Recent content by Decimal

Thank you for the response HAYAO, I have since discussed this problem with my supervisors as well. Your point on the spin orbit coupling was actually very helpful, I had (stupidly) not considered yet that this is the actual reason ##l## and ##s## can no longer be considered good quantum numbers...

I have a question related to the following passage in the quantum mechanical scattering textbook by Taylor, Here Taylor makes the choice to use a basis of total angular momentum eigenvectors instead of using the simple tensor product given in the first equation above (6.47). I understand that...

Oh yeah I see, I got a little carried away inputting the identity matrices. Indeed it should be, $$\int d^3 r \ |\mathbf{r}\rangle \langle \mathbf{r}| \otimes \mathbb{I}_{\mathrm{spin}},$$ and, $$V = \int d^3 r \int d^3 r' \ |\mathbf{r}\rangle \langle \mathbf{r}| V |\mathbf{r}'\rangle \langle...

Great! This is exactly what I was hoping for. Indeed for arbitrary particle operators if you want the full matrix element as just a c-number you will need ##\langle\mathbf{r}, s, \nu|\hat{A}| \mathbf{r}', s',\nu'\rangle##. But perhaps what Newton means here is that he actually computes the...

Okay yes I understand all of this, but this is also all consistent with my notation so it would seem there is no problem there. But then I am just back at my original question again. Why is the author allowed to insert a set of position states only, instead of inserting position-spin states. It...

Alright, I see how that works for the scattering state yes, but he specifically states that ##\psi_0(\mathbf{k}s\nu;\mathbf{r})## is a plane wave state. For example see the following passage, It could very well be (in fact I am starting to think its likely) that my confusion is just simply...

I have been thinking about this more as well. I do not mean a scalar product of ##| \mathbf{r}\rangle## with ##|\mathbf{k, \sigma}\rangle##, which as you said does not make sense. Rather what I do mean is if I view ##\langle\mathbf{r}|## as a projection operator, then I can surely form an...

Yes this is definitely a way to solve my issue, but it does not really help me understand what actually happens in these derivations I am reading. Perhaps it would help to give a more specific example, consider the Lippmann-Schwinger equation, $$| \psi, \nu^{(+)}\rangle = |\mathbf{k}, \nu\rangle...

Hello, Throughout my undergrad I have gotten maybe too comfortable with using Dirac notation without much second thought, and I am feeling that now in grad school I am seeing some holes in my knowledge. The specific context where I am encountering this issue currently is in scattering theory...

I am having a hard time understanding the discussion of chaotic sets on invariant manifolds as given in Chaos in Dynamical Systems by Edward Ott. If the invariant manifold of a particular system contains a chaotic attractor ##A##, then the transverse Lyapunov exponent ##h## will generally...

@fresh_42 Thank you! That helps a lot, I feel like my intuition still needs a lot of work in this area but it's getting much clearer already.

I am not a star at drawing ;) , but I can try to give of an overview of my current intuitive understanding of these functions. Please correct me if something is wrong. On the manifold ##M##, which I envision as a "curved sheet", there exist points ##\mathbf{P}##. The chart ##\phi_{\alpha}##...

After looking into this a little more I agree with you that the equation is actually a definition and not a result. This still confuses me though, what are you actually defining? Aren't all the functions in this case already explicitly defined, so what is there left to choose? It seems odd to me...

I am trying to understand the following derivation in my lecture notes. Given an n-dimensional manifold ##M## and a parametrized curve ##\gamma : (-\epsilon, \epsilon) \rightarrow M : t \mapsto \gamma(t)##, with ##\gamma(0) = \mathbf{P} \in M##. Also define an arbitrary (dummy) scalar field...

Hello, I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, ##f(x) = 4\lambda x(1-x)##. Let me then compare 1,2 and 4 iterations of this map on fixed points. I assume that ##\lambda## is large enough such that two period doublings...