Questions tagged [hilbert-space]
This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete or closed. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.
4,553
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How to know which states are entangled from a state vector? [closed]
consider the following state vector of three qubits
$$(1/2)|000⟩+(1/2)|011⟩+(1/2)|101⟩+(1/2)|110⟩.$$
how to know which qubits are entangled with respect to their basis states, in other words, how do ...
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0
answers
72
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Applying an operator on both sides of an equation [migrated]
I am doing a quantum mechanics question involving the positivity of the norm. So I'm using the fact that the norm will be greater than zero but i want to apply an operator onto the ket on one side of ...
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0
votes
1
answer
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Time evolution of operators in the Heisenberg picture
In the Schrodinger picture, a state $|{\psi_{S}(t)}\rangle$ at a time $t$ is given by applying the time-evolution operator $\hat{U}(t)=e^{-\frac{i\hat{H}t}{\hbar}}$ to the state $|{\psi_{S}(0)}\rangle$...
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1
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Confusion about Deriving Momentum Operator and Hamiltonian Operator
In Sakurai's quantum mechanics, the derivation of momentum operator and Hamlitonian operator is based on spatial translation and time translation as below,
for spatial translation and momentum ...
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1
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What are some examples of bounded momentum/Hamiltonian operators in infinite dimensional Hilbert spaces?
It is well known that one of the operators satisfying the Canonical Commutation Relation $[x,p]=i$ must be unbounded. In most cases I have seen, either both are unbounded or only $p$ is (e.g. Particle ...
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0
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Why assume no degeneracy in Spin operator?
The Stern-Gerlach experiment measures a physical property of a quantum system. We hence associate an operator $\hat{S_z}$ with the observable, in accodance with the postulates of QM.
The eigenvalues ...
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0
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34
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Does positive-definite Hamiltonian for a fermion make sense? [closed]
I have been told that positive-definite Hamiltonian for a fermion doesn't make sense. Can anyone explain why is that the case?
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Question about spin-$½$ particles
Spin-½ particles needs to rotate 720º to return to its original state. If you rotate it 360º, its state will become opposite, for example $\left| ↑ \right>$ to $-\left| ↑ \right>$. This is my ...
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1
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0
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Convergence of series of elements in a quasi-local algebra
I am studying the quasi-local algebra on Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics, but there is one thing that is not clear to me at the moment. Let's say that the ...
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1
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Projectors of unbounded operators in *-algebra
Let's suppose we have a Hilbert space $\mathcal{H}$ and the C*-algebra of the set of bounded operators $\mathcal{B}(\mathcal{H})$. For what I have understood, unbounded operators as for example ...
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0
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Is really hermiticity necessary to be a physical observable? What about larger class of operators like PT invariant operators or pseudo hermitian one?
It's really necessary for an observable represented by an operator acting in a Hilbert space to be hermitian?
It's known that not only hermitian operators have real eigenvalues and that also normal ...
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2
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How to apply operator on spin wavefunction of pauli spin matrices? [closed]
There are singlet and triplet spin wavefunctions which are shown below:
For singlet: |↑↓−↓↑> and
For triplet: |↑↑>, |1/√2(↑↓+↓↑)>, |↓↓>
If we want to apply an operator σz which represents ...
2
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1
answer
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The Expectation Value of Momentum Operator
So I had found two question based on the title one was talking about momentum operator in bound state and the other was a more general. Where in the first bound state calculation they had related $\...
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0
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Central potential and tensor [closed]
when we solve the equation of Schroedinger with central and tensor potential for nuclear reaction on two-body model, we have to use Clebsch-gordon coefitsent. but i can't understand why it happens, ...
0
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1
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Expectation Value of Operator [closed]
In the book Modern Quantum Mechanics by Sakurai the expectation value of A is defined as
$$ \left <A\right> = \left<\psi\right|A\left|\psi\right>$$
And, we can also write it as (as ...
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Nuclear spectral theorem in Rigged Hilbert Spaces (Gelfand-Maurin theorem)
Before I go into the question, I would like to mention that I am a physicist with some formal mathematical knowledge, but not expert in functional analysis.
In physics, we very often say: Let $|x \...
