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
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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:
$$L^2Y_l^m(\theta, \...
- 175
2
votes
1
answer
92
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About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
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1
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1
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Moller Operator and the Determination of Bound States in Quantum Scattering Theory
I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states.
...
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94
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Using density argument of Schwartz functions in $L^2$ to show an operator is symmetric
The position operator $X$ is defined as multiplication by $x$, i.e.
$$(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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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
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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 ...
- 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 $$\mathcal S(H):...
- 428
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311
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Limit of a particular trace norm.
I have the following problem.
Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
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3
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79
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Whether $[A,B]=0$ if $\langle[A,B]\rangle=0$ for all states in the Hilbert space?
Let $\hat{A}$ and $\hat{B}$ are two self-adjoint operators in quantum mechanics corresponding to two dynamical variables $A$ and $B$. If $$\langle[\hat{A},\hat{B}]\rangle\equiv \langle\psi|[\hat{A},\...
- 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
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1
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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(...
- 461
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1
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70
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If $0\leq A\leq B$ and $C\geq0$ then does $0\leq AC\leq BC$?
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 $0\leq A\leq B$ and $C\geq0$ then does $0\leq AC\...
- 461
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34
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Meaning of "smooth finite-rank operators $Q$ in $L^2(\mathbb R^d)$"
What does it mean for a finite-rank operator to be "smooth"? How can one finite-rank operator be smoother than another? For me, a finite-rank operator on $L^2(\mathbb R^d)$ has an expression
...
- 461
1
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1
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84
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Operator theory: nonnegative operator $Q$ less than an orthogonal projection $P$ satisfies $PQ=QP$
Suppose that we have a Hilbert space $\mathcal{H}$, an orthogonal projection $P$ (i.e. $P=|\Psi\rangle\langle\Psi|$ for some $\Psi\in\mathcal{H}$ with $\|\Psi\|=1$) and another non-negative bounded ...
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If $A$ is trace-class on $L^2(\mathbb R^2)$ then $\mathbb 1(-\Delta\leq 1)A\mathbb 1(-\Delta\leq 1)$ is finite-rank. [closed]
I am trying to prove that if $A$ is trace-class on $L^2(\mathbb R^2)$ then $\mathbb 1(-\Delta\leq 1)A\mathbb 1(-\Delta\leq 1)$ is a finite-rank operator, where $\mathbb 1(-\Delta\leq 1)$ is defined on ...
- 461
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1
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65
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An inequality for Schatten norms with a compact self-adjoint operator
Let $Q$ be a compact self-adjoint operator on $L^2(\mathbb R,\mathbb C)$. Notice that $(1-\Delta)^{s}$ is a positive (and hence self-adjoint) operator on $H$ for any $s>0$.
We denote by $\mathfrak ...
- 461
3
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1
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130
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Commutator bertween shift operator and "position" operator
I was studying the shift operator in quantum mechanics from wikipedia and in the commutator section explains that for a shift operator $\hat T(x)$ and a position operator $\hat x$, the commutator $\...
- 495
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1
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59
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In what sense is $\breve{g}(x-y)/(2\pi)^n$ the integral kernel of $g(-i\nabla)$?
It is said in Trace Ideals and Their Applications by Barry Simon that the integral kernel of the operator $g(-i\nabla)$ on $L^2(\mathbb R^n)$ is given by $\breve{g}(x-y)/(2\pi)^n$.
Indeed, for any $\...
- 461
1
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1
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80
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Hartree equation in density matrix formalism is equivalent to a system of Hartree equations
I want to prove that:
If the density matrix $$\gamma(t):=\sum_{j=1}^N|u_j(t)\rangle\langle u_j(t)|=\sum_{j=1}^N\langle u_j(t),\cdot\rangle u_j$$ solves $$i\partial_t\gamma(t)=[-\Delta+w*\rho,\gamma],$...
- 461
4
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1
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145
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Definition of ground state?
In quantum mechanics, a ground state is an eigenstate of the hamiltonian with the minimal eigenvalue and its existence is guaranteed by appropriate theorems.
At least that's how it's defined in ...
- 305
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1
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42
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Find self-adjoint of $P=|a \rangle \langle b |$
Let $P$ be an operator s.a. $P=|a \rangle \langle b |$ and $P|f \rangle = \langle b | f \rangle | a \rangle $.
Find the self-adjoint and the $P^2$ operator.
My attempt:
We know to find the self ...
- 463
4
votes
1
answer
175
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Are quantum state (density) operators unique?
For context, the following question is related to my question on Physics SE. Now for the question.
Consider the following requirements on an operator $\rho$ on a Hilbert space (or on a rigged Hilbert ...
- 1,143
1
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1
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69
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For projection matrices, which vector of the outer product should be complex conjugated?
I have a conjugation wrong somewhere in my definitions, and I can't work out where it is. I want to define the standard matrix for a projection operator. If you can provide correct and standard ...
- 1,849
2
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1
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320
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On the usage of derivative in operator theory
In quantum mechanics, we work with linear operators on Hilbert spaces $\mathscr{H}$.
Suppose I have two bounded ones, defined on the same space $A, B: \mathscr{H}\to\mathscr{H}$.
It seems to me there ...
- 141
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1
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437
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Show that the adjoint of two operators is the sum of the adjoints
Problem
Show that for any two operators $\hat{A}$ and $\hat{B}$, the adjoint $(\hat{A} + \hat{B})^\dagger = \hat{A}^\dagger + \hat{B}^\dagger$. Do so using the integral form of the definition of ...
- 1,381
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1
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55
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Positive linear operator inequality
I'm stuck proving the following inequality. Let $H$ autohermitian operator such that
$$\langle u|H|u\rangle\geq0\qquad\forall |u\rangle$$
Proof that
$$|\langle u|H|v\rangle|^2\leq\langle u|H|u\rangle\...
- 59