Quantum Mechanics and a Trip To Blackwell In Oxford

In the last post I described how I was beginning to learn Quantum Mechanics again. To that end, I watched the first two lectures of the Theoretical Minimum Quantum Mechanics course.

Since then I’ve managed to watch the next four and I would say I’m starting to gain a reasonable intuition for quantum systems. It’s particularly fascinating that the topic of states, which in Classical Mechanics barely requires 10-20 minutes of explanation, requires almost half the lecture course to describe in Quantum Mechanics!

Susskind has concentrated primarily on the single qubit system for the majority of the lectures so far, but has now started to discuss correlated singlets and the mathematics of entanglement. We’ve also had a good look at the time-dependent Schrödinger equation and (briefly) the time-independent Schrödinger equation (although I suspect we will look at the latter substantially later).

In previous Quantum Mechanics courses that I have studied, the path taken (excuse the pun) towards understanding of the key topics such as the uncertainty principle, entanglement, angular momentum and such was very “experimentally” oriented. That is, some key phenomena was described and then some theory was quickly “rushed in” to provide a model for the effect.

This course is fundamentally different since it teaches QM in a theoretical manner. The mathematical formalism (Hilbert spaces and observables, eigenstates etc) was outlined almost from the beginning. The implications of the mathematical formalism were fully explored for the simplest possible system. In my opinion this is a much better way to go since it helps break down the assumptions about behaviours that hold over from the classical world.

I also decided to take a brief trip to the famous Blackwell bookshop in Oxford yesterday and picked up both of the current Theoretical Minimum books – Classical Mechanics and Quantum Mechanics, as well as Binney and Skinner’s The Physics of Quantum Mechanics and (rather ambitiously) Quantum Field Theory for the Gifted Amateur by Lancaster and Blundell. I decided to skim the latter on the train home.

The book has really helped explain the whole idea of QFT for me, which I had not previously had a good grasp on, as well as the concept of Creation/Annihilation operators and the quantum harmonic oscillator. Having peeked ahead in the Theoretical Minimum text on QM, I see that these topics are both covered in the final section of the book.

I’ve now got four lectures remaining for the Theoretical Minimum QM course and afterwards I hope to find a book (or three) where I can practice QM questions, just as I’ve done with the books on Lagrangian/Hamiltonian Mechanics. I find I only truly gain insight by doing. As one of my mathematics Professors at the University of Warwick was fond of saying “Mathematics is NOT a spectator sport.”

Beginning Quantum Mechanics

Having finished the Theoretical Minimum‘s Classical Mechanics course last week, I decided to start the course on Quantum Mechanics. I was initially hesitant in that I wanted to continue with Classical Field Theory (e.g. Special Relativity, Electromagnetism and General Relativity). However, since the courses were ordered this way I was nervous about missing out!

I’ve watched the first two lecturers so far. The pace is about right and gives you time to consider the concepts in detail (there’s also the pause button for that!). However, a lot of time was given over to introducing the concept of a (complex) vector space. Since I studied mathematics at undergraduate level, this material was quite straightforward for me and was simply revision from my linear algebra courses. Professor Susskind also briefly mentioned the concept of an infinite-dimensional function space, which again I’m quite familiar with, having studied linear analysis.

The key difference for me was finally seeing a solid application of functional space theory, something which is not often discussed in a formal mathematical setting. We were introduced to the concept of a “qubit” or “spin” and the counter-intuitive behaviour of preparation/measurement of quantum mechanical systems. It was shown that complex vector spaces naturally encode the behaviour of quantum systems.

In the next lecture I believe we will be covering Hermitian matrices/observables and the Pauli spin matrices. This will hopefully fill in some significant gaps in my knowledge that I became aware of when I last attempted to learn Quantum Field Theory.

Once again I highly recommend these courses. The pace is very good and the questions by the audience often probe issues you might otherwise not consider. I’m very much looking forward to continuing the journey!

Finishing the Theoretical Minimum’s Classical Mechanics Course

I’ve just finished watching the lecture series on Classical Mechanics at the Theoretical Minimum.

The final two lectures concerned the motion of a charged particle in time-independent electric and magnetic fields. The majority of the lectures concerned a uniform magnetic field as the electric field was shown to be straightforward, by virtue of the previous material studied.

The magnetic field was much more interesting. Ultimately the plan was to derive the Lorentz Force Law, using a Lagrangian procedure. The thrust of the derivation involved the use of vector potentials and gauge invariance. Professor Susskind was apt to highlight the extreme importance of the principle of gauge invariance within theoretical physics, so I’m looking forward to learning more once we consider particle physics.

This is actually my first “formal” (i.e. lectured) introduction to the world of electrodynamics. The only previous exposure has been through sporadic motivating examples in the more mathematical setting of vector analysis, as well as a few chapters in general relativity texts. Hence I am very much looking forward to learning about Classical Field Theory in the months ahead. In order to accomplish this I will be following the Special Relativity and Electrodynamics course at the Theoretical Minimum, as well as ordering Volume II of the acclaimed physics series of Landau and Lifshitz, named The Classical Theory of Fields.

