All Questions
Tagged with quantum-mechanics operator-theory
48
questions with no upvoted or accepted answers
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
6
votes
0
answers
299
views
Two Body Schrodinger Equations
I have a question involving the eigenvalues of a two-body Schrodinger equation. Let
$$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$
over the Hilbert space $L^2(\...
- 701
5
votes
0
answers
971
views
Proof of Stone's Theorem on unitary groups
I dont understand a particular step in the proof of Stone's Theorem [ B.C. Hall, "Quantum Theory for Mathematicians",p.210-213]. Let me state the Theorem and explain where I got stuck.
Stone'...
- 389
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 ...
3
votes
0
answers
78
views
What do we lose by going from self adjoint to essentially self adjoint
I am reading Glimm and Jaffe to try and get a more formal understanding of quantum mechanics and they keep insisting that operators are essentially self-adjoint. I understand from QM we want some ...
- 427
3
votes
0
answers
143
views
Given the commutator of two operator,how to relate their eigenvectors?
Given A and B is two linear, self-adjoint operators of a (separable) Hilbert space. Now if commutator of A and B is known, the eigenvalues and eigenvectors of A and B is known as well. Is there any ...
- 1,249
3
votes
0
answers
96
views
Is it possible to give one general definition of the normal ordering symbol?
In Quantum Field Theory one usually defines the Normal Ordering Symbol by means of examples and a description of its action: the normal ordering $N$ applied to one expression will be the expression ...
- 26.9k
3
votes
0
answers
220
views
Pureness of Vector States
How does one show that irreducibility is equivalent to a vector state being pure?
In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let $\...
- 802
3
votes
0
answers
697
views
$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection
I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
- 31
2
votes
0
answers
68
views
Why are operators often written in 3s?
In a lot of my quantum mechanics and linear algebra books, operators are often defined as
$M=IMI$
and many operations on operators are often done in similar fashion, for eg. operator $M$ might be ...
- 151
2
votes
1
answer
256
views
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 ...
2
votes
0
answers
122
views
Show that a $N$-body Schrodinger operator has compact resolvent
I have a $N$-body Sch\"odinger operator of the following form
\begin{equation}
H := \sum_{j=1}^{N} (-\Delta_{x_j}+U_{trap}(x_j))+\sum_{i<j} V(x_i-x_j),
\end{equation}
where $U_{trap}$ is a positive ...
- 697
2
votes
0
answers
98
views
A complicated OPERATOR equation
I ended up in the following equation:
$J=2F^{\dagger}JF-F^{\dagger}FJ-JF^{\dagger}F$
where J is a known operator and F is unknown.
I wanna specify that my operator are infinite dimensional. How do ...
- 123
2
votes
1
answer
961
views
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: $$T_\...
- 29
2
votes
0
answers
89
views
Doubt about the spectrum of an operator
I consider the Laplacian operator
$$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
- 21