MSc started

I’ve started my MSc! And this is my last blog post – I’m going on a 18 month Internet detox. I would advise others to try it as well since if everybody in the world decreased their web surfing by an hour a day, and used this time to think there should be some good ideas generated by that extra billion hours or so of human thought….imagine the possibilities.

Question: In your previous week of web surfing how much can you remember? How about last month? Last year?? If you can’t remember anything of it then the time was wasted!

MSc courses chosen

This is a study log and it’s been a while since I’ve updated it. So here goes…

• Worked through half of Goldstein’s classical mechanics while using a couple of other texts that introduced flows on manifolds etc.
• Completed some of the 1B mathematical methods course concentrating on  PDEs and Fourier analysis.
• Worked through a complex analyis text up to the residues\poles section.
• Lately it’s been chapter 1 of Shankar’s Principle’s of Quantum Mechanics in conjunction with the first 100 pages or so of Dirac’s Quantum Mechanics. I’ll then progress through the next few chapters of Shankar’s book until I’m feeling ‘comfortable’ with the basics of QM.

After a few meetings with staff at my university I’ve been able to nail down a course selection (which means I can focus my preparation even better):

Semester 1, 2009

•  Quantum Physics  (using Sakurai’s Modern Quantum Mechanics)
•  Complex Analysis
•  Algebra & Number Theory (mainly group theory)
•  Electromagnetic Theory II (using Jackson)

Semester 2, 2009

• Algebraic Methods of Math Physics (Lie algebras & superalgebras; quantum groups & algebras; Hopf & quasi-Hopf algebras; affine & Kac-Moody algebras)
• Adv Hamiltonian Dyn & Chaos  (Hamilton-Jacobi methods. Perturbation theory, integrable systems, KAM theorem, area-preserving maps, Pioncare-Birkoff theorem …)  
• Master’s Project or Thesis (1st half – select topic, prepare summary, read relevant papers …)

Semester 1, 2010

• Advanced Quantum Theory  (Introduction to Quantum Field Theory)
• Advanced EM Theory  (More Jackson)
• Project or Thesis (2nd half including poster, presentation, finalize thesis)

 My weakness in my preparation is EM – I’ll have to fix this later in the year with the help of Griffiths’ Introduction to Electrodynamics.

Part 1B Bookshelf

There have been many hits on my old bookshelf post (not sure why but any hit is appreciated) so here is a more recent one showing the books I have ready for my Part 1B study. Click the image to obtain a better quality picture so that the book spines can be read more easily.  This photo was taken a while ago – For Linear Algebra II which I have completed, I stopped using Linear Algebra by Curtis and instead used Hoffman and Kunze which I feel is much better. I have also added Serge Lang’s Undergraduate Analysis to the shelf.

Mini book reviews

I’ve just noticed that my library’s computer system has a new feature that can produce a list of books previously borrowed. This makes it much easier to reborrow a pre-borrowed book with a forgotten title. It also makes it easier to write a blog post about this list (cut and paste) – so here is a mini review on each book (about 3 months worth of borrowings). These reviews will help me remember which books were helpful and may assist others as well…

•  Lie groups for pedestrians.  Lipkin, Harry J.    A classic and often borrowed out from the library. 
   
• A second course in mathematical analysis / by J. C. Burkill and H. Burkill.  Burkill, J. C. (John Charles).    Suggested reading for Analysis II. A good book with some solutions in the back.
   
• Principles of mathematical analysis / Walter Rudin.     A concise classic.
   
• Mathematical methods for physicists.  Arfken, George Brown    Another classic. I like the classics. Great for improving ones ‘methods’
   
• Calculus on manifolds : a modern approach to classical theorems of advanced calculus.  Spivak, Michael   An advanced undergraduate calculus book. Read this one to learn calculus the correct way.
   
• Tensor calculus.  Spain, Barry  Is supposed to be a classic but the Dover book on the same topic is better.
   
• Sources of quantum mechanics / edited with a historical introduction by B.L. van der Waerden.    Interesting history of QM.
   
