All Questions
Tagged with quantum-mechanics operator-theory
181
questions
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Kraus operators
Suppose we have a POVM given by the family of positive, hermitian operators $\{E_i\}_{i\in I} \in \mathcal{H}$.
From the Neimark dilation theorem we know that the given POVM can be obtained from ...
- 75
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1
answer
961
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Quantum translation operator
Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$.
Where $P=-i\hbar \nabla$.
Here's what I've gotten: $$T_\...
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- 1
1
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1
answer
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Moller Operator and the Determination of Bound States in Quantum Scattering Theory
I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states.
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1
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Spectrum of a positive definite differential operator
When solving the two body problem in three dimensions in my QM lectures, you end up, essentially, with the Legendre associated equation. In terms of the angular momentum operator:
$$L^2Y_l^m(\theta, \...
- 3,234
2
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1
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92
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About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
- 61
4
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1
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903
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Stone's theorem and the spectral theorem
I am struggling to formally derive the expression found in the Stone's theorem for one-parameter unitary groups. I am aware that this can be done by using the spectral theorem. I am mostly interested ...
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10
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1
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300
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Is the failure of $\mathcal{B}(H)\simeq H\otimes H^*$ in infinite dimensions the reason for non-normal states in quantum information?
In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of ...
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Using density argument of Schwartz functions in $L^2$ to show an operator is symmetric
The position operator $X$ is defined as multiplication by $x$, i.e.
$$(X\psi)(x):=x\psi(x).$$
We take the domain $D(X)\subset L^2(\mathbb R)$ for $X$, which we can take
$$D(X):=\{\psi \in L^2(\mathbb ...
- 5,601
2
votes
4
answers
121
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The value of $\langle p^2 \rangle$ for $\psi(x)=e^{-|x|}$, where $\hat{p}:=i\frac{d}{dx}$ is the momentum operator.
Consider the Hilbert space $H$ of square-integrable complex functions on the real line equipped with the inner product $$\langle \phi,\psi\rangle:=\int \phi^*\psi d\mu,$$ where $\mu$ is the Lebesgue ...
- 3,234
6
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177
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Convergence from spectrum of the Choi matrix?
Suppose I have a linear operation of the form $T(C)=\frac{1}{m}\sum_i^m A_i C A_i$ for a set of $m$ symmetric $d \times d$ matrices $A_i$.
I construct Choi matrix below with $e_i$ referring to ...
- 3,407
2
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1
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202
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Spectrum of a sum of self-adjoint operators
This is a "sequel" to that question where I explain why I need the spectrum of an operator given as the sum of a convolution and a function multiplication. Here, I am considering the ...
- 209k
2
votes
2
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266
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Compactness of subset of trace-class operators on a Hilbert space
Consider an infinite-dimensional, complex and separable Hilbert space $H$ and let $\mathcal I(H)$ denote the space of trace-class operators.
The set of density operators is defined by $$\mathcal S(H):...
- 2,140
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311
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Limit of a particular trace norm.
I have the following problem.
Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
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3
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79
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Whether $[A,B]=0$ if $\langle[A,B]\rangle=0$ for all states in the Hilbert space?
Let $\hat{A}$ and $\hat{B}$ are two self-adjoint operators in quantum mechanics corresponding to two dynamical variables $A$ and $B$. If $$\langle[\hat{A},\hat{B}]\rangle\equiv \langle\psi|[\hat{A},\...
- 15.5k
1
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Coefficient conditions for square root of the Laplacian
I am confused about what's going on in the attached picture (from the introduction of Friedrich's Dirac Operators in Geometry). The author claims that for an operator $P$ to satisfy $P = \sqrt{\...
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