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location United Kingdom
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visits member for 2 years, 8 months
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Currently doing a PhD in Controlled Quantum Dynamics at Imperial College and the University of Oxford.


1d
comment Are the electrons at the centre of the Sun degenerate or not?
Could you specify exactly what it is you want to calculate? Whether or not you have a good approximation is always dependent on what you are trying to calculate in the first place.
1d
comment Random walk recurrence term and the self-energy
Yes, I am pretty sure that there is an extensive mathematical literature on similar topics beyond Pierre-Louis' work, so you might have some luck mining that.
1d
comment Random walk recurrence term and the self-energy
Sorry, the third link I posted was incorrect: here is the paper I meant. Its value is probably more in the slightly more physical discussion of what path-sums actually mean in the context of an example model. By the way, these papers are seriously, seriously mathematical. They will probably need some commitment if you want to understand them properly (I do not myself).
1d
comment Random walk recurrence term and the self-energy
Feelings 100% mutual :) I do try to encourage more people with an interest in quantum stochastic problems to join SE whenever I can, it's a shame there are not more users here who share our interests.
1d
comment Random walk recurrence term and the self-energy
An old colleague of mine wrote his PhD on a method to compute elements of the many-body perturbation expansion in terms of walks on graphs. Your question reminds me quite a lot of this. The basic idea is to represent quantum dynamics as a random walk on the graph whose adjacency matrix is the Hamiltonian. Unfortunately, his thesis is apparently not available online, but you might find some insights in his papers, here, here and here (I helped a bit with the last one, but just with numerics).
1d
comment Local explanation of the Aharonov-Bohm effect in terms of force fields
Also relevant: arXiv:1507.00068.
Jun
30
revised A problem with indistinguishable fermions and the order of applying operators
edited body
Jun
23
answered How to experimentally create an atom in a superposition of ground and excited states?
Jun
21
answered Phase on Aharonov-Bohm effect doubts
Jun
18
comment Which bipartite entangled states violate the CHSH maximally?
@CuriousOne That may be, but some physical principles are more commonly and easily stated in a manifestly operational way than others. Hence there is pedagogical value in trying to find a way to state them all in a manifestly operational manner. By the way, if you ever dare to speak to anyone outside of the high-energy community you might find the world of physics outside is a bit less segregated between theorists and experimentalists than you imagine.
Jun
18
comment Which bipartite entangled states violate the CHSH maximally?
@CuriousOne You think "a quantum state is represented by a ray in a Hilbert space" is a physical principle? I think we are talking about different things here. The principles people should be interested in are operational, e.g. all observers moving with constant velocity w.r.t. each other measure light to move at $c$. I don't know what interests the author of that paper since, like you, I don't waste my time reading such work. In any case, my comment simply suggested there may be some pedagogical value in quantum foundations (read "mysticism"), I'm not foolish enough to try convincing you.
Jun
18
comment Which bipartite entangled states violate the CHSH maximally?
@CuriousOne The usual analogy here is with special relativity, which can be "derived" from essentially symmetry arguments. While I agree with you that such a derivation is not necessary since observation is the only true foundation of scientific theories, it is certainly not inconceivable that such an argument could exist, and it would at least have pedagogical value. So I don't think it is quite fair to rubbish that research goal out of hand.
Jun
9
comment Periodically connected QHO's
Yeah that seems like it could give you some insight.
Jun
9
comment Periodically connected QHO's
The k value is just an index labelling the eigenvalues. This only corresponds to a momentum in the translation-invariant case, where you know the analytical solution anyway. This is why I said it is not strictly a dispersion relation when the system is not translation-invariant. Close to translation-invariance, your Fourier transform idea is a good one, although it will need to be interpreted with some caution since momentum is not a good quantum number.
Jun
9
comment Periodically connected QHO's
I am not saying "analytical", just "exact" ;) I would diagonalise the matrix in MATLAB or equivalent for a large system. The eigenvalues give you the dispersion relation, and you're done. Note that it's not really a "dispersion relation" when the system is not translation invariant. However, close to translation-invariance you should be able to calculate e.g. group velocities and so on by taking numerical derivatives. Obviously you can also do this all with pen and paper if you are really feeling masochistic... But yes I would say these two questions are identical and should be merged.
Jun
9
comment Periodically connected QHO's
So this is the same as your other question, right? If you choose the coupling to be as you have written, the problem is identical to your other question. A more natural coupling for oscillators is $\hat{x}_i\hat{x}_j$, which is slightly more tricky because of the $a^\dagger_i a^\dagger_j$ terms. Nevertheless, any problem bilinear in bosonic operators for $N$ oscillators can be solved exactly in principle simply by diagonalising matrices of size $N\times N$.
Jun
7
comment Capturing (perturbatively) non-equilibrium field theory effects using “elementary” methods
@zudumathics How necessary is it that the interaction term be $\chi^2\psi^2$? I think you will find it much easier to consider $\chi\psi^2$, since then the equation of motion for $\chi$ can be exactly solved and substituted into the equation of motion for $\psi$.
Jun
7
comment Constructing a dispersion relation from the Hamiltonian
Ah, sorry, I misunderstood you. In the translation-invariant case the momentum is given basically by $p_k = \hbar 2\pi k/(\Delta N)$, where $\Delta$ is the lattice spacing. Regarding the $N$, I think you should get a factor of $1/N$ from the product of two $b$ operators, which should each carry a $1/\sqrt{N}$ normalisation. This will cancel with a $N \delta_{kk^\prime}$ when you perform the summation over lattice sites.
Jun
6
comment Capturing (perturbatively) non-equilibrium field theory effects using “elementary” methods
The simplest thing would be to just give $\chi$ a kinetic term $(\partial\chi)^2$ in the Lagrangian. You want to keep the Lagrangian for $\chi$ quadratic. Then integrating out $\chi$ should give you an additional interaction term for $\psi$, so that the non-interacting $\psi$ vacuum is no longer an eigenstate. I don't really understand what you mean by the "canonical" approach in your comment. I am not talking about "canonical quantisation", I am using the word canonical as in "usual", "well-studied" etc. As I said there are many canonical approaches in this sense.
Jun
6
comment Constructing a dispersion relation from the Hamiltonian
I do not think you should have any prefactor $N$, since this would mean that the bandwidth increases with the number of lattice sites, which is not the correct behaviour. About finding the momentum $k$, for sure you could Fourier transform the eigenfunctions. However, note that in the absence of translation invariance it does not make too much sense to assign a momentum to the eigenstates, since it is not a good quantum number.