Tagged Questions
1
vote
1answer
18 views
Expectation value of pure state in quantum mechanics
It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
2
votes
0answers
39 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 $A$ ...
6
votes
1answer
90 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 ...
0
votes
1answer
167 views
Show that $A^{\dagger^{\dagger}} = A $
How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that ...
1
vote
1answer
131 views
Commutator relationship proof $[A,B^2] = 2B[A,B]$
I'm trying to find the condition necessary for this commutator relationship equality:
$$[A,B^2]=2B[A,B]$$
So far I've done this:
\begin{align*}
[A,B^2] & = B[A,B] + [A,B]B \\
...
3
votes
1answer
86 views
What are the requirements for a “test” function to show operators commute?
To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$.
My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...
10
votes
1answer
283 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 ...
2
votes
2answers
91 views
Perturbation theorem of Weyl
Does anyone know where to find something about the perturbation theorem of Weyl, preferably
on the internet. The theorem I'm talking about states:
let $A$ be a self-adjoint operator on a Hilbert ...
0
votes
0answers
289 views
Exponential of an operator
The definition of the exponential of an operator is given by the following relation:
\begin{equation}
e^U\equiv\sum_{n=0}^\infty\frac{U^n}{n!}
\end{equation}
This definition is a relation in the ...
