All Questions
Tagged with quantum-mechanics operator-theory
181
questions
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70
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If $0\leq A\leq B$ and $C\geq0$ then does $0\leq AC\leq BC$?
Let $H$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. We say $A\geq 0$ if $\langle Ax,x\rangle\geq0$ for all $x\in H$. If $0\leq A\leq B$ and $C\geq0$ then does $0\leq AC\...
- 461
1
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0
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34
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Meaning of "smooth finite-rank operators $Q$ in $L^2(\mathbb R^d)$"
What does it mean for a finite-rank operator to be "smooth"? How can one finite-rank operator be smoother than another? For me, a finite-rank operator on $L^2(\mathbb R^d)$ has an expression
...
- 461
1
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1
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84
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Operator theory: nonnegative operator $Q$ less than an orthogonal projection $P$ satisfies $PQ=QP$
Suppose that we have a Hilbert space $\mathcal{H}$, an orthogonal projection $P$ (i.e. $P=|\Psi\rangle\langle\Psi|$ for some $\Psi\in\mathcal{H}$ with $\|\Psi\|=1$) and another non-negative bounded ...
- 85
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40
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If $A$ is trace-class on $L^2(\mathbb R^2)$ then $\mathbb 1(-\Delta\leq 1)A\mathbb 1(-\Delta\leq 1)$ is finite-rank. [closed]
I am trying to prove that if $A$ is trace-class on $L^2(\mathbb R^2)$ then $\mathbb 1(-\Delta\leq 1)A\mathbb 1(-\Delta\leq 1)$ is a finite-rank operator, where $\mathbb 1(-\Delta\leq 1)$ is defined on ...
- 461
1
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1
answer
65
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An inequality for Schatten norms with a compact self-adjoint operator
Let $Q$ be a compact self-adjoint operator on $L^2(\mathbb R,\mathbb C)$. Notice that $(1-\Delta)^{s}$ is a positive (and hence self-adjoint) operator on $H$ for any $s>0$.
We denote by $\mathfrak ...
- 461
3
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1
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130
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Commutator bertween shift operator and "position" operator
I was studying the shift operator in quantum mechanics from wikipedia and in the commutator section explains that for a shift operator $\hat T(x)$ and a position operator $\hat x$, the commutator $\...
- 495
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59
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In what sense is $\breve{g}(x-y)/(2\pi)^n$ the integral kernel of $g(-i\nabla)$?
It is said in Trace Ideals and Their Applications by Barry Simon that the integral kernel of the operator $g(-i\nabla)$ on $L^2(\mathbb R^n)$ is given by $\breve{g}(x-y)/(2\pi)^n$.
Indeed, for any $\...
- 461
1
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1
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80
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Hartree equation in density matrix formalism is equivalent to a system of Hartree equations
I want to prove that:
If the density matrix $$\gamma(t):=\sum_{j=1}^N|u_j(t)\rangle\langle u_j(t)|=\sum_{j=1}^N\langle u_j(t),\cdot\rangle u_j$$ solves $$i\partial_t\gamma(t)=[-\Delta+w*\rho,\gamma],$...
- 461
4
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1
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145
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Definition of ground state?
In quantum mechanics, a ground state is an eigenstate of the hamiltonian with the minimal eigenvalue and its existence is guaranteed by appropriate theorems.
At least that's how it's defined in ...
- 305
1
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1
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42
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Find self-adjoint of $P=|a \rangle \langle b |$
Let $P$ be an operator s.a. $P=|a \rangle \langle b |$ and $P|f \rangle = \langle b | f \rangle | a \rangle $.
Find the self-adjoint and the $P^2$ operator.
My attempt:
We know to find the self ...
- 463
4
votes
1
answer
175
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Are quantum state (density) operators unique?
For context, the following question is related to my question on Physics SE. Now for the question.
Consider the following requirements on an operator $\rho$ on a Hilbert space (or on a rigged Hilbert ...
- 1,143
1
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1
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69
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For projection matrices, which vector of the outer product should be complex conjugated?
I have a conjugation wrong somewhere in my definitions, and I can't work out where it is. I want to define the standard matrix for a projection operator. If you can provide correct and standard ...
- 1,849
2
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1
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321
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On the usage of derivative in operator theory
In quantum mechanics, we work with linear operators on Hilbert spaces $\mathscr{H}$.
Suppose I have two bounded ones, defined on the same space $A, B: \mathscr{H}\to\mathscr{H}$.
It seems to me there ...
- 141
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1
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437
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Show that the adjoint of two operators is the sum of the adjoints
Problem
Show that for any two operators $\hat{A}$ and $\hat{B}$, the adjoint $(\hat{A} + \hat{B})^\dagger = \hat{A}^\dagger + \hat{B}^\dagger$. Do so using the integral form of the definition of ...
- 1,381
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1
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55
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Positive linear operator inequality
I'm stuck proving the following inequality. Let $H$ autohermitian operator such that
$$\langle u|H|u\rangle\geq0\qquad\forall |u\rangle$$
Proof that
$$|\langle u|H|v\rangle|^2\leq\langle u|H|u\rangle\...
- 59