• Quantum mechanics : symbolism of atomic measurements / Julian Schwinger   The first couple of chapters on measurement algebra is good reading, plus provides insight into Schwinger’s unique way of thinking.
   
• Mechanics / Translated from the 4th German ed. by Martin O. Stern.  Sommerfeld, Arnold  A quirky mechanics book – examples about gyroscopes at sea and bicycles.
   
• Theoretical concepts in physics : an alternative view of theoretical reasoning in physics / Malcolm S. Longair   A ‘must read’ for every physics undergraduate – excellent.
   
• Differential forms : with applications to the physical sciences.  Flanders, Harley.   I only had a quick look. Looks like a good introduction.
   
• Calculus.  Apostol, Tom M.   This text is mentioned a lot on physicsforums.com so I decided to take a look. A good text requiring hard work by the reader.
   
• An introduction to probability theory and its applications.   I’ve been neglecting probability. I read the first chapter. And I’m still neglecting probability. But it is a good book and I’ll be sure to pick it up again once I start learning some statistical mechanics. 
   
• Inward bound : of matter and forces in the physical world / Abraham Pais.    Very detailed history of 20th century physics. Take this one on a long cruise.
   
• A first course in mathematical analysis.  Burkill, J. C.  A gentle introduction. Feels somewhat too easy so I’m using this in conjuction with Serge Lang’s ‘Undergraduate Analysis’
   
• Dirac : a scientific biography / Helge Kragh.    Interesting biography but I feel it somehow seeks to diminish his achievements. 
   
• Space-time-matter / Translated from the German by Henry L. Brose.  Weyl, Hermann   I’m sorry, but reading about the philosophical implications of space-time is a bit weird.
   
• Introductory statistical mechanics / Roger Bowley, Mariana Sanchez.    A good introduction – I borrowed this book for an early look.
   
• Radiative processes in astrophysics / George B. Rybicki, Alan P. Lightman.    Was highly recommended in Longair’s book above. Clear, interesting applications of EM theory.
   
• Consistent quantum theory / Robert B. Griffiths.    Avoiding the Copenhagen interpretation while teaching QM is like avoiding determinants while teaching linear algebra.
   
• Foundations of mathematical physics / Sadri Hassani.  - Another excellent methods book.
   
• Undergraduate analysis / Serge Lang.   – In my last post I wrote more about this book. I’m hoping to move onto Lang’s Complex Analysis book afterwards.
   
• Sin-itiro Tomonaga : life of a Japanese physicist / edited by Makinosuke Matsui ; English version edited and annotated by Hiroshi Ezawa ; translated from Japanese by Cheryl Fujimoto and Takako Sano.   Feynman won the Nobel prize together with Schwinger and Tomonaga. I’ve read a few Feynman biographies but have always been annoyed that while Schwinger’s career is described in detail, often Tomonaga only gets a paragraph or two.  This fixes this imbalance.
   
• Geometry, topology, and physics / Mikio Nakahara.  Very highly recommended on Amazon but out of my league. Perhaps in a couple of years I’ll be able to get past chapter 1.
   
• Quantum field theory / Eberhard Zeidler   See previous post about this one. One of the best books on this list.
    
• Thinking like a physicist : physics problems for undergraduates : a collection of problems and solutions / written by the staff of the Physics Department of the University of Bristol and edited by N. Thompson.      Landau expected that his students of physics should be completely comfortable with undergraduate mathematics (so that mathematical details don’t hinder physical understanding). After learning mathematics for a few years and it’s time to start learning physics again try this book and see how little everyday physics you may know.
   
• Beautiful models : 70 years of exactly solved quantum many-body problems / Bill Sutherland.  Exactly solvable models are a research area at my local university so I borrowed this book to get an introduction. I really need more QM study before looking at this book again.
   
• Lie groups, Lie algebras, and representations : an elementary introduction / Brian C. Hall.   Great introduction but have had no time to start serious reading.
   
• Quantum field theory / Lewis H. Ryder.   Borrowed this one for a look-see. 
   
