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Apr
28
answered Two definitions of the density matrix?
Apr
27
comment Complexity of quantum simulation
@CuriousOne I'm not sure I understand exactly what you mean, nor am I really qualified to answer, but I'll have a crack. As far as I know, no quantum algorithm has been proven to lie outside of P. The best example of an algorithm which is expected to be hard is probably boson sampling. There is also no known example of a (local) quantum Hamiltonian whose simulation is provably outside P, since this would be a quantum algorithm not in P. I do expect various contrived classical algorithms have been proven to lie outside of P (and also thus outside NP), however.
Apr
27
comment Complexity of quantum simulation
It is well known that simulating quantum models is not necessarily hard, e.g. stabilizer quantum mechanics, Gaussian continuous-variable quantum mechanics, or one-dimensional spin systems satisfying an area law, can all be simulated with polynomial overhead. (I suppose Feynman was not aware of these examples when he made his famous proposal.) Computer scientists have various reasons to believe that there should exist quantum simulations which are hard (i.e. not in $P$), but there is still no proof.
Apr
22
comment How to calculate the second functional derivative of the action of a one-particle system?
Integration by parts will solve all your troubles.
Apr
21
comment Is there a quantum computing model accounting for uncertainty of a qubit state?
It would be hilarious and pretty embarrassing if physicists hadn't thought of this ;) It's called quantum error correction and a large number of ways have been proposed or developed to implement it.
Apr
19
comment Deriving the correlation function of a system interacting with a bath of harmonic oscillators
Let us continue this discussion in chat.
Apr
19
comment Deriving the correlation function of a system interacting with a bath of harmonic oscillators
Try to express $2n+1$ in terms of hyperbolic trig functions.
Apr
19
comment Deriving the correlation function of a system interacting with a bath of harmonic oscillators
Think about the trace. The trace can be performed in any basis, in particular the eigenbasis of $b^\dagger b$.
Apr
18
comment Relating $C_j(t-t') = \left<\hat{B}_j(t)\hat{B}_j(t')\right>$ to $\left<\hat{B}_j(t)^2\right>$
It's simpler than you think, just set $t = t'$. This makes sense because your state is stationary (otherwise $C(t,t')$ would depend on both $t$ and $t'$ separately, rather than their difference), so the result for $\langle B_j(t)^2\rangle$ should be $t$-independent.
Apr
18
comment Deriving the correlation function of a system interacting with a bath of harmonic oscillators
Hint: first consider the single-mode case. Try to prove that $\mathrm{Tr}[b^\dagger b \mathrm{e}^{-\beta\omega b^\dagger b}]/\mathrm{Tr}[\mathrm{e}^{-\beta\omega b^\dagger b}] = (\mathrm{e}^{\beta\omega} - 1)^{-1}$. Why does this result make sense given what you know about Bose-Einstein statistics? You should then tackle the multi-mode case.
Apr
13
comment Going to the interaction picture in the Jaynes–Cummings model
@JDH Actually that is the right answer, although you're not quite there yet. You need to use the fact that an operator $A$ commutes with its exponential $e^{A t}$.
Apr
13
comment Fermi-level of lattice model
Note that you can use MathJax for typesetting equations. This is generally necessary to make a post readable. I have done this for you, you can peruse the edit history to see how this is done.
Apr
13
revised Fermi-level of lattice model
added 71 characters in body
Apr
11
comment How can absorbtion of a photon in an atom take place?
@PeterDiehr You are of course right, although in my field of (AMO) physics, nowadays people routinely do experiments on single- or few-atom systems, close to zero temperature or otherwise confined, such that there really isn't any significant broadening apart from the natural line width.
Apr
11
comment How can absorbtion of a photon in an atom take place?
"The transition energy for an electron in an atom has a sharp defined value", this is simply wrong. The transition energy is never sharp, but has a finite uncertainty $\delta E$ (called the line width). This is given by $\delta E \sim \hbar\Gamma$, where $\Gamma$ is the rate of decay, consistent with heuristic considerations of energy-time uncertainty.
Apr
11
comment What do the wave functions associated to the Fock states of each mode of a bound state system mean?
This is a nice question. I guess the answer is going to take one of two forms: 1) identification of a physical observable whose probability amplitude is Gaussian-distributed in the vacuum state, 2) proof that there exists no such observable, because the analogy between the Fock spaces of many-body QM and the harmonic oscillator is purely formal. I suspect 2) is true when dealing with conserved particles, i.e. fermions or their composites like atoms. This is because the field quadrature will look schematically like $a + a^\dagger$ (where $a$ annihilates atoms) which surely isn't an observable.
Apr
10
comment Going to the interaction picture in the Jaynes–Cummings model
@JDH I have edited the answer with some hints for you.
Apr
10
revised Going to the interaction picture in the Jaynes–Cummings model
added 266 characters in body
Apr
8
comment Why don't we call the fermions in the standard model force carriers?
Matter is made of fermions. Forces are a Newtonian concept used to describe the dynamics of classical matter. Hence it (sort of) makes sense to talk about forces between matter particles (fermions).
Apr
8
answered Going to the interaction picture in the Jaynes–Cummings model