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## New answers tagged hilbert-space

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After watching through the youtube lectures by the author of the book mentioned in the comments (link, min. 57), I can post a more thorough reply: There is, indeed, a relation between $U_1 = \sum_i |... • 281 0 votes ### Contradicting Results from Exact Diagonalization of Many-Body Hamiltonian An elegant way of solving this problem is by using hole transformation: $$h_i=c_i^\dagger,\\ h_i^\dagger=c_i,$$ where the whole creation operators act on a state filled with electrons: $$|full\... • 60.4k 7 votes ### One doubt about Dirac notation Your last equation is wrong, it is incompatible with bras being the hermitian conjugate of kets \langle\psi|=\left(|\psi\rangle\right)^{\dagger}. \langle U^{\dagger}n| = \langle n|U = \left(U^{\... • 618 1 vote ### Why can non-differentiable solutions to the Schrödinger equation be ignored? This is a response to supplement Valter Moretti's answer and to elaborate on my comments on his answer. The key idea is as follows. We're interested in operators of the form H_0 = -\Delta\psi+V\psi, ... • 710 10 votes Accepted ### How to tell if a state written in second quantization is a Slater determinant? Let \mathfrak h denote a single-particle Hilbert space, and H_N:=\wedge^N \mathfrak h the Hilbert space of N identical fermions. We call a normalized vector \psi \in H_N a Slater determinant ... • 8,244 4 votes ### How to tell if a state written in second quantization is a Slater determinant? A many-fermion state that can be written as a single Slater determinant if and only if it satisfies the Plucker relations. • 54.7k 6 votes Accepted ### Why can non-differentiable solutions to the Schrödinger equation be ignored? This is an issue of mathematical nature, however it has physical implications so that it deserves to be tackled. Spectral theory as used in QM applies to so-called normal operators. In the concrete ... • 74.5k 2 votes ### Showing that the angular momentum operators and Laplace-Runge-Lenz operator together are generators of SO(4) This is not the math site. Just follow the strategy of wikipedia and, noting that H commutes with L and A, it can be moved out of their commutators just like a constant number. (In its diagonal ... • 64.5k 2 votes Accepted ### Contradicting Results from Exact Diagonalization of Many-Body Hamiltonian My guess is that the OP missed some negative signs when computing the matrix of the Hamiltonian in the two-particle sector that appear when anti-commuting the operators past each other. Here is an ... • 8,359 18 votes Accepted ### All QFTs are Finite The problems in QFT do not begin or end with the perturbative expansion. There are various difficulties relating to the procedure you're proposing. On a high level, there is a difficulty in defining ... • 710 4 votes ### All QFTs are Finite This is a short answer that doesn't discuss the full issues, but may shed light on the fact this does not work. Consider a free Klein-Gordon theory in flat spacetime. We know very well how to ... • 22.3k 5 votes ### All QFTs are Finite Constructing rigorously a quantum field theory is a very complicated endeavour, so there are many points at which the naive approach fails. At point (1), the difficulty is not finding the Hilbert ... • 6,039 1 vote ### All QFTs are Finite There are several problems in what you describe, so much that one could write a book out of it. The most fundamental one is the assumption that the normalizable states exist and that they belong to ... • 1,633 4 votes ### If a unitary operator is close to the identity, will it leave any state it acts on unchanged? Here is a very simple lower bound for the fidelity, maybe this helps in addition to the existing answer: Let U=\exp(-i\delta G), then using Duhamel's formula one can show the inequality || \exp(A) -... 11 votes Accepted ### If a unitary operator is close to the identity, will it leave any state it acts on unchanged? I assume that the spectrum of G=G^* stays in [0,+\infty) and \Lambda \leq +\infty is an upper bound of it.$$\langle \psi |e^{-i\delta G} \psi\rangle = \langle\psi|\psi\rangle+ \langle \psi |(e^... • 74.5k 1 vote ### What is fundamental object we are looking for in QFT? Quantum field theory is a formalism ideally suited to study fundamental dynamics. So what it calculates is a probability (amplitude) for certain events to take place. These events are often quantified ... • 15.2k 1 vote ### How do operators on kets and wavefunctions correspond? Is it by showing that this holds for position and momentum operators, and then have the result follow from any observable having to be a function of these? Yes, naturally. Your text or instructor ... • 64.5k 1 vote ### Unitarily Inequivalent Representations For a succinct explanation of the orthogonality of the vaccua for the van Hove model (which seems to be what the initial pdf might have been referring to), see Section 2 of this paper. Tne answer by ... • 11 4 votes Accepted ### Some confusion about understanding the relativistic quantum mechanics The group is still the Poincare group (Lorentz+ translations). The tricky thing is that we need to find a way for that group to act on the Rays in such a way that it preserves the probability. There ... • 510 2 votes Accepted ### The eigenvectors associated to the continuous spectrum in Dirac formalism Dirac is dead, we cannot ask him what he really meant by this. However, the pretension that these eigenstates exist in some sense is prevalent in many texts on quantum mechanics. There are several ... • 127k 1 vote ### Operator's definition in Dirac picture I have a question about the definition of quantum operators in the Dirac picture. The definition is: $$A=\sum_i \sum_j \vert i \rangle A_{ij} \langle j \vert.\tag{1}$$ By deplacing the ket vector I ... • 22k 4 votes ### Operator's definition in Dirac picture How did you get from the 1st equation to the second equation? The Second equation will be a number, not an operator. You can't just move the ket to the right. The top equation is an outer product but ... • 510 2 votes ### The eigenvectors associated to the continuous spectrum in Dirac formalism As long as you allow elements of the associated rigged Hilbert space the answer is "yes." • 54.7k 0 votes ### Hermitian conjugation in Radial Quantization I wish I could make this clear. In the Euclidean QFT, we use the variable $$z=e^{ix-\tau}$$ to label the Eucludean quantum field$\phi_{E}(z)$, which is related to the ... 2 votes Accepted ### The meaning of a representation in one-dimensional quantum mechanics In abstract terms the meaning of representation in these cases is the ordinary meaning of representation in the sense of representation theory. Concretely, you are representing the Heisenberg algebra ... • 127k 2 votes Accepted ### What is the dimension considered in the Schmidt Decomposition? It is the complex dimension. So for an orthonomal basis of$\mathbb{C}^2$, you would choose two vectors which need not be real. Actually this is the case not just for the Schmidt decomposition but ... • 618 1 vote ### Quantum: why linear combination of vectors (superposition) is described as "both at the same time"? Suppose you're travelling with a bearing of 45 degrees to North. We call this direction "Northeast". Are you travelling North? Some might say yes. Are you travelling East? Some might say yes.... • 14.4k 1 vote ### Quantum: why linear combination of vectors (superposition) is described as "both at the same time"? In quantum theory the evolution of a measurable quantity is described by a linear operator called and observable. The eigenvalues of that observable represent the possible outcomes of a measurement of ... • 8,835 10 votes ### Quantum: why linear combination of vectors (superposition) is described as "both at the same time"? “Both at the same time” is actually bad language, especially as this is a basis-dependent statement. For instance, the eigenstate$\vert \uparrow\rangle_x$of$\sigma_x\$ is one state. If you make a ...
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