• Elementary applied differential equations : with Fourier series and boundary value problems / Richard Haberman.    Takes a gently-gently approach that sends me to sleep. I’d rather have a more concise book.
   
• Beginning functional analysis / Karen Saxe     I liked this book, and then I didn’t like it, and then got mixed vibes. I think some important technical details were glossed over in an attempt to provide a smooth introduction which is good for some. But, I would have to go back and fill in the gaps anyway so I’d prefer a more rigorous text.
   
• Green’s functions / G.F. Roach.    I didn’t realize there was enough about Greeen’s functions to fill a book! No prizes for guessing what color this book is. Green’s functions are so important to mathematical physics that I’d think it would be well worth the time to work through this book.
   

Still progressing and still here

Lately my blog posting frequency has been averaging about once a month or worse. But study has been progressing faster than ever. It seems like the more trouble my study has that the more I post about it. Anyway, this is a ’study log’ so I’ll update it with my current status…

Tripos 1B Topics

• Linear Algebra II – I’m now up to the last 2 chapters of Hoffman and Kunze (the important ones!). Schaum’s ‘Linear Algebra’ is a great companion to this book and gives some variety to the topic. I’ve happily found that I can now breeze through the chapters in my quantum mechanics books that introduce the bra-ket notation and operators on Hilbert space topics. This motivated me to borrow some functional analysis books from the library which at the moment is becoming my favorite ‘pure’ math subject (probably because it’s directly beneficial to impure math subjects).
• Topology and Metric Spaces – I only found Linear Algebra interesting once I got past all the theorems and proofs regarding elementary topics and started learning about various operators (decomposition, projections, spectral theorem etc.). So I’m now hoping that Topology will become interesting once I get past the basics as well. So in other words, I’m forcing myself to memorize the basic definitions and theorems regarding continuity, compactness and connectedness so that I can move on to the move interesting stuff. Will also be of direct benefit to complex analysis later as well (not to mention those functional analysis books)
• Analysis II – I’ve got Serge Lang’s analysis book waiting on my shelf. I’m still undecided whether this book is suitable for a first or second book on the topic.

Bridging knowledge from undergrad to grad

After studying undergraduate mathematics for a couple of years, browsing math\physics web sites and reading math books it’s natural to have a very sketchy idea about a large number of terms and topics. Things like…

• Differential Forms
• Homotopy
• Renormalization
• Asymptotic freedom
• Compactification
• Functors
• Geometric Langlands
• etc. etc.

Often the basic idea behind these topics only becomes obvious after committing much time to studying them. By ‘basic idea’ I mean an intuitive mental model that serves as a base to build upon and that has links to other mathematical concepts so that it is anchored in one’s mind. But sometimes a kind author provides a summary of their decades of experience so that students are pointed in the right direction quickly. Lately i’ve come across some material such as this – the most useful book has been…

 Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicistsby Eberhard Zeidler

I was disappointed with ‘The Road To Reality’ – I really wanted to like it but couldn’t because I was expecting something just like Zeidler’s book instead. Zeidler introduces just about every topic in mathematical physics you could think of within about 1000 pages. There are also hundreds of interesting quotes from famous mathematicians and physicists. Many hundreds of references are also given so that given topics can be explored further.

Amazingly this 1000 page volume is meant to be an ‘overview’ with Zeidler exploring topics in depth in further volumes that are yet to be published.

Two month study summary – Part 1A finished

Lack of posts on this blog have been caused by a succession of good books and by using every spare moment for some study. So here is my study log entry in point form for the past 2 months:

• Finished reading Theoretical Concepts in Physics: An Alternative View of Theoretical Reasoning in Physicsby Longair. I wish I had read this book during my undergraduate degree – if you are thinking of quiting studying physics then read this book! If you think thermodynamics is boring then read this book. If you think Einstein was just lucky to think of this ‘thought experiment’ first and then followed a logical progression in ideas to general relativity then read this book too. Every student of physics should read this book after their first undergraduate year. Where else can you read about how to determine the kiloton yield of a bomb by simply viewing a movie of a test and then using dimensional considerations?

