The hilbert-space tag has no wiki summary.
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1answer
31 views
Vector $\vec{z}$ and its conjugate transpose $\overline{\vec{v}^\top}$ - is it the same as $\left|z\right\rangle$ and $\left\langle z \right|$
Lets say we have a complex vector $\vec{z} \!=\!(1\!+\!2i~~2\!+\!3i~~3\!+\!4i)^T$. Its scalar product $\vec{z}^T\!\! \cdot \vec{z}$ with itself will be a complex number, but if we conjugate the ...
11
votes
3answers
139 views
Hilbert space of harmonic oscillator: Countable vs uncountable?
Hm, this just occurred to me while answering another question:
If I write the Hamiltonian for a harmonic oscillator as
$$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$
then wouldn't one set of ...
1
vote
1answer
84 views
Some Dirac notation explanations
Equation for an expectation value $\langle x \rangle$ is known to me:
\begin{align}
\langle x \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x
\end{align}
By the definition we ...
4
votes
0answers
74 views
+50
Coherent U(N) intertwiners in LQG and a measure on the Grassmanian
This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by Freidel and Livine (2011 - archive). It is very specific but related to finding a measure on a quotient ...
3
votes
2answers
191 views
QM formalism is one big confusion - lack of geometrical explaination with images
I have been trying to learn QM and it went well (all untill harmonic oscilator) until i had to face the formalism:
Hilbert space- As a novice to QM i am very sad that in none of the books i have ...
0
votes
1answer
58 views
What is inner product of the vacuum state with itself?
If $|0 \rangle$ is the vacuum state in quantum mechanics and $\alpha$ is any complex number, what is $\langle 0 | \alpha | 0 \rangle$? I need to have that $\langle 0 | \alpha | 0 \rangle = \alpha$, ...
6
votes
1answer
175 views
Entangled or unentangled?
I got a little puzzled when thinking about two entangled fermions.
Say that we have a Hilbert space in which we have two fermionic orbitals $a$ and $b$. Then the Hilbert space $H$'s dimension is just ...
-1
votes
1answer
80 views
Operators in quantum mechanics
According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$?
If so then how?
And if we transfer to integrate formulation then how it will look like?
2
votes
2answers
114 views
Vector representation of wavefunction in quantum mechanics?
I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
3
votes
1answer
104 views
The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
2
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1answer
76 views
Nonseparable Hilbert space
What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. ...
10
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3answers
309 views
How to tackle 'dot' product for spin matrices
I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
3
votes
1answer
128 views
How does a state in quantum mechanics evolve?
I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
$$
I am ...
3
votes
5answers
223 views
Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
1
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1answer
64 views
Notational techniques for dealing with creation operators on Fock space
This question is trying to see if anyone has some simple notation (or tricks) for dealing with operators acting on coherent states in a Fock space. I use bosons for concreteness; what I'm interested ...
2
votes
1answer
170 views
Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $
I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that
$$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
2
votes
2answers
142 views
Inner Product Spaces
I am trying to reconcile the definition of Inner Product Spaces that I encountered in Mathematics with the one I recently came across in Physics. In particular, if $(,)$ denotes an inner product in ...
5
votes
3answers
215 views
What is a dual / cotangent space?
Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
0
votes
3answers
117 views
What is this state as a matrix?
In QM I have the state $\lvert 00 \rangle \langle 00 \rvert$. Can anyone tell me what this would look like as a matrix? I know that
$$ \lvert 00 \rangle = \begin{pmatrix} 1 & 1 \\ 0 & 0 ...
8
votes
1answer
187 views
What really are superselection sectors and what are they used for?
When reading the term superselection sector, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-)
But now I have read in this answer, that for ...
4
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1answer
94 views
Spontaneous symmetry breaking: How can the vacuum be infinitly degenerate?
In classical field theories, it is with no difficulty to imagine a system to have a continuum of ground states, but how can this be in the quantum case?
Suppose a continuous symmetry with charge $Q$ ...
3
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1answer
83 views
State space of QFT, CCR and quantization, and the spectrum of a field operator?
In the canonical quantization of fields, CCR is postulated as (for scalar boson field ):
$$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$
in analogy with the ordinary QM commutation relation:
...
1
vote
1answer
54 views
Can I prove boundedness of an operator without checking it for its whole domain?
(I don't have a direct reference so this is a little fishy and I'll delete it if nobody recognises what I'm talking about, but I though for starters I'll ask anyway)
I've heard at university that if ...
4
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2answers
112 views
Space of states in quantum mechanics
A state in quantum mechanics I think is just a vector in a complex Hilbert space. As the physical properties are defined up to a phase $e^{i\theta}$ then this Hilbert space is invariant under the ...
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1answer
143 views
Differences between orthogonality and Kronecker delta function? [closed]
If $i$ and $j$ are two variables then Kronecker delta is written as
$$\delta_{i,j}~:=~\begin{cases}1 \hspace{3mm} \mbox{if} \hspace{3mm} i=j,\\
0 \hspace{3mm}\mbox{if} \hspace{3mm}i \neq ...
1
vote
1answer
172 views
What does it mean for something to be a ket?
