Recent content by Couchyam

The complete density matrix would have no repeated indices, or would include a sum over both ##V## indices as well as ##V^*## indices: $$ \hat\rho_0 = \rho_{a,i,j; b,k,\ell} |a,i,j\rangle\langle b, k, \ell| \equiv \sum_{a,i,j,b,k,\ell} \rho_{a,i,j; b,k,\ell} |a, i, j\rangle\langle b, k, \ell| $$...

##\rho_3## should be the "partial trace" of the original density matrix over the ##i##, ##j## subsystem. It's sort of like how the Ricci tensor isn't the full trace of the Riemann curvature tensor (which would give the scalar curvature.) So ##\rho_3## should be a 2 by 2 matrix, and at a...

In short, I'm interested in working on a web-app to make landmark papers in theoretical physics and mathematics more broadly accessible, especially to undergraduate and graduate students who are looking to catch up to modern topics (without sacrificing rigor or exactness of understanding), and...

At the heart of the theory of open quantum systems is the idea that the measurement statistics of many-body systems can be expressed in terms of a reduced density matrix, obtained by tracing over degrees of freedom that are irrelevant to the system of interest. In general, given a pure state...

Exactly.

We might be misinterpreting each other. How broad/encompassing is your understanding/definition of decoherence?

Well, there are the cyclic groups of order ##N##, dihedral groups, and the symmetries of Platonic solids, which I think McKay noticed correspond (tantalizingly) to the (possibly extended) A, D, and E Dynkin diagrams respectively. https://en.wikipedia.org/wiki/McKay_graph

I think I can see where you're coming from: for example, it's hard to recognize interference effects in single-count measurements (such as the lived experience of a typical person), and the observed phenomenon of wave function collapse could be an illusion of sorts (maybe related to the illusion...

Thank you for that tactfully administered dose of reality. Now I see where you and @vanhees71 are coming from (after all, H3 is an unstable molecule), and that my earlier contribution to this thread could be more than just a little bit misleading. There is still a (admittedly somewhat trivial...

It might also help to explain that the focus in math shifted to algebraic geometry and more general algebraic groups. Also, I think there might still be at least some finite, nonzero level of mathematical interest in the mysterious and unexpected connection between semi-simple Lie groups (or...

@SteveMaryland: forget about the heat death of the universe for the moment. Are you familiar with energy conservation? (That orbits of closed classical systems are confined to a constant energy "level set"?) From a purely classical perspective, the Boltzmann distribution emerges from a sort of...

From a certain perspective it's only true point-wise in "##\vec k##" space, so it might be misleading. I can't think of any setting off the top of my head where that equation (i.e. ##\mathscr{F}\left[\hat f_1(\vec k) f_2(\vec x)\right] = \hat f_1(\vec k)\hat f_2(\vec k)##) specifically would be...

You might want to double check your algebra for changing variables (e.g. do the bounds of the integral over ##x## make sense?)

But Hydrogen atoms aren't fermions.

Perhaps... one could argue that the strength of interactions between internal degrees of freedom play a stronger role in determining evident boson/fermion behavior, and the magnitude of interactions depends strongly on screening. For example, if the Coulomb interaction were attractive rather...