• Finished reading Dirac: A Scientific Biography by Helge Kragh. Not as interesting a read as Feynman’s biography of course but after reading the biographies of Schwinger and Schrodinger I’m running out of scientists. Anecdotes about his unique personality are always interesting. Surprising items were that he was married (In my mind I imagine the a conversation like ‘Have you ever watched the full moon rise dear?’. ‘No. I did see the moon once 3 degrees above the horizon so I was 12 minutes late’) and that he was an avid traveller often visiting the USA from the UK only to return home the long way around while stopping off to climb some Russian mountains. Another interesting aspect written about in the book was how collaboration was carried out between scientists using snail-mail. Some researchers would find out only months later that their ‘breakthrough’ had already been worked out in another country.

• Finally completed my Part 1A study (example sheets, past exams, everything). I’m now onto part 1B Linear Algebra II and Analysis II.

• And best of all I have just moved house – I’m now only 10 minutes walk away from the University of Queensland and this library. While taking an afternoon walk, I dropped in to the library to browse and found an English translation of a book about Tomonaga – more about that in another post.

Extra books

I have a 12 book limit at my university library and last week I only needed to borrow 6 books for my part 1B study. So with a free hour to spare before my IT project meeting I decided to get a few extra books. I always keep a list of interesting books handy for just such a situation and here are the results:

Theoretical Concepts in Physics: An Alternative View of Theoretical Reasoning in Physics by Longair

The Amazon reviews give it 5 stars and so do I. This book surveys all the major areas of physics using case-studies that highlight the historical background, scientific method and mathematics used. It’s pitched at the 2nd/3rd year undergraduate physics student which is exactly where I’m at. In Longair’s words the book contains what the physics degree often ’squeezes out’ – which in this case is a whole lot of interesting physics both modern and historical.

Quantum Mechanics: Symbolism of Atomic Measurements by Schwinger

I’ve run out of Feynman material to read so Schwinger is the next logical choice. I’ve previously checked out Quantum Kinematics and Dynamics – but found the notation archaic even though the text regarding ‘Measurement Algebra’ was interesting. What I like about Schwinger’s style is that he doesn’t try to dance around for 30 pages talking about blackbody radiation etc. to gently lead the reader to the way QM works – he tells it like it is. He says (in my words) ‘The world is quantum mechanical. That’s just how it is. If a violent measurement is made on a particle how can you expect it to remain in the same state? Nature can’t be subdivided into infinitely small parts – there is a limit. The effects of this limit can’t be ignored when measurements and predictions are made.’

Longair talks a lot about ‘model building’ in physics. Schwinger in his 25 page prologue to this book helps the reader build the correct model for quantum mechanics and is great reading.

For readers who like learning about the lead-up to the formulation of quantum mechanics in all it’s tedious detail I recommend the first 2 chapters of Quantum Mechanics: An Introduction by Greiner.

Mechanics: Lectures on Theoretical Physics by Arnold Sommerfield 

I was alerted to Sommerfield by an old 1994 sci.physics post that said ‘Sommerfeld is God for mathematical physics’. Edward Teller on this page describes him as a ‘terribly stiff, formal individual’ – by looking at the portrait inside the book’s front cover I’m not surprised. But flicking through the book in the library I think he has a unique way of explaining physics and is worth a read. I’ve also neglected mechanics too much in my past studies and want to change that. Of course I’ve got a copy of Goldstein’s Mechanics on my book shelf for when I get serious. By the way, in Sources of Quantum Mechanics (see next book) it is mentioned that Dirac learned Hamiltonian Mechanics from one of Sommerfield’s other books, Atomic Structure and Spectral Lines 

Sources of Quantum Mechanics by Van Der Waerden

Why would I read this book when I’ve called the lead-up to the formulation of quantum mechanics ‘tedious’ ? Well, I found it highly recommended from somewhere on the net while surfing and therefore added it to my book list. The book gets to the point quickly without a lot of mathematical derivations. The book’s sole reviewer on Amazon gives it 5 stars and I agree with his comments.