0
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0
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31
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2D Quantum Hamonic oscillator in magnetic field with a shiftted position
Background
Consider a hole in a 2D parabolic potential in a magnetic field which is generate by the following gauge:
$$
\vec{A} = \left( - \frac{B_z y}{2}, \frac{B_z x}{2},0\right)
$$
Our quantum ...
2
votes
2
answers
55
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Can you apply non-unitary operators to a qubit?
I am wondering if it is possible to apply continuous, invertible transformations to a qubit which are not linear, i.e. not elements of $U(N)$ where $N=2^n$ where we have $n$ qubits.
Consider $n=1$. ...
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1
answer
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A limit of a particular Quantum Fidelity
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 ...
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votes
1
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369
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Relation between Stone von Neumann Theorem and Bargmann's theorem
I am trying to understand a relation between the two theorems stated in the title. What I observed so far is that since $H^{2}(\mathbb{R},U(1))=\{e\}$, using Bargmann's theorem, we have that ...
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2
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How can we figure out what fraction of pure states in a Hilbert space are entangled? [duplicate]
The full Hilbert space of a quantum system will generally contain entangled states, and thus when entanglement is lost through decoherence, parts of Hilbert space become inaccessible. Is there a ...
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0
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What exactly is the definition of the representation of an operator in position or momentum space?
I apologize for this kind of silly question, I haven't brushed up on QM for a while.
I was looking at a problem today, essentially I'm given some operator $V = \lambda |{\xi}\rangle \langle{\xi}|$ ...
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1
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2
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Does the linear combination of basis functions, need to use eigenfunctions as basis?
Given a trial function like this one:
$$\lvert\hat{\Psi}\rangle = \sum_i c_i\lvert\psi_i\rangle$$
where the trial function is expanded using exact solutions $\psi_i$ to the Time Independent ...
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0
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63
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What is meant by "bases" not belonging to state space and how are they possible?
Section A3 of Chapter II on mathematical basis of QM in Cohen-Tannoudji’s "Quantum Mechanics" book has me a bit confused. The section is named ...
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0
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40
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Wave function square integrable [duplicate]
In quantum mechanics, when showing that the momentum operator is Hermitian operator, we use the fact that the wave function and its derivative go to zero at infinity from the assumption that the wave ...
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votes
1
answer
42
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Do all representations of state space hold the same information?
I've started taking an advanced quantum mechanics course and I'm having some trouble understanding some of the concepts related to state space representation. As I understand, $|\psi\rangle\in\mathcal{...
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1
answer
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Are there any other useful representations of state space apart from position and momentum?
As I understand, $|\psi\rangle\in\mathcal{H}$, where $\mathcal{H}$ is the Hilbert state space, is a general representation of the wave function of a system. It is a vector that, in itself, is ...
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1
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How does Weinberg definition of particle states from standard momentum work?
In his first volume, part 2.5, Weinberg define one particle states $Φ_{p,𝜎}$ ($p$ is the momentum and $𝜎$ another quantum number) in the following way :
Choose a Standard momentum $k$
Find a ...
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How do Poincare group act on Classical field, Quantum field operator, Field configuration states, Fock space states?
I will try to make each of my statement as clear as possible, if any of the statements are wrong prior to my question, please point them out:)
For simplicity, we work in free QFT with scalar field.
...
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Scalar Product Calculation and Identical Particles in Quantum Mechanics
In the book "Nolting, Theoretical Physics Part 5/2" (German), on Page 264, Formula 8.80, the author introduces second quantization in the case of identical particles. One considers the ...
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1
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Dimensionality of state space of TQFTs
As the title suggests, I am wondering about the dimensionality of state spaces in $d$-dimensional TQFTs. As of yet I have mostly been concerned with the mathematical, functorial definition of TQFTs as ...
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1
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Derivative of inner product [closed]
Consider the following product rule.
\begin{equation*}
\frac{\partial}{\partial\theta}\bigl(\langle b|a\rangle\bigr)
=\left(\frac{\partial}{\partial\theta}\langle b|\right)|a\rangle
+\langle b|\left(\...
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1
answer
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Concrete understanding of QFT Hilbert space for spinor
I'm trying to understand the concepts of a spinor field in QFT. I naively understand there are two values at each spatial position $\vec{r}$: a probability amplitude and a spinor value. Is there a ...
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2
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What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?
For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've ...
