# Presentation - A study on the Dyson series

1. Dec 13, 2008

### diegzumillo

Presentation - "A study on the Dyson series"

Hi there,
I'm not sure this is the right place to talk about this. If not, please indicate me where I can take this. :]

I wrote a dissertation about the Dyson series, as a requirement for my bachelor degree. It's not very long, but for the presentation I have to filter something! I'm listing here the topics covered, and if anyone has any suggestions on what to talk about, I'll sure apreciate it! :]

-Introduction: Nothing really substantial here. Just a really brief history of the series, motivations for quantum mechanics, etc.

-Quantum mechanics fundamentals: Hilbert space, Dirac notation and postulates.

-Time evolution operator and the Dyson series: Definition and properties like unitarity and convergence.

-Schrödinger, Heisenberg and interaction picture.

-Transition probability: The point here is how to obtain approximate results using the Dyson series. Furthermore, there is a physical interpretation and specification to Hamiltonian eigenstates.

-Constant perturbation (or step perturbation): We calculate the transition amplitude for a system with a constant perturbation (turned on after t=0).

-Harmonic oscilattor with a harmonic perturbation: We calculate the transition amplitude. The difference is that the calculations are really, really extensive!

For now, I'm preparing the presentation skipping the fundamentals: Evolution operator, transition amplitude and the applications.

So... What would you like to hear? If you were forced to sit down and listen to this for 50 minutes, or something.

2. Dec 13, 2008

### CompuChip

Re: Presentation - "A study on the Dyson series"

Unlike your dissertation, don't start with the mathematical fundamentals. You probably want to give a simple example first (the constant perturbation may be a good idea) where you explain some concepts along the way ("This is the Hamiltonian. Since we know a lot about free theory, we will split off the non-interacting part. [...] Now we start with some guess, if we plug it into the series we will get the next term. In principle, we can get the exact answer by doing this infinitely often, but in practice, we will stop after two or three times. ... Born series, which you all know ... See, now this is happening, .... This is what we call a Dyson series").
If you think it is an interesting example you can then show some stuff from the harmonic oscillator. Just point out analogies with the simple case you just discussed and keep stressing the basis ("look, here we plug this solution back into the formula to get the next step - note how the formulas look more complicated but it is conceptually the same").

Any time around that is usually well-spent giving some historical feeling, showing a list of problems for which the method is very handy or even required and showing interesting facts you found out which will keep the audience interested.

Depending on the knowledge of your audience you can go into as much or little detail as you want. But I always say: it's more important to make them hear what you say, than to make them struggle through your formulas. A good presentation doesn't contain formulas on every sheet! Writing: "Let H be the Hilbert space of square integrable functions on the set C defined by ..." is good for a dissertation, but in a presentation you will usually prefer something like "So I'm forgetting about the Hilbert space here, but that's just some space of square integrable functions as you are used to" (if they care, they will ask you). Finding a good balance about rigour and quantity is up to you.

3. Dec 14, 2008

### diegzumillo

Re: Presentation - "A study on the Dyson series"

Thanks compuchip! :]
I have a clearer idea of what to do now.