However, in the short term I really need to make sure I fully understand the Principle of Least Action, the Euler-Lagrange Equations, Hamiltonian’s Formulation of Mechanics and Poisson Brackets. Other areas that were left out of the Theoretical Minimum course are Chaos, Hamilton-Jacobi Theory and the Lagrangian procedure in continuous systems. The latter I expect to cover in depth in courses on Classical Field Theory.

Once I’m confident that I can apply the Lagrangian/Hamiltonian frameworks to simpler problems in classical mechanics, I will switch tack and carry out the Quantum Mechanics course.

It is going to be an extremely interesting next few months! I will keep everybody abreast of my progress. In addition, if anybody else is also going through this course, it would be great to hear from you. Please write about your experiences in the comments.

Continuing with Classical Mechanics

In the previous post I discussed Professor Susskind’s Theoretical Minimum course. I’ve been continuing with the Classical Mechanics lectures and delving into alternative formulations of classical mechanics.

It has been absolutely fascinating, from the position of an undergraduate mathematician who went directly into engineering research, to study formulations of the laws other than the traditional “F=ma” approach of Newton. Progressing from the Newtonian method, through to the Principle of Least Action, then onto the Hamiltonian approach and finally through to Poisson Brackets has really filled in a substantial part of the missing pieces in my knowledge of this part of theoretical physics.

For instance, I was always confused about the importance given to commutating angular momentum operators in Quantum Mechanics. It felt extremely abstract and I was struggling to find a rationale when I studied it before. In addition, I picked up a Quantum Field Theory textbook (by Ryder) recently and tried to work my through. I was very rapidly thrust into the world of generators and continuous rotational symmetry. While I was happy working through these in a mathematical sense, I just did not have the physical insight.

Now that Poisson Brackets have provided such an insight, I feel very confident that I’ll be able to work through the Theoretical Minimum Quantum Mechanics (and ultimately QFT!) courses without too much difficulty.

Another benefit of learning these alternative approaches is that I feel like I can really attack a completely different set of (classical mechanical) problems now, that I otherwise considered extremely difficult to achieve under the Newtonian force-based approach. In that vein I have ordered some new books, which provide a large amount of interesting solved problems to work through.

One aspect that I want to clear up (and I will likely ask this in the Physics Stack Overflow portal) is whether or not the equations of motion for fluid dynamics are ever derived from an Lagrangian/Hamiltonian approach. From what I could briefly gather from the Theoretical Minimum lectures, since viscous fluids have velocity dependent forces associated with them, they cannot be derived from a Lagrangian or Hamiltonian.

I now only have two more lectures on the Classical Mechanics course to complete. These are both on applying all of the techniques learned to date to electromagnetism. No doubt the talks will as fascinating as the previous lectures have been.

The Theoretical Minimum with Leonard Susskind

I’m extremely fortunate to have come across a set of courses run by Stanford University and the renowned physicist Leonard Susskind, called The Theoretical Minimum. The core courses are a set of theoretical physics modules at an approachable undergraduate level that (eventually) help a mathematically-inclined viewer gain an understanding of the current frontiers of theoretical physics.

The courses follow a logical order of introducing Classical Mechanics (particularly the Lagrangian and Hamiltonian formations), followed by Quantum Mechanics, Special Relativity and Electrodynamics. General Relativity and Cosmology are then treated leading to a final course on Statistical Mechanics.

I can’t recommend these courses enough. Leonard Susskind is an exceptional lecturer. In equal parts humble and but also highly experienced and intuitive with the material. He doesn’t waste precious lecture time with extensive derivation, but provides resources to do so elsewhere (or encourages you to have a go!).

However, the courses are not full accounts of each topic. After all, they have been designed as the “theoretical minimum” needed to be studied prior to gaining an understanding of modern physics. Each course will require significant study outside of the lectures themselves. This is by design – it allows the student to get to grips with the main themes, even at a reasonable degree of mathematical sophistication, without being confused by extensive details.

I’m currently working through the modules on Classical Mechanics. To date I have made notes on lectures 3-6, which begin with the Principle of Least Action (really it should be Stationary Action!), then the Langrangian and Euler-Lagrange equations, symmetry and conservation laws (including Noether’s Theorem) and finally Hamiltonian Mechanics and conservation of energy.

I decided to skip lectures 1 and 2, since I was relatively familiar with the material from undergraduate mathematics. Mathematicians spend a good deal of time considering ordinary differential equations, which form a big component in Newtonian mechanics. That being said, it has been some time since I considered complex phase-space dynamics so after lecture 7 I think I will go back and watch the first two for completeness.

If you wish to gain an insight into the current state of theoretical physics then you should really consider working through the courses.