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4

Causality requires that $\chi(t) \propto \Theta(t)$, where $\Theta(t)$ is the Heaviside step function. In other words, $\chi(t-t') = 0$ for $t'>t$, so that only past influences from times $t'\leq t$ affect the system response at time $t$. This leads to constraints on $\chi(\omega)$ viewed as a function of complex frequency: it must be analytic in the ...


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In answer to the title question: no, you can't always decompose an $L_2$ function in terms of only the bound spectrum of hydrogen. This is because there are orthogonal functions to all bound states, which naturally represent the free states of the electron. The quickest examples are of course the Coulomb-wave eigenfunctions $|\chi_{E,l,m}⟩$ of the ...


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As has been pointed out in the comments, it's not entirely clear how you intend to specify a metric without the use of some set of coordinates. That said, a couple of common GR texts have non-standard approaches to the Schwarzchild metric that you might find interesting. Misner, Thorne, and Wheeler's Gravitation has a fairly detailed sidebar (Box 23.3, ...


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While a funny-looking coincidence, this is not a valid alternative expression for entropy in general, since the entropy of a probability distribution (which are what rigorously hides behind the strange word "macrostate") is more generally given by $$ S = - k_B \sum_i p_i\ln(p_i) \tag{1}$$ and becomes only $$S = k_B \ln(\Omega) \tag{2}$$ in the case of a ...


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Ok, I do not know the Sage algorithm but I am going to offer a conjecture of what is happening. You have to verify the conjecture by further numerical investigations. I assume that the Sage algorithm works optimally for bifurcations of a single equilibrium and can run into problems such as we see here when dealing with equilibria (AKA fixed points) ...


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The space of semi-infinite forms is basically the name used by mathematicians for the fermionic Fock space please see for example: Friedrich Wagemann lecture, page 8. Given an infinite dimensional vector space with spanned by: $\{ e_i, i\in \mathbb{Z}\}$, let its dual space be spanned by $\{ f_i, i\in \mathbb{Z}\} (\langle f_j, e_i \rangle = \delta_{ij}$) ...


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This subtlety is related to the fact that the momentum operator $\hat{P}$ (unlike the Hamiltonian $\hat{H}=\frac{\hat{P}^2}{2m}$) has no eigenfunctions compatible with the Dirichlet boundary conditions, and $\hat{P}$ is not a self-adjoint operator. This is essentially Example 4 in F. Gieres, Mathematical surprises and Dirac’s formalism in quantum mechanics, ...


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I don't think we need Sokhotski-Plemelj for this. Think of $E_j - E_i$ as a fixed value $E$. Then the formula is re-written as $$\frac{\hbar \omega}{E - \hbar \omega - i \eta}\, .$$ Now let $x \equiv \hbar \omega$ and you get $$\frac{x}{E -x - i \eta} \, .$$ This integral is dominated by the part where $x \approx E$ so let's try shifting the variables $y ...


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Noting the simple relation $$ \frac{\Delta E_{i,j}}{\Delta E_{i,j}-\hbar\omega-i\eta}-\frac{\Delta E_{i,j}-\hbar\omega-i\eta}{\Delta E_{i,j}-\hbar\omega-i\eta}=\frac{\hbar\omega+i\eta}{\Delta E_{i,j}-\hbar\omega-i\eta} $$ and by Sokhotski-Plemelj theorem $$ \lim_{\eta\rightarrow 0^+} \frac{i\eta}{\Delta E_{i,j}-\hbar\omega-i\eta}=0 $$ because, if the limit ...


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From a mathematical perspective this means that it is not differentiable. The problem is that you need the discreteness to be able to count states. If you replace the discreteness by something smooth you get something differentiable, but your definition of entropy no longer makes sense. This is just one of the points where mathematicians cringe, but it works ...


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This is typically said of solutions to differential equations which make no sense. For example when calculating skin effect you find that the solution to the field in the conductor is a sum of an exponentially growing term and an exponentially attenuating term. You throw away the first one because it is unphysical*. * Or because it does not satisfy ...


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Define $D=\{(x,y)\in\mathbb R^2:\|(x,y)\|\leq R\}$ as the disk of interest. There are two spaces of interest here: the space of square-integrable functions on $D$, $L_2(D)$, and the space of such functions with Dirichlet boundary conditions, $\mathcal H=\{\psi\in L_2(D):\psi(p)=0\:\forall p\in \partial D\}$. You're interested in the hamiltonian ...



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