Finishing Quantum Mechanics at the Theoretical Minimum

Yesterday I finished the final lecture in the Theoretical Minimum Quantum Mechanics course. I very much enjoyed the final two lectures as we finally began discussing the one-dimensional particle case. It was fascinating to see the interplay between the position space and momentum space, which was simply a manifestation of the Fourier Transform. I’m also gaining a good ability to calculate using the logic of QM, namely that of infinite-dimensional vector spaces. This isn’t surprising, since this was a big part of my undergraduate mathematics degree, but it was rather exciting to see it applied to less abstract situations.

While I’ve now finished the lectures, I still have to finish the accompanying text. I’m actually now onto the material covered in the final two lectures so am not too far behind the video content. The book does include some discussion on the quantum harmonic oscillator, as I alluded to in previous posts, so this will hopefully prepare me somewhat for tackling -very basic- introductory Quantum Field Theory texts.

Again, as I’ve alluded to in previous posts I’m now going to move onto the Special Relativity and Electrodynamics course, which goes into significant detail on Classical Field Theory via the Lagrangian/Hamiltonian approach. Prior to tackling this, however, I might brush up on Lagrangian Mechanics for continuous systems (at least for one dimension). The usual example given here is the derivation of the wave equation via the one-dimensional constrained vibrating string. There’s a good discussion of it in A Student’s Guide to Lagrangians and Hamiltonians.

Concurrently to the Theoretical Minimum lectures I am going to try working my way through Landau & Lifshitz Vol II (The Classical Theory of Fields). I’ve had some suggestions as to which sections are worthwhile to pursue. In addition I’ve been recommended the Feynmann Lectures on Physics as a more intuitive guide when L&L becomes a bit terse! Although the latter is somewhat pricey, so it may have to wait.

How to Create a Dedicated Study Day

My current employment situation means that I have a reasonable degree of flexibility when it comes to allocating my time. I’m going to assume that the majority of you have one or two free days every weekend, and/or a number of evenings over the week, that are not directly taken up with employment.

One of the ways in which I have rapidly accelerated my physics learning recently was to carve out a dedicated study day per week. In particular, I have chosen to work on physics every Monday, without fail (up to emergencies, of course!). Monday was chosen simply because it gave me something highly enjoyable to look forward to after the weekend. I cannot emphasise enough how useful this approach has been in being able to actually make significant progress.

The most important part of attempting to create dedicated study time is to make a commitment to yourself that you will actually carry out the study every week, without fail, at that particular time. Consistency is the key to making good progress. Obviously, this is not straightforward when one has a busy lifestyle and family obligations! I have found that because I look forward to this particular day every week, that I never postpone it. I guess this is simply a question of motivation!

So what should you do on a study day? This will clearly depend upon your personal situation, but I have found that watching one or two lectures (between 1-2 hours in length) and making extensive notes is extremely helpful. I don’t overwhelm myself with trying to learn too much in a day and I make consistent progress over the longer term. For instance, at the time of writing, I’ve almost finished my second course on the Theoretical Minimum after only a couple of months.

The key is really to be consistent. Even a single day taken every week, over the course of a few years, can add up to a lot of study.

Quantum Mechanics and The Next Steps

This week’s study day consisted of Lectures 8 and 9 from the Theoretical Minimum Quantum Mechanics course. Susskind has heavily emphasised Entanglement, since it is clearly a concept that causes significant confusion! I myself have been working through both the lecture courses and the accompanying book. The book is significantly more detailed than the lecture courses and includes many topics not covered in the videos. In particular, there is a brief treatment of the Quantum Harmonic Oscillator as well as Creation/Annhilation Operators.

Lecture 9 began the discussion of the (quantum) continuous particle in one dimension. Thus we are finally beginning to discuss position and momentum operators, as well as the time-dependent Schrödinger equation. Susskind spent a good deal of time discussing the Dirac Delta Function (and briefly Distribution Theory) and Fourier Series. Thankfully, these are concepts that I’m quite familiar with having studied Linear Analysis and Fourier Analysis the past in a mathematical setting. It was particularly pleasing to see the Fourier treatment applied to quantum mechanics.

I met up with a practising String Theorist a few nights ago, who happened to be working on the Ads/CFT Correspondance. It was fascinating to discuss Quantum Field Theory and how it relates to String Theory, at a level which was well beyond my current understanding! It was recommended that I take a look at the seminal two-volume work by Cohen-Tannoudji on Quantum Mechanics. I’ve also just received the first two volumes of Landau & Lifshitz in the post today, namely Mechanics and The Classical Theory of Fields.

The key question now is how to proceed upon completion of the Quantum Mechanics lectures on the Theoretical Minimum. The Special Relativity and Electrodynamics is an obvious next step, as is Landau & Lifshitz Vol II. However, I would like some slightly less daunting texts on electromagnetism and electrodynamics, prior to L&L Vol II. Griffiths is supposed to be a good undergraduate introduction, but from what I gather shies away from the Lagrangian/Hamiltonian approach, which is exactly what I need in order to tackle more advanced material.

Irrespective of the subject matter being studied, I definitely need to start tackling some actual questions. There are plenty of Classical Mechanics and Quantum Mechanics questions to consider. Also, I really want to start putting together my own notes into Classical Mechanics, that will ultimately become a course that will live on this site. I think I’m going to devote some of the next study day to putting together a brief syllabus.

Quantum Gravity or Numerical Relativity for Research?

For my study day this week I spent a lot of time working through Lecture 7 of the Theoretical Minimum Quantum Mechanics course. The theme of the lecture is Quantum Entanglement. At this stage we’re limiting ourselves to considering a singlet state, as I described in the previous post.

The thrust of the lecture involved Density Matrices and ultimately Bell’s Theorem. These are deep results and are not something which come easily, hence the need to repeat the material and carry it out over two lectures. It is interesting that having a bit of a background in stochastic processes and financial engineering certainly helps with the probabilistic concepts.

I’m glad that I’m finally beginning to get a solid grasp of elementary QM concepts. Once the course is over I’m going to work my way through the Special Relativity and Electrodynamics course. At this stage I really want to make sure I’m up to scratch on Classical Mechanics, QM, SR and EM by virtue of working through some textbooks and questions on the respective topics. Essentially, if I’m happy with the Landau & Lifshitz topics for Vols I (Classical Mechanics), II (Classical Theory of Fields) and III (Non-Relativistic Quantum Mechanics), then I feel I am in a good position to have a go at more complicated topics.

I am currently deciding on whether to pursue a deeper study of Quantum Field Theory or General Relativity, as a means of getting closer towards independent research. At this stage I am drawn towards both Quantum Gravity and Numerical Relativity as potential research areas. Perhaps I need not restrict myself to either. Nevertheless I am going to require a solid education in QFT, String Theory and GR in order to continue.

The next logical step is to “parallelise” learning of QFT and GR via the Theoretical Minimum lecture series, which has a specific GR module, and study QFT via some of the introductory textbooks. This is because there isn’t a specific QFT course at the Theoretical Minimum, although a lot of the necessary material is covered in the Advanced Quantum Mechanics course.

However, once I cover General Relativity via a number of textbooks (and even a workbook), I will have to decide whether to push hard on Numerical Relativity. This is a logical continuation of my PhD research into compressible fluid dynamics, so I should be able to begin researching more rapidly. But I am still very interested in the prospect of theoretical research into Quantum Gravity. Watch this space!

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.”