All Questions
Tagged with quantum-mechanics operator-theory
181
questions
12
votes
1
answer
616
views
Quantization of angular momentum: is Dirac's proof wrong?
I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
- 32.8k
10
votes
1
answer
300
views
Is the failure of $\mathcal{B}(H)\simeq H\otimes H^*$ in infinite dimensions the reason for non-normal states in quantum information?
In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of ...
- 211
8
votes
3
answers
524
views
Computing the product $(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n$
I want to compute the product
$$
(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n,
$$
for a natural number $n$. For $n$ equal to 0 or 1, the computation is very simple but for such a low number as 2 the brute ...
- 1,589
8
votes
1
answer
349
views
How to find interesting operators for a quantum system?
How can we find "interesting" operators for a quantum mechanical system?
I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow V$,...
- 29.9k
8
votes
1
answer
687
views
Application of Schwartz Kernel Theorem to Quantum Mechanics
I am currently reading Quantum Mechanics for Mathematicians and have a question about a statement made in the book:
Remark. By the Schwartz kernel theorem, the operator B can be represented by an ...
- 81
6
votes
1
answer
323
views
Spectral theorem in Quantum Mechanics
In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
user66906
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
6
votes
1
answer
985
views
Kernel of adjoint operator
This problem is puzzling me, even though it should be really simple.
Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
- 329
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
1
answer
589
views
If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?
I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
- 12.4k
5
votes
1
answer
263
views
Definition of outer product
I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon.
I can understand the transition in (3.12):
$$(|\psi\rangle \...
5
votes
1
answer
747
views
The completeness relation from QM in terms of inner products
I remember from QM that the completeness relation says
$$ \sum_{n=1}^\infty |e_n\rangle \langle e_n | = I$$
so that $\langle x\mid y\rangle =\sum_{n=1}^\infty \langle x\mid e_n\rangle \langle e_n \...
- 1,015
5
votes
1
answer
672
views
A question about Moyal product
In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form
$f\star g = fg + \sum_{n=1}^{\infty} \...
- 375
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