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3
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Confusion about outer product in QM
For reference, I am a beginning graduate students. I am familiar with Griffiths and am currently working through the beginning of Sakurai.
I am kind of confused about what the operator $ |a\rangle\...
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0
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What is the single-particle Hilbert space in the Fock space of QFT?
In Quantum field theory, the fields are operator-valued functions of spacetime. So for a scalar (spin $0$) field $$\psi: \mathbb{R}^{3,1} \rightarrow O(F),$$ where $O(F)$ is the space of operators on ...
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2
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Does the momentum operator applied to a position state vanish?
In quantum mechanics we have
\begin{equation*}
\langle x|p\rangle=C\exp\left(\frac{ipx}{\hbar}\right)
\end{equation*}
where $C$ is a normalization constant.
It follows that
\begin{equation*}
-i\hbar\...
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0
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Reference for mathematics of quantum mechanics with infinite degrees of freedom?
I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
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2
answers
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Is it valid to say that if two wavefunctions are not orthogonal, they must be one and the same?
I'm working on a homework problem that states the following:
My first thought was that the negative sign would simply add an additional phase of pi to the wavefunction, and hence the two ...
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1
answer
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How is the Wigner little group representation of Poincaré group Unitary?
From Weinberg's QFT Vol.1, eq(2.5.11):
$$U(\Lambda)\Psi_{p,\sigma}=({N(p)\over N(\Lambda p)})\sum_{\sigma'}D_{\sigma'\sigma}(W(\Lambda,p))\Psi_{\Lambda p ,\sigma '}.\tag{2.5.11}$$
However, this is not ...
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1
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Link between the charge and the phase in a superconducting circuit
I have a question related to superconducting quantum circuits. Especially regarding the derivation of the transformation of $\cos(\phi)$ in the charge basis.
In this question, a user states that ...
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votes
1
answer
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Does quantum mechanics need projective representations only due to the Born rule?
In quantum mechanics, physical states don't live in the Hilbert space, but rather on the equivalence class of rays on the Hilbert space. This is called a projective space. This is the reason why when ...
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2
votes
1
answer
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Overlap between eigenstates of angular momentum operators
Consider the states $\left|j,m_x\right>_x$ and $\left|j,m_z\right>_z$ with total angular momentum $j$ and the angular momentum operators $\hat{S}_x$ and $\hat{S}_y$. In particular, assume that ...
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1
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Relative "volume" of entangled vs product states [duplicate]
A system containing $n$ qubits is described by a $2^n-$dimensional Hilbert space. Some of these states can be decomposed as product states, but not all of them. The remaining ones are called entangled ...
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0
votes
2
answers
72
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Kraus Operator for two-qubit basis
Let A and B each be a single qubit so that $\mathbb{H_{AB}}$ is a two-qubit system. In the basis {$|\uparrow\uparrow>,|\uparrow\downarrow>,|\downarrow\uparrow>,|\downarrow\downarrow>$, the ...
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1
answer
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Eigenvalues of spin operator [duplicate]
I am currently referring Sakurai. He introduces spin states and operators from general arguments and experimental evidence but ad hoc introduces that the eigenvalues of the pure states as $\pm \frac{\...
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votes
0
answers
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Calculation about fermions via quantum field theory
I want to ask a specific question occurred in my current learning about neutrinos.
What I want to calculate is an amplititude:
\begin{equation}
\langle\Omega|a_{\bf k m}a_{\bf pj}a_{\bf qi}^{\dagger}...
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0
votes
1
answer
72
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Finding a complete eigenbasis for an "entangled" Hamiltonian?
Suppose we have a tensor product Hilbert space $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$ and we have a Hamiltonian defined thereon which is given by
$H = H_{1e} \otimes I+ H_1 \otimes H_2$. ...
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1
answer
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Closed form for number of dimension of angular momentum eigenspace
I am trying to construct the totally uncorrelated state for a subsystem $i$ given the information that a spin has total angular momentum eigenvalue $s_i$. I therefore write that the state operator ...
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6
votes
0
answers
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The unitarity of the $\delta(x)$ potential
One of the common potentials to solve in quantum mechanics is when $$H=\frac{p^{2}}{2m}+\delta(x).$$ Is this Hamiltonian considered to produce unitary evolution?
In particular, I'm not sure what is ...
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