Questions tagged [quantum-mechanics]
For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.
110
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Does every operator have a hermitian adjoint?
If we think of operators as matrices, every matrix can be transposed and its elements can be complex-conjugated. But the identification of the hermitian adjoint with the transpose conjugate comes from ...
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Deriving Rodrigues Formula and Generating function of Hermite Polynomial from $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$
There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, ...
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Show that a set of projectors summing to the identity implies mutually orthogonal projectors
The general setting is the study of positive operator measures in quantum mechanics,
instead of the projector operator measures.
Going from the PVM to the POVM is just saying that our bunch of ...
- 71
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Is the Unitary Group of a Hilbert Space a Lie group?
Let $H$ be an infinite-dimensional complex Hilbert space. Then the set of unitary operators on $H$ forms a group, known as the unitary group or Hilbert group. My question is, is this group a Lie ...
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How does one define the complex distribution $1/z$?
I have read the following formula in a quantum mechanics book, supposedly attributed to Dirac
$$ \lim_{y\,\searrow\, 0} \frac 1 {x+iy} = \operatorname{p.v.} \left(\frac 1 x\right) - i \pi\delta (x)$$
...
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1
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Possible significant error in proof of the spectral theorem, Brian C Hall, Quantum Theory for Mathematicians
Note/Edit: Read the below paragraphs for context. I think I found a counterexample, though I don't have the energy to work through it now. Suppose $\mathcal{H} = L^2([0,1],m) \oplus L^2([0,1],m)$ ...
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Geometric quantization: not understanding the curvature form and Weil's theorem
I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
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Expected Values of Operators in Quantum Mechanics
I've recently started an introductory course in Quantum Mechanics and I'm having some trouble understanding what the expectation of an operator is. I understand how we get the formula for the ...
- 1,403
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1
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Does the split-step operator method work for a PDE in cylindrical coordinates?
I am trying to numerically solve the 3D Schrodinger equation with an extra non-linear term (the Gross-Pitaevskii equation) using the split-step operator method. This is a follow up to a previous ...
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Given product and convolution of pair of functions can you find original pair?
Suppose you have two functions $f,g: \mathbb{R} \rightarrow \mathbb{C}$ and you have their product and convolution
$$ h_1(t) = fg$$
$$ h_2(t) = \int_{-\infty}^{\infty} f(t-\tau)g(\tau) d\tau $$
Is it ...
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4
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Understanding the Expression $tr\Big(\rho(X\otimes I)\Big)=\sum_{a,b,a',b'} \rho_{ab,a'b'}X_{a,a'}\delta_{b,b'}$
How do I make sense of
$$
tr\Big(\rho(X\otimes I)\Big)=\sum_{a,b,a',b'} \rho_{ab,a'b'}X_{a,a'}\delta_{b,b'}
$$
If this does not involve any tensor products it would be a simple elementwise matrix ...
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Why does $\Omega_{ij} = \overline{\Omega}_{ji}$ only imply $\Omega$ is Hermitian in the finite dimensional case?
1 Question for Bounty
In the context of an infinite-dimensional vector space, below I present a supposed "proof" – designated by $(*)$ – that
$$
\Omega_{ij} = \overline{\Omega}_{ji}
$$
...
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Proving that the Laguerre polynomials do indeed solve the differential equation
I am trying to show that the Laguerre differential equation, given in my homework problem as
$xL''_n(x)+(1−x)L'_n(x)+ nL_n(x) = 0$,
is indeed solved by the Laguerre polynomials in their closed sum ...
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Dirac Delta function identities
I'm using the definition of the one dimensional dirac-delta function, $\delta(x)$, being, $$\int_{-\infty}^{+\infty}\delta(x)f(x)dx = f(0) \tag1$$ and I'm doing a question which asks me to (via ...
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When does this integral vanish, which appears in the derivation of the quantum virial theorem?
In the derivation of the quantum virial theorem by Slater (appendix of this article), the following term $A$ appears, which is said to vanish upon integration ($\bar \psi$ is the complex conjugate of $...
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