All Questions
Tagged with quantum-mechanics operator-theory
181
questions
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
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
0
answers
65
views
On the Spectrum of $-\Delta+V$ and the Image of $V$
Both the fundamental examples of Schrodinger Operators (Harmonic Oscillator, Hydrogen Atom ones) and the physical intuition suggests that the discrete spectrum of $-\Delta+V$ is always a subset of the ...
0
votes
1
answer
207
views
Square of the squeezing operator
What is the square of the squeezing operator $S(z)=\exp[\frac12\left((z(a^{\dagger})^{ 2}−z^\ast a^2\right)]$?
I mean, with $z \in \mathbb{R}$, what is
$S(Z)S(Z)$?
Is there any formula ?
- 11
1
vote
1
answer
106
views
How to prove that frame functions on Hilbert spaces are additive?
Let $\newcommand{\calH}{\mathcal{H}}\newcommand{\eff}{\operatorname{Eff}(\calH)}\calH$ be some separable Hilbert space, and denote with $\eff$ the set of effects on $\calH$, that is, the set of ...
- 7,095
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
5
votes
1
answer
477
views
Does the split-step operator method work for a PDE in cylindrical coordinates?
I am trying to numerically solve the 3D Schrodinger equation with an extra non-linear term (the Gross-Pitaevskii equation) using the split-step operator method. This is a follow up to a previous ...
- 243
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 ...
0
votes
1
answer
73
views
Deficiency index of symmetric operator is locally constant
Let $A: D_A \rightarrow H$ be a symmetric linear operator defined on $D_A \subseteq H$. Define the deficiency index of $A$ at $z \in \mathbb{C} \setminus \mathbb{R}$ to be $\dim( \ker (A^* - \bar{z}))$...
- 5,456
1
vote
0
answers
75
views
Trotter product formula with complex exponent
If $A$ and $B$ are bounded operators, one version of Trotter product formula is:
$$e^{A+B} = \lim_{n\to \infty}\bigg{(}e^{i\frac{A}{n}}e^{i\frac{B}{n}}\bigg{)}^{n}$$
where the limit is with respect to ...
- 4,045
0
votes
1
answer
142
views
Prove convergence of series under trace-norm topology
Let $H$ be a separable Hilbert space, $T(H)$ the set of trace-class operators, and $D(H)\subset T(H)$ the set of density operators (i.e., positive and having trace 1).
For any unit vector $\vert \...
- 1,254