Tuesday, 2/9, 15.15-17.00. After some general information about the course I gave a brief review of the beginnings and development of quantum mechanics. I emphasized that you shouldn't worry too much about not being able to ``make sense'' of quantum mechanics (``...no one understands quantum mechanics.'', R. Feynman). We're going to embark on a journey into a different way of thinking, radically breaking away from ``common sense'' and our everyday way of looking at the world! It's sure going to feel strange! In the rest of my lecture I outlined the basics about linear vector spaces, the mathematical formalism that underpins quantum mechanics. Most of the stuff you have seen before in your linear algebra courses, the new thing (at this stage) being that the vector spaces relevant to quantum mechanics are defined over complex numbers. Also, the notation (Dirac notation) is probably new to you. However, it's simple to learn and use. And it's really useful, as I tried to show you via some simple examples.

Wednesday, 3/9, 10.00-11.45. I continued my exposition of linear vector spaces, running through a fairly large number of definitions, theorems and examples. I then gave a preview of the four postulates of (non-relativistic) quantum mechanics. I tried to convey to you the beauty and simplicity of how quantum mechanics is built: Hilbert spaces + 4 postulates! That's all! You'll find all you need in Sakurai, chapter 1, and hopefully, with my lecture as a background, you should rather quickly be able to get a hang on this.

Tuesday 16/9, 15.15-17.00. After a quick review of what we've done so far I introduced the concept of a(n infinite-dimensional) Hilbert space via an example, letting the dimension of a specific vector space (discrete functions on a finite interval) go to infinity. I outlined the consequences, and then spent some time to discuss the (very important!) differential operator, including its role as representing momentum in quantum mechanics. I showed you some standard manipulations using the Dirac notation, and then went back to discuss the postulates of quantum mechanics and how to use them when thinking about (how to predict) the outcome of an experiment.

Tuesday 23/9, 15.15-17.00. I started up by reviewing how to do an ideal measurement in quantum mechanics (and what to watch out for). In this context I went through some stuff on compatible/incompatible observables, expectation values, uncertainties, etc.... and then proved the general Heisenberg uncertainty relation. Note how easy and clean the uncertainty relation fell out, thanks to the powerful formalism that we have introduced! I then tried to get to the heart of the measurement problem by doing an experiment with a "quantum dice", telling you that the probabilistic nature of quantum mechanics is very different from how we are used to think about probabilities. I continued with a discussion of the the various attempts to solve the "measurement problem" ("What really happens when we collapse a state by doing an observation?"): The Copenhagen interpretation, Hidden variable theories, The many worlds interpretation, The "consistent histories" approach and the latest fashion, Decoherence theory. I'm afraid I may have gotten carried away here, and you probably didn't get much substantial out of my rambling thoughts. These are (very!) difficult (and fascinating!) questions to which we shall return later when we have developed some more tools and concepts...

Tuesday 30/9, 15.15-17.00. I started the lecture by discussing the fourth postulate and its implications: "How to find the time evolution of a quantum mechanical system?". Most of what I did you probably recognized from earlier courses, the possible new element being the prominent role played by the time evolution operator, or propagator. After a discussion of the Schrödinger vs. the Heisenberg picture we took a break, and after that I tried to explain why the harmonic oscillator is (one of) THE paradigm(s) for applications of quantum mechanics to real systems. I then sketched how to obtain its propagator (and hence, time evolution!) by the conventional route, going to coordinate space to solve the time-independent Schrödinger equation. Given the solution I discussed some of its properties (which are fairly generic): quantized energy levels from the condition of a physical Hilbert space, the interpretation of the spectrum in terms of fictituos particles ("quanta"), quantum fluctuations, symmetry properties, tunneling beyond the classical turning points, and the meaning of Bohr's correspondence principle. I then showed you how one can bypass the coordinate representation (and discard wave functions altogether!) and instead work directly in the energy basis, using creation- and destruction operators. This is an important formalism, which can be generalized to a large class of problems.

Tuesday 7/10, 15.15-17.00. Basics of the quantum mechanical theory of angular momentum. (Sebastian Eggert)

Tuesday 14/10, 15.15-17.00.

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This page was last updated on September 30, 2003.