Right, I was going to ask how that relates to how big O is used in computation. But I guess in this case it just means that the remaining terms are on the order of $(\delta\varepsilon)^2$
But really we use this machinery of PDEs implicitly all the time, just no one proves the relevant stuff. At the very least when you write down something like a Green's function in classical EM you are doing something implicitly with PDEs and distributions and, yes, perhaps Sobolev spaces :P
@TobiasFünke we work with Markov operators which when restricted to L^2 becomes the usual projection operator. I guess I wrote an answer on this last year.
@User1865345 if you find it, consider to link it. I guess I won't understand much, but I like taking a glance at new topics, so that I have at least heard of a few things :)
At the outset, there's nothing special about $A$ in $ P_A.$
To briefly (over)simplify, the concept of orthogonal projection stems from the necessity of finding the "closest" point of an element in a certain suitable space.
Formally,
Result $ 1:$ Let $\mathscr A$ be a closed, convex subset (e.g. c...
@Allie nah, I am not (idk if you addressed this to me anyway). I just have the right amount of understanding that I don't know 99% of math, and if I ever run in trouble I know that it could be either due to my sloppiness or due to the lack of rigor I employ
e.g. (d2/dx2 +k)x=0 is the oscillator equation. one can write the Greens function of this operator and use it to solve perturbation interactions added to the oscillator
@Relativisticcucumber I would prefer to have a bit more time to think about it myself, but sure. Let's use the backup room since there's currently a different conversation going on in here I don't want to interrupt.
@Slereah i have written the idea in this post. there are a few properties of these operators which make me think that they are creation-annihilation operators. e.g. they satisfy $[a,H]=-\omega a$ with the full Hamiltonian $H$
@TobiasFünke I have a question! Hope you still don't mind me asking
So, when writing the energy functional in DFT, they typically write it in terms of a universal functional $F$ plus a potential energy term $\int \rho(\mathbf{r})V(\mathbf{r}) d\mathbf{r}$
@Slereah i usually think of it as that the average person had been deprived of an education that they should have received, that would have otherwise have made their lives better in significant ways
And the reasoning they give for not including that last term is that it depends on the system (namely, its potential) while all the other terms are independent of the system
But, by HK1, isn't the $V$ uniquely determined by your choice of $\rho$?
barring in mind the usual assumptions (which could be lifted), like non-degeneracy of the ground state and so on, the HK proves a one-to-one correspondence between the ground state density and the potential (modulo additive constant)
we've discussed this a few days ago. IIRC, I've told you that you have to define the domains of the functionals. The "original" HK functional is defined only for $v$-representable densities, i.e. densities which come from (non-degenerate) ground states of some external potential (things can be made rigorous)
let's call the set of such densities $\mathcal V$. Then for every $n\in \mathcal V$, you can evaluate the function $E_v[n]:=F[n]+\int n v$ for a given external potential $v$.
but yeah, as you said, the universal functional is universal in the sense that it does not depend on the external potential
so what you will learn any minute is that for a given external potential $v$ under consideration, the ground state of the functional I've defined above is achieved by the ground state density $n_0$, i.e. $E_0(v)=E_v[n_0]$, where I've made it explicit that you fix the external potential from the outset, i.e. you specify the problem you want to solve, so to speak. (which type of molecule, for example)
I again would like to point out to the book/notes by H. Eschrig on DFT (until recently, these notes were accessible for free on his homepage; I don't know if they are still...he died)
he discusses the functional stuff very nicely
@RyderRude ah. $L^p$ spaces are also used in DFT :p
...so in view of my answer: I think that you can see your proposed functional $F[n]+\int v[n] n$ as giving the ground state energy corresponding to the ground state density $n$ (after you somehow sorted out the problem that $v$ is determined by $n$ only up to an additive constant, which should be doable I think). but as I tried pointed out, I think this most likely is not what people mean when they write $F+\int v n$ or so
I am currently studying the motion of relativistic charged particles in electromagnetic fields. More exactly, we first derived the equation of motion in the 4-vector formalism.
I was a bit confused when my teacher talked about acceleration in special relativity. In which cases are we allowed to t...
Because (after long university absence) I recently came across field operators again in my QFT lectures (which are not necessarily Hermitian):
What problem is there with observables represented by non-Hermitian operators (by observables, I obviously don't mean the tautological meaning "Hermitian ...
This question is going to look a lot like a duplicate, but I've read dozens of related posts and they don't touch the subject. Here we go.
Why are observables represented by hermitian operators?
Because then we'll measure real stuff, since the eigenvalues are real;
Because hermitian operators ...
For that... Since imaginary numbers emerge from square roots of negative numbers, does that mean existence of objects with negative mass must be premised before existence of tachyons?
Nevermind, this math doesn't check out; Positive numbers have a positive sqrt and a negative sqrt.
This question is going to look a lot like a duplicate, but I've read dozens of related posts and they don't touch the subject. Here we go.
Why are observables represented by hermitian operators?
Because then we'll measure real stuff, since the eigenvalues are real;
Because hermitian operators ...
