All Questions
Tagged with quantum-mechanics operator-theory
181
questions
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What is the most general Self-Adjoint operator acting on $L^{2}(\mathbb{R})$?
What is the most general Self-Adjoint operator acting on $L^{2}\mathbb{R}$?
My hypothesis is that the aswer to my question will be that the most general Self-Adjoint operator $A$ acting on $L^{2}(\...
- 71
1
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105
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On the structure of the set of Self-Adjoint operators acting on $L^{2}(\mathbb{R})$.
Let us consider the $Hilbert$ Space $L^{2}(\mathbb{R})$ and let $SA(L^{2}(\mathbb{R})$ be the space of all self adjoint operators acting on $L^{2}$. I have worked with operators such as $X$, $P$, $X^{...
- 71
2
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1
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370
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If $\hat{A}$ commutes with $\hat{r}^2$, does $\hat{A}$ commute with $\hat{r}$?
Let $\hat{x}, \hat{y}, \hat{z}$ be the position operators in 3D space and $\hat{r}^2=\hat{x}^2+\hat{y}^2+\hat{z}^2$ as usual.
If one operator $\hat{A}$, commutes with $\hat{r}^2$, i.e.
$$\Big[\hat{A}, ...
- 51
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Question regarding conjugate operators and the harmonic operator.
Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$
I'...
1
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1
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145
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Given $[A,B]=0$ then $[A,f(B)]=0$
Given $A$ and $B$ two operators that commute ($[A,B]=0$) then $A$ commutes with an arbitrary function of $B$
I recently saw this property of the commutators on a quantum mechanics course. We didn't ...
- 541
1
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1
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67
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Equivalence between two definitions of hermitian adjoint
Given the two definitions of hermitian adjoint:
$(1): <\Psi|A|\Phi>=(<\Psi|A^+|\Phi>)^*$
$(2): <\Psi|A\Phi>=<A^+\Psi|\Phi>$
I want to show that they are equivalent
However I ...
- 541
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62
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Integral over operators equal -When are operators equal?
This is a physics motivated question from quantum mechanics. When I have two operators $\hat{A}$ and $\hat{B}$ acting on the Hilbertspace $\mathbb{C}-L^2(\mathbb{R})$ with
$$\int\psi^*(x)\hat{A}\psi(x)...
- 320
1
vote
1
answer
42
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What can I infer from this scalar product identity?
Suppose I have shown for a single vector $\lvert 0 \rangle$ that
$$\langle 0 \rvert U^\dagger \phi U \lvert 0 \rangle=\langle 0 \rvert \phi \lvert 0 \rangle$$
where $\phi$ is a certain operator and $U$...
- 121
5
votes
1
answer
191
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Problem on a rank 1 perturbation of an self-adjoint operator
With my teacher we have been trying to solve for a while a problem that consists of two parts, and each one in three sections. The first part, which has to do with the problems that I was able to ...
- 160
1
vote
1
answer
74
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Action of an operator-valued function of the momentum operator $\hat{p}$ and its unboundedness
I am currently dealing with an operator-valued function
$f(\hat{T})$ of the following kind:
$$f(\hat{T}) =\sqrt{1 + b\hat{T}^2} $$
where $b$ $\in \mathbb{R}$ and $\hat{T}$ is a linear operator acting ...
- 31
1
vote
1
answer
94
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What is the meaning of the operator projection in quantum mechanic
Let's have a vector $B_1$ which lies in the $xz$ plane forming an angle $\theta$ with $z$ axis. Let $S_\theta$ be the projection of the operator S along $B_1$.
Question 1:
How can i write $S_\theta$ ...
- 315
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1
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33
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On derivatives of tensor products in QM
Does the derivation follow a sort of Leibniz's Rule for a tensor product?
$ \displaystyle
\frac {\partial} {\partial t} \left ( A_t \otimes B_t \right) \overset ? =
\frac {\partial A_t} {\...
- 141
4
votes
2
answers
99
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Can I conclude these weak derivatives are strong?
I'm trying to show the momentum operator $P : \mathcal D(P)\ \dot{=}\ C^\infty_0(\Bbb R)\to L^2(\Bbb R)$ with $P=-i\hbar\partial_x$ is essentially self-adjoint, which is equivalent to saying that it ...
- 7,532
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192
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proof of the product: x times m derivative of delta function.
The question is in one dimension and is : Prove that $$x\delta^{(m)}=-m\delta^{(m-1)},\ m \in \mathbb{N},$$
where $\delta^{(m)}$ is the $m$-derivative of $\delta.$
As I know, I got through this way:
$...
- 77
2
votes
1
answer
82
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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\...
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