I better finish this post and get back to my example sheets – I think I last saw them a couple of days ago under this pile of books somewhere…

Pesky Integrals

Integrals are a first year topic in the majority of mathematics degrees and therefore should be easy. And they are easy – the problem is that there are many varieties to memorize. And once memorized (or crammed) for the exam they are often promptly forgotten.

I remember in my 3rd year at university, occasionally the professor would be working through equations at the board and would arrive at an innocent looking integral. The professor would then say something like ‘and that integral should be no problem for the students here so we’ll move on’. The integral would look vaguely familiar but the technique used to solve it would be long forgotten – many blank stares would result…

I found myself in the same position while working through some vector calculus example sheets. I successfully applied what I had remembered from the lecture notes only to be stuck with an integral that contained the square root of a trigonometric expression.  I tried a couple of different methods and failed. The next day I tried again, gave up and finally allowed myself to check the answer in my calculus textbook. Surprisingly, I found that the method used to solve that integral has quite ’stuck’ in my mind and I don’t think I will forget how to solve it again. I therefore have a new (for me anyway) study method to remember integrals:

1. Find an integral you don’t know how to solve. Do not look it up in a table of integrals!

2. Try 2 or 3 different methods of solution – substitution, integration by parts etc.

3. If still unsuccssful resist the urge to immediately look up the answer – sleep on it and try again tomorrow.

4. Repeat step 2. Don’t look up the answer! Try again…feeling frustrated? Do you really, really want to know how to solve that integral? If no, goto step 2, if yes – you are ready for step 5.

5. Look up the method of solution and see where you went wrong. Slap your forehead – yell out ‘of course!’. I guarantee that you’ll remember how to solve that integral for much longer compared to if you had looked up the integral immediately.

The above study method isn’t useful if an exam is looming in a couple of weeks. But, if you come across the occasional integral that you have memorized and then forgotten repeatedly it should help. 

My 12 month study plan

My Part 1A example sheet review has been proceeding very slowly due to the amount of programming I’ve been doing. I’ve completed my Groups & Geometry revision (except for the group part). I figure I’ll forget my group theory in another 3 months anyway so it would be more efficient to wait until I’m doing something that requires it.

 I’ll then move on to Vector Calculus revision this week which should be quick (famous last words). Finally, I’ll then review my Analysis and may even work through the first half of Rudin’s, Principles of Mathematical Analysis in preparation for the Part 1B Analysis II course.

Then the plan is to immediately move on to the Part 1B courses (didn’t I say something like this 2 months ago??). I’ll do the Michaelmas courses first since it’s obvious that the courses are structured to be done this way. So the order will be…

1. Analysis II
2. Linear Algebra
3. Methods
4. Metric and Topological Spaces

I’m giving myself 3 months for the above and I’ll do some concurrently. I’ve already done some multi-variable calculus and have read ahead about inner-product spaces etc. in preparation for the QM course. I’ve also been reading Sutherland’s, Introduction to Metric and Topological spaces to spice up my study (I can hear the chuckles). Then it will be on to…

1. Geometry – using Do Carmo’s book, Differential Geometry of Curves and Surfaces
2. Complex Analysis
3. Complex Methods

I’m not sure how much time will be needed for these but I guess it will be around Christmas 07 before I’m finished. I’ve done complex analysis and methods back in my B.Sc. degree so it will depend on if my mathematical memories have been suppressed or deleted by the events of the past decade. But hopefully I can revise these topics quickly so I can move on to …

1. Quantum Mechanics – wave functions, operators …
2. Special Relativity – time dilation, length contraction and 4-vectors. Also part of my B.Sc.
3. Electromagnetism – I’ll work through Griffith’s book.

Again, some of the above I’m already familiar with but I’ll be testing myself to make sure using book problems and example sheets.

Why am I doing all this? Well, I’ve just been accepted into a M.Sc. course at my university to start in August 08. So if that’s not enough motivation then I don’t know what is. Now if I can just stop procrastinating by reading Spivak’s, Calculus on Manifolds and Arnold’s Ordinary Differential Equations and get some real study done.