All Questions
30
questions
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
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 $$\mathcal S(H):...
- 428
0
votes
3
answers
79
views
Whether $[A,B]=0$ if $\langle[A,B]\rangle=0$ for all states in the Hilbert space?
Let $\hat{A}$ and $\hat{B}$ are two self-adjoint operators in quantum mechanics corresponding to two dynamical variables $A$ and $B$. If $$\langle[\hat{A},\hat{B}]\rangle\equiv \langle\psi|[\hat{A},\...
- 693
1
vote
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
2
votes
1
answer
321
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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
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
$$
\left\langle x\left\...
- 35
3
votes
1
answer
417
views
If $\omega$ is a faithful state, then the corresponding GNS representation is faithful.
This question is from Pieter Naaijken's "Quantum Spin Systems on Infinite Lattices": Let $\mathcal{U}$ be a C*-algebra and $\omega$ a faithful state on $\mathcal{U}$, meaning that $\omega(A^*...
0
votes
1
answer
47
views
If $\sum_{x=1}^{|A|} |\psi_x\rangle\langle \psi_x|=I$, then $|\psi_x\rangle\langle \psi_x|$ is a basis projectoer.
Let $\left\{E_{x}\right\}_{x=1}^{|A|}$ is a rank 1 matrix in $\operatorname{Herm}(A)$ which $\sum_{x=1}^{|A|} E_{x}=I^A$. $A$ is our Hilbert space and $I^A$ is the identity matrix.Then I want to show ...
1
vote
1
answer
226
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Translation operator unitarily equivalent to multiplication by exponential
This is part of a problem from Hall's book "Quantum Theory for Mathematicians".
Determine the unitary operator $U:L^2(\mathbb{R^n})\to L^2(\mathbb{R^n})$ (unique up to a constant) such that
...
- 1,054
0
votes
1
answer
210
views
Position representation of an operator
$$\langle x| M|x'\rangle=M(x)\langle x|x'\rangle=M(x)\,\delta(x-x')$$
I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general operator $M $?
$\...
- 39
1
vote
1
answer
262
views
Momentum operator in the position basis
J.J Sakurai shows in the section of ' Momentum operator in the position basis' as
$P$$\lvert\alpha\rangle$=$\int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi}$ $\partial\over\partial x{'}$$ \...
- 39
2
votes
1
answer
178
views
Is it possible to take square root of the operator $e^{a\frac{d}{dx}}$?
Is it possible to take square root of the operator $e^{a\frac{d}{dx}}$ where $a$ is a real or complex constant? Actually in physics one can take the square root of $(\Box +m^2)$ operator associated ...
- 693
4
votes
0
answers
158
views
Domain issues with Weyl quantization
Most algebras of observables from quantum mechanics are closed. For example, fix a separable Hilbert space $H$, and consider the algebra of bounded operators on it. This is a Banach space.
In ...
4
votes
2
answers
349
views
Why is the magnetic Schrödinger operator positive?
In the book Schrödinger Operators by Cycon et al. they prove that the magnetic Schrödinger operator (as well as the Pauli operator) have essential spectrum $\sigma_{ess} = [0,\infty)$ if $B$ has decay ...
- 1,405
2
votes
1
answer
256
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Real eigenvalues of continuum spectrum of a self-adjoint operator
Is my understanding that if you assume eigenvectors of a self-adjoint operator are in Hilbert space, then is easy to prove that the eigenvalues must be real. However, it could happen that such ...
