What is Operators on hilbert space: Definition and 25 Discussions
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm
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A
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HS
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A
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{\displaystyle \|A\|_{\operatorname {HS} }^{2}=\sum _{i\in I}\|Ae_{i}\|^{2},}
where
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{\displaystyle \|\cdot \|}
is the norm of H,
{
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{\displaystyle \{e_{i}:i\in I\}}
an orthonormal basis of H. Note that the index set need not be countable; however, at most countably many terms will be non-zero. These definitions are independent of the choice of the basis.
In finite-dimensional Euclidean space, the Hilbert–Schmidt norm
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HS
{\displaystyle \|\cdot \|_{\text{HS}}}
is identical to the Frobenius norm.
I have a problem understanding the motivation behind why all observables are represented via a hermitian operator.
I understand that from the eigenvalue equation
$$ \hat A\ket{\psi} = A_i\ket{\psi}$$
after requiring that the eigenvalues be real, the operator ##\hat A## needs to be hermitian...
I am not able to use Latex for some reason. It is very glitchy and if I do one backspace then it fills my whole screen with multiple copies of the same equation. Thus I am pasting a screenshot of handwritten equations instead. Apologies for any inconvenience.
In Introduction to Quantum...
I considered an operator ##X \in \mathcal{L}(\mathcal{X} \otimes \mathcal{K})##, that is positive, ##X \geq 0##. And I defined it as it follows:
##X = \sum_{i,j} a_{ij} ∣x_i \rangle \langle x_i ∣ \otimes ∣k_j \rangle \langle k_j∣ ##
Where ##x_i## are basis for ##\mathcal{X}## and ##k_j## basis...
According to this Wikipedia entry a quantum pure qbit state is a ray in the Hilbert space ##\mathbb H_2## of dimension 2. In other words a qbit pure quantum state is a point in the Hilbert projective line.
Now my question: is an arbitrary vector in ##\mathbb H_2## actually a "mixed" state for...
Hi,
I'm aware of the wave function ##\Psi## of a quantum system represents basically the "continuous components" of a quantum state (a point/vector in the infinite-dimension Hilbert space) in a basis. If we take the ##\delta(x - \bar x)## eigenfunctions as basis on Hilbert space then the wave...
So, I have a hamiltonian for screening effect, written like:
$$ H=\sum_{k}^{}\epsilon_{k}c_{k}^{\dagger}c_{k}+ \frac{1}{\Omega}\sum_{k,q}^{}V(q,t)c_{k+q}^{\dagger}c_{k} $$
And I have to find an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}##.
I...
I've been reading this book, in which the author expresses the vacuum projection operator ##\vert 0\rangle\langle 0\vert## in terms of the number operator ##\hat{N}=\hat{a}^{\dagger}\hat{a}##, where ##\hat{a}^{\dagger}## and ##\hat{a}## are the usual creation and annihilation operators...
Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check
<ψ|Qψ>=<Qψ|ψ>
But what about dr, dphi and d theta
I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it.
1. Homework Statement
Given an infinite base of orthonormal states in the Hilbert space...
I am trying to decompose some exponential operators in quantum optics. The interesting thing is that the operators includes operators from Su(1,1) algebra $$ [K_+,K_-]=-2K_z \quad,\quad [K_z,K_\pm]=\pm K_\pm.$$
For example this one: $$ (K_++K_-)^2.$$ But as you can see they are squares of it.
I...
Hi all, so I'm not sure if what I'm asking is trivial or interesting, but is there any general or canonical way to interpret say, The follwing operator? (Specifically in the study of quantum mechanics):
A = 1/(d/dx) (I do not mean d-1/dx-1, which is the antiderivative operator )
How would...
In the momentum representation, the position operator acts on the wavefunction as
1) ##X_i = i\frac{\partial}{\partial p_i}##
Now we want under rotations $U(R)$ the position operator to transform as
##U(R)^{-1}\mathbf{X}U(R) = R\mathbf{X}##
How does one show that the position operator as...
Hey guys,
Am facing an issue, we know that x and y operators take the same form due to isotropy of space, but sir if we destroy the isotropy, then what form will it take?
Can u pleases throw some light on this!
Thanks in advance
Hi Guys,
at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...
Can anybody give a natural interpretation of operators and states in the Heisenberg Picture? When I imagine particles flying through space, it seems that the properties of the particles are changing, rather than the position property itself. Is there any way I should be thinking about these...
I'm trying to prove the following relation $$\langle\psi\lvert \hat{A}^{\dagger}\rvert\phi\rangle =\langle\phi\lvert \hat{A}\rvert\psi\rangle^{\ast}$$ where ##\lvert\phi\rangle## and ##\lvert\phi\rangle## are state vectors and ##\hat{A}^{\dagger}## is the adjoint of some operator ##\hat{A}##...
I'm a bit confused as to what is meant by instantaneous eigenstates in the Heisenberg picture. Does it simply mean that if vectors in the corresponding Hilbert space are eigenstates of some operator, then they won't necessarily be so for all times ##t##, the eigenstates of the operator will...
Hello, could you please give me an insight on how to get through this proof involving operators?
Proof: Given an eigenvalue-eigenvector equation, suppose that the vectorstate depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t...
Homework Statement
I am trying to find the possible measurement results if a measurement of a given observable ##Q=I-\left|u\right\rangle\left\langle u\right|## is made on a system described by the density operator ##\rho={1 \over 4}\left|u\right\rangle\left\langle u\right|+{3 \over...
Hi there,
I was wondering, which is the space of (not necessarily linear) mappings from ##L^2## to itself? If you have an element ##f(x) \in L^2##, then a nonlinear mapping could be ##g(\cdot)##. Then if ##g## is bounded the image is in ##L^2##, does that mean that the space of linear and...
Homework Statement
Two identical spin-1/2 particles of mass m moving in one dimension have the Hamiltonian $$H=\frac{p_1^2}{2m} + \frac{p_2^2}{2m} + \frac{\lambda}{m}\delta(\mathbf r_1-\mathbf r_2)\mathbf s_1\cdot\mathbf s_2,$$ where (pi, ri, si) are the momentum, position, and spin operators...
Hello! I am reading about the creation and annihilation operators and I don't get how you find the creation operator from the annihilation one. The creation one is
\hat{a}=\sqrt{\frac{m \omega}{2 \hbar}}\left( \hat{x}+\frac{i \hat{p}}{m \omega}\right)
and the annihilation operator is...
I am looking at the derivation of the Heisenberg Uncertainty Principle presented here: http://socrates.berkeley.edu/~jemoore/p137a/uncertaintynotes.pdf
and am confused about line (21)...
I do not understand why AB and BA are complex conjugates of each other... (I'm still in high school so I...
Might be simple but I couldn't see. We can easily derive momentum operator for position space by differentiating the plane wave solution. Analogously I want to derive the position operator for momentum space, however I am getting additional minus sign.
By replacing $$k=\frac{p}{\hbar}$$ and...
Hello all!
I have the following question with regards to quantum mechanics.
If ##H## is a Hilbert space with a countably-infinite orthonormal basis ##\{ \left | n \right \rangle \}_{n \ \in \ \mathbb{N} }##, and two operators ##R## and ##L## on ##H## are defined by their action on the basis...