All Questions
Tagged with quantum-mechanics operator-theory
181
questions
1
vote
0
answers
33
views
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
0
votes
1
answer
34
views
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:
$$L^2Y_l^m(\theta, \...
- 175
2
votes
1
answer
92
views
About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
- 61
1
vote
1
answer
62
views
Moller Operator and the Determination of Bound States in Quantum Scattering Theory
I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states.
...
- 854
1
vote
0
answers
94
views
Using density argument of Schwartz functions in $L^2$ to show an operator is symmetric
The position operator $X$ is defined as multiplication by $x$, i.e.
$$(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
views
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
votes
1
answer
177
views
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
votes
1
answer
202
views
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
votes
2
answers
266
views
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 $$\mathcal S(H):...
- 428
6
votes
0
answers
311
views
Limit of a particular trace norm.
I have the following problem.
Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
- 71
0
votes
3
answers
79
views
Whether $[A,B]=0$ if $\langle[A,B]\rangle=0$ for all states in the Hilbert space?
Let $\hat{A}$ and $\hat{B}$ are two self-adjoint operators in quantum mechanics corresponding to two dynamical variables $A$ and $B$. If $$\langle[\hat{A},\hat{B}]\rangle\equiv \langle\psi|[\hat{A},\...
- 693
1
vote
0
answers
53
views
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
1
vote
0
answers
104
views
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
1
vote
0
answers
43
views
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
votes
1
answer
77
views
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
0
votes
1
answer
70
views
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
1
vote
0
answers
34
views
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
vote
1
answer
84
views
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 ...
- 85
0
votes
1
answer
40
views
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
1
vote
1
answer
65
views
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
votes
1
answer
130
views
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
0
votes
1
answer
59
views
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
vote
1
answer
80
views
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
votes
1
answer
145
views
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
1
vote
1
answer
42
views
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
views
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
vote
1
answer
69
views
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
votes
1
answer
321
views
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
0
votes
1
answer
437
views
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
0
votes
1
answer
55
views
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
$$|\langle u|H|v\rangle|^2\leq\langle u|H|u\rangle\...
- 59
0
votes
0
answers
63
views
What is the most general Self-Adjoint operator acting on $L^{2}(\mathbb{R})$?
What is the most general Self-Adjoint operator acting on $L^{2}\mathbb{R}$?
My hypothesis is that the aswer to my question will be that the most general Self-Adjoint operator $A$ acting on $L^{2}(\...
- 71
1
vote
0
answers
105
views
On the structure of the set of Self-Adjoint operators acting on $L^{2}(\mathbb{R})$.
Let us consider the $Hilbert$ Space $L^{2}(\mathbb{R})$ and let $SA(L^{2}(\mathbb{R})$ be the space of all self adjoint operators acting on $L^{2}$. I have worked with operators such as $X$, $P$, $X^{...
- 71
2
votes
1
answer
370
views
If $\hat{A}$ commutes with $\hat{r}^2$, does $\hat{A}$ commute with $\hat{r}$?
Let $\hat{x}, \hat{y}, \hat{z}$ be the position operators in 3D space and $\hat{r}^2=\hat{x}^2+\hat{y}^2+\hat{z}^2$ as usual.
If one operator $\hat{A}$, commutes with $\hat{r}^2$, i.e.
$$\Big[\hat{A}, ...
- 51
0
votes
0
answers
27
views
Question regarding conjugate operators and the harmonic operator.
Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$
I'...
1
vote
1
answer
145
views
Given $[A,B]=0$ then $[A,f(B)]=0$
Given $A$ and $B$ two operators that commute ($[A,B]=0$) then $A$ commutes with an arbitrary function of $B$
I recently saw this property of the commutators on a quantum mechanics course. We didn't ...
- 541
1
vote
1
answer
67
views
Equivalence between two definitions of hermitian adjoint
Given the two definitions of hermitian adjoint:
$(1): <\Psi|A|\Phi>=(<\Psi|A^+|\Phi>)^*$
$(2): <\Psi|A\Phi>=<A^+\Psi|\Phi>$
I want to show that they are equivalent
However I ...
- 541
1
vote
0
answers
62
views
Integral over operators equal -When are operators equal?
This is a physics motivated question from quantum mechanics. When I have two operators $\hat{A}$ and $\hat{B}$ acting on the Hilbertspace $\mathbb{C}-L^2(\mathbb{R})$ with
$$\int\psi^*(x)\hat{A}\psi(x)...
