All Questions
Tagged with operator-theory quantum-mechanics
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
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34
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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:
$$L^2Y_l^m(\theta, \...
- 175
2
votes
1
answer
92
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About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
- 61
1
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Moller Operator and the Determination of Bound States in Quantum Scattering Theory
I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states.
...
- 854
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94
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Using density argument of Schwartz functions in $L^2$ to show an operator is symmetric
The position operator $X$ is defined as multiplication by $x$, i.e.
$$(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
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4
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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 ...
6
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1
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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 ...
- 5,477
2
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1
answer
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 ...
- 1,139
2
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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 $$\mathcal S(H):...
- 428
6
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311
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Limit of a particular trace norm.
I have the following problem.
Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
- 71
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3
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79
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Whether $[A,B]=0$ if $\langle[A,B]\rangle=0$ for all states in the Hilbert space?
Let $\hat{A}$ and $\hat{B}$ are two self-adjoint operators in quantum mechanics corresponding to two dynamical variables $A$ and $B$. If $$\langle[\hat{A},\hat{B}]\rangle\equiv \langle\psi|[\hat{A},\...
- 693
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53
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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{\...
- 330
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104
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Power rule for derivative operator $D^n$
I was playing with commuting the derivative operator $D$ and the scalar multiplication of $x$ applying to a function $f$. If $[\cdot,\cdot]$ denote de commutator, then:
$$[D,x]f=D(xf)-xDf=f+xDf-xDf=f$$...
- 495
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43
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If $A\geq 0$ and $0\leq B\leq C$, then $0\leq \sqrt{B}A\sqrt{B}\leq \sqrt{C}A\sqrt{C}$ [duplicate]
Let $H$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. We say $A\geq 0$ if $\langle Ax,x\rangle\geq0$ for all $x\in H$. If $A\geq 0$ and $0\leq B\leq C$, then does $0\leq \...
- 461
2
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77
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If $0\leq A\leq B$ then the corresponding densities satisfy $0\leq\rho_A\leq \rho_B$
Let $A,B$ on $L^2(\mathbb R^d,\mathbb C^d)$ such that $0\leq A\leq B$ AND $A,B$ are trace-class. Then their densities are given by $$\rho_A(x):=\sum_jA\varphi_j(x)\overline{\varphi_j(x)},\qquad \rho_A(...
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