The Theoretical Minimum

The Theoretical Minimum series by Leonard Susskind et al.

My favourite books by a long way.  They do exactly what it says on the tin: Provide the minimum level of theory to start doing Physics properly.

Prior to reading these books I had read a lot of pop-science and was always left with an uneasy feeling that I'd been duped. Having chosen to keep the audience broad the authors of most pop science struggle to communicate the concepts, and usually fall back on analogy - or wonder! - creating ambiguity and misunderstanding. (There is one notable exception - Feynman's QED: The Strange Theory of Light and Matter.)

The Theoretical Minimum series side steps the problems faced by most science communicators by assuming the readers do have some tools under their belt.  Specifically basic calculus.  It's then able to take the quickest possible route to the edge of science.  (Okay, maybe not the edge, but it feels like it.)  Just don't skip any bits because there's no fat on it.

Book I: Classical Mechanics

I bought this with Book II (Quantum Mechanics) and TBH I was more interested in the QM book at the time.  In fact I thought that I already had a handle on classical mechanics, and I probably only read Book I because Book II implied it was a pre-requisite.  However, I'm really glad I did, as I learned a lot of stuff I didn't know I didn't know.

The whole Lagrangian approach to mechanics was an eye opener.  It opens up the way to using arbitrary coordinate systems to solve problems, which is useful for GR.  But more importantly it formulates classical mechanics in a way that makes it easier to see the parallels with quantum mechanics.  Or to put it another way, to see how classical mechanics is a limiting case of quantum mechanics.

For me the big new concept was phase space: the idea that the whole state of a classical system can be described as a point in $\mathbb{R}^n$, where n is the number of particles, multiplied by the number of spatial dimensions, multiplied by 2 (for the position and momentum coordinates).  Each point in phase space is associated with an arrow that describes how system evolves from there.  This left me with a mental image of a marble rolling around a (complicated) track as a model for the evolution of the universe.

Book II: Quantum Mechanics

I launched into this with gusto, but by the end of Chapter 3 I was struggling with the level of abstraction.  The stumbling points were the "principles" of quantum mechanics, namely that states are vectors in vector spaces over the complex numbers, observables are hermitian operators, and the values of observables are the eigenvalues of the operators.  At least the authors recognize how difficult this stuff is:

I realize that this is the kind of hopelessly abstract statement that makes people give up on quantum mechanics and take up surfing instead

A couple of weeks after accepting defeat I discovered that all the original lectures that led to the books were available online.  I then watched all of the QM lectures and this time I had much more luck.  I think there are some concepts which are particularly difficult to get across in writing, but in a lecture it is possible to really slow down for these bits and emphasize: this is the difficult bit.  Having completed the course I re-read the book, this time managing to finish it.  After that the physical object became to me very much like a trophy.

The highlight of the book comes at the end, with a complete mathematical treatment of the quantum simple harmonic oscillator.  You feel like you've finished the work Einstein left half done in 1905 when he showed that the photoelectric effect could be explained by assuming light was quantized in units of $\hbar \omega$.  The bit left undone was to explain why it was quantized and  why in those units.

I've now read this book twice, watched the lectures twice, and I'm on my 2nd time through the Advanced QM lectures that follow them (but aren't covered by the book).  I'm getting a bit obsessed. 

Book III: Special Relativity and Classical Field Theory

Top Marx!

As with Book I, I thought I already had a handle on both these subjects.  But again I learned a lot of stuff I didn't know I didn't know.  Top of the list was how the magnetic and electric fields change under the Lorentz Transformation.  Much like how time and space coordinates in $S$ get mixed together to become the time and space coordinates in $S'$, components of the magnetic and the electric fields get mixed together.  So in one frame there may be no magnetic field and in another it appears.  Also the tensor versions of Maxwell's field equations are pretty neat.  If you had those on your T-shirt you'd definitely have a right to look down on anyone with the original version on theirs.

My notes, taken in Handwrite Pro:

Book IV: (In Progress) General Relativity

The online lectures are already available, but I'm waiting for the book.  It's a bit like punting: they give you a paddle, but you want to see if you can get by without it.