- 320
1
vote
1
answer
42
views
What can I infer from this scalar product identity?
Suppose I have shown for a single vector $\lvert 0 \rangle$ that
$$\langle 0 \rvert U^\dagger \phi U \lvert 0 \rangle=\langle 0 \rvert \phi \lvert 0 \rangle$$
where $\phi$ is a certain operator and $U$...
- 121
5
votes
1
answer
191
views
Problem on a rank 1 perturbation of an self-adjoint operator
With my teacher we have been trying to solve for a while a problem that consists of two parts, and each one in three sections. The first part, which has to do with the problems that I was able to ...
- 160
1
vote
1
answer
74
views
Action of an operator-valued function of the momentum operator $\hat{p}$ and its unboundedness
I am currently dealing with an operator-valued function
$f(\hat{T})$ of the following kind:
$$f(\hat{T}) =\sqrt{1 + b\hat{T}^2} $$
where $b$ $\in \mathbb{R}$ and $\hat{T}$ is a linear operator acting ...
- 31
1
vote
1
answer
94
views
What is the meaning of the operator projection in quantum mechanic
Let's have a vector $B_1$ which lies in the $xz$ plane forming an angle $\theta$ with $z$ axis. Let $S_\theta$ be the projection of the operator S along $B_1$.
Question 1:
How can i write $S_\theta$ ...
- 315
0
votes
1
answer
33
views
On derivatives of tensor products in QM
Does the derivation follow a sort of Leibniz's Rule for a tensor product?
$ \displaystyle
\frac {\partial} {\partial t} \left ( A_t \otimes B_t \right) \overset ? =
\frac {\partial A_t} {\...
- 141
4
votes
2
answers
99
views
Can I conclude these weak derivatives are strong?
I'm trying to show the momentum operator $P : \mathcal D(P)\ \dot{=}\ C^\infty_0(\Bbb R)\to L^2(\Bbb R)$ with $P=-i\hbar\partial_x$ is essentially self-adjoint, which is equivalent to saying that it ...
- 7,532
0
votes
1
answer
192
views
proof of the product: x times m derivative of delta function.
The question is in one dimension and is : Prove that $$x\delta^{(m)}=-m\delta^{(m-1)},\ m \in \mathbb{N},$$
where $\delta^{(m)}$ is the $m$-derivative of $\delta.$
As I know, I got through this way:
$...
- 77
2
votes
1
answer
82
views
Scaled Dirac Delta function: $ \delta (xe^r - y) $
I was reading on squeezed Gaussian states and stumbled upon this paper:
Equivalence Classes of Minimum-Uncertainty Packets. II.
It is mentioned after Eq. $\left(2\right)$ that
$$
\left\langle x\left\...
- 35
2
votes
1
answer
105
views
Consequences of Deficiency indices theorem (Von Neumann theory)
Let $T: \operatorname{dom}(T) \rightarrow \scr H$ be a symmetric operator.
$T$ admits self-adjoint extensions $\iff$ $d_+ = d_-$, where $d_\pm = \dim \ker(T^\dagger \pm i \mathbb{I})$
If $d_+ = d_-$,...
- 141
0
votes
1
answer
91
views
Translation on X operator
I need to show that given $[x, p_x]=i\hbar$ then the following is true:
$$
e^{iap_x/\hbar}f(x)e^{-iap_x/\hbar}=f(x+a)
$$
for a general function $f(x)$. I've tried using Taylor Series for both ...
0
votes
2
answers
58
views
Operator norm of X
I've been trying to prove that the operator norm of $\hat{X}\phi(x) = x\phi(x)$ with $\phi(x) \in L^2[0,1]$ is given by:
$$
||\hat{X}|| = \sup_{||\phi(x)||=1} ||\hat{X}\phi(x)||=1
$$
However i havent ...
1
vote
1
answer
204
views
Hermiticity of operator on square integrable functions
I'm trying to solve this problem for my Quantum Mechanics class but its my first time dealing with infinite dimensional spaces so I'm a little lost here. I need to show that the operator defined by
$$
...
1
vote
2
answers
49
views
Algebra involving operator of x and p
For operators x and p in QM with $p= {h\over i}{d \over dx} $, how can I find the combination of operator such as
$$(xp-px)^2 $$ or $$(x+p)(x-p) $$
Can I just expand them by using normal algebra such ...
- 13