Ok so I will provide the following example, which I am choosing at random from Sabio et al(2010):
$$\psi(r,\phi)~=~\left[
\begin{array}{c}
A_1r\sin(\theta-\phi)\\
...
1
vote
1answer
261 views
Wave function and Dirac bra-ket notation
Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation?
How do you get from one to the ...
1
vote
1answer
128 views
Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?
I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did ...
2
votes
2answers
149 views
In Dirac notation, what do the subscripts represent? (Solution for particle in a box in mind)
So the set of solutions for the particle in a box is given by
$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L}).$$
In Dirac notation $<\psi_i|\psi_j>=\delta_{ij}$ assuming $|\psi_i>$ ...
0
votes
3answers
215 views
Normalisation factor $\psi_0$ for wave function $\psi = \psi_0 \sin(kx-\omega t)$
I know that if I integrate probabilitlity $|\psi|^2$ over a whole volume $V$ I am supposed to get 1. This equation describes this.
$$\int \limits^{}_{V} \left|\psi \right|^2 \, \textrm{d} V = 1\\$$
...
1
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2answers
127 views
What does the quantum state of a system tell us about itself?
In quantum mechanics, quantum state refers to the state of a quantum
system. A quantum state is given as a vector in a vector space, called
the state vector. The state vector theoretically ...
8
votes
3answers
227 views
A confusion about base states of a quantum system
I have been told that the eigenkets of a operator of a space form a basis for the state of the quantum system. The eigenbasis obtained from the position operator $\textbf{x}$ is the (continuously) ...
2
votes
2answers
133 views
Why must quantum logic gates be linear operators?
Why must quantum logic gates be linear operators? I mean, is it just a consequence of quantum mechanics postulates?
12
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1answer
370 views
Intuitive meaning of Hilbert Space formalism
I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:
The observables are given by self-adjoint operators on the ...
3
votes
3answers
160 views
Banach Space representations of physical systems
I think most physicists mostly model physical systems as some kind of Hilbert space.
Hilbert spaces are a strict subset of Banach spaces.
Questions:
Can physical systems really have non-compact ...
4
votes
3answers
684 views
Don't understand the integral over the square of the Dirac delta function
In Griffiths' Introduction to Quantum Mechanics he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being
$$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right).$$
...
1
vote
2answers
334 views
What's the physical significance of the inner product of two wave functions in quantum region?
I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the inner product of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the ...
2
votes
0answers
88 views
The state of Indefinite metric in Quantum Electrodynamics
I faced difficulties to grasp why indefinite metric is introduced from no where in QED, after searching internet I found that this is a problem in QED, because one needs it to preserve theory's ...
1
vote
0answers
153 views
Representing a polarization vector for light as a 'manifold of two state'
Explain me these projections please
Context: I was reading a paper (Phys. Rev. A 68, 052307) which involved mesoscopic coherent states of light. There, in order to calculate the uncertainty of a ...
4
votes
1answer
190 views
Existence of adjoint of an antilinear operator, time reversal
The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a ...
4
votes
1answer
136 views
Scattering states of Hydrogen atom in non-relativistic perturbation theory
In doing second order time-independent perturbation theory in non-relativistic quantum mechanics one has to calculate the overlap between states
$$E^{(2)}_n ~=~ \sum_{m \neq n}\frac{|\langle m | H' ...
7
votes
5answers
305 views
The role of representation theory in QM/QFT?
I need help understanding the role of representation theory in QM/QFT. My understanding of representation theory in this context is as follows: there are physical symmetries of the system we are ...
5
votes
3answers
250 views
Takhatajan's mathematical formulation of quantum mechanics
So I began skimming L. Takhatajan's Quantum Mechanics For Mathematicians, and saw the mathematical formulation of QM that he uses (page 51). (The PDF file is available here.)
I've only taken a basic ...
5
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1answer
251 views
Rigged Hilbert space and QM
Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
2
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3answers
95 views
How much space to simulate a small Hilbert space?
I'm thinking about trying to do a numerical simulation of some very simple QM problems.
How much space do I need? To simulate the Hilbert space?
I'd like to eventually simulate the absorption or ...
6
votes
2answers
689 views
Difficulties with bra-ket notation
I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
4
votes
2answers
145 views
Kugo and Ojima's Canonical Formulation of Yang-Mills using BRST
I am trying to study the canonical formulation of Yang-Mills theories so that I have direct access to the $n$-particle of the theory (i.e. the Hilbert Space). To that end, I am following Kugo and ...
0
votes
2answers
156 views
When and how do you represent a two body state as a tensor product?
I have read that in quantum mechanics, compound systems are constructed as tensor products.
But on page 177 of Griffith, for example, a two body wavefunction is introduced as
Psi ...
3
votes
3answers
305 views
Why we use $L_2$ Space In QM?
I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ...
7
votes
1answer
246 views
When we define the S-matrix, what are “in” and “out” states?
I have seen the scattering matrix defined using initial ("in") and final ("out") eigenstates of the free hamiltonian, with
$$\left| \vec{p}_1 \cdots \vec{p}_n \; \text{out} \right\rangle
=
S^{-1}
...
