that this is related to the electric field. That is






So, to make the divergence of Ampere's law vanish, Maxwell added this term. The missing term is the "displacement current" and it leads to a coupling of the electric and magnetic fields-giving us traveling electromagnetic waves or radiation. To sum up Maxwell changed Ampere's law to






These "laws" of vector calculus are abstract mathematical laws--supposedly laws of pure thought. At first sight one might not expect that they would hold precedence over experimental observation. But it turns out they do. Maxwell used the laws to determine what form Ampere's law should really have, and in the process discovered something that was unknown at the time-radio waves.
This is just one small example of the interplay between math and physics. Later we'll explore connections between abstract mathematics and quantum theory which describes every last detail of atomic behavior.