The hilbert-space tag has no wiki summary.
30
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214 views
Physical interpretation of different selfadjoint extensions
Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
17
votes
2answers
89 views
What Shannon channel capacity bound is associated to two coupled spins?
The question asked is:
What is the Shannon channel capacity $C$ that is naturally associated to the two-spin quantum Hamiltonian $H = \boldsymbol{L\cdot S}$?
This question arises with a view ...
13
votes
3answers
186 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 ...
12
votes
1answer
372 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 ...
11
votes
1answer
51 views
Stabilizer formalism for symmetric spin-states?
This question developed out of conversation between myself and Joe Fitzsimons. Is there a succinct stabilizer representation for symmetric states, on systems of n spin-1/2 or (more generally) n higher ...
10
votes
3answers
325 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] + ...
10
votes
2answers
889 views
Applications of the Spectral Theorem to Quantum Mechanics
I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
8
votes
2answers
338 views
What Hermitian operators can be observables?
We can construct a Hermitian operator $O$ in the following general way:
find a complete set of projectors $P_\lambda$ which commute,
assign to each projector a unique real number $\lambda\in\mathbb ...
8
votes
2answers
953 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
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 ...
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) ...
7
votes
2answers
157 views
Under what assumptions can we split a Hilbert space into subspaces?
I was thinking about an apparently simple question about quantum mechanics, if I am looking at a quantum system described by a Hilbert space $\cal{H}$ under what hypothesis can I define A and B as ...
7
votes
5answers
307 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 ...
7
votes
1answer
75 views
Representation on Hilbert space of the product of two symmetry transformations
We know by Wigner's theorem that the representation of a symmetry transformation on the Hilbert space is either unitary and linear, or anti-unitary and anti-linear.
Let $T$ and $S$ be two symmetry ...
7
votes
1answer
251 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}
...
6
votes
2answers
695 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 ...
6
votes
2answers
175 views
Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian
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 ...
6
votes
1answer
177 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 ...
5
votes
3answers
221 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 ...
5
votes
2answers
109 views
Quantum Mechanical Operators in the argument of an exponential
In Quantum Optics and Quantum Mechanics, the time evolution operator
$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$
is used quite a lot.
Suppose $t_i =0$ for simplicity, and say the ...
5
votes
3answers
251 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
votes
1answer
263 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.
5
votes
3answers
485 views
What is a basis for the Hilbert space of a 1-D scattering state?
Suppose I have a massive particle in non-relativistic quantum mechanics. Its wavefunction can be written in the position basis as
$$\vert \Psi \rangle = \Psi_x(x,t)$$
or in the momentum basis as
...
5
votes
1answer
272 views
Why is the Haar measure times the volume of the eigenvalue simplex considered a good measure of Hilbert space volume?
In particular, why do we need both of these to find the volume? And should I be thinking of it as an actual volume or not?
This Hilbert space volume is talked about in this paper. It says
There ...
4
votes
3answers
707 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
$$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right).$$
...
4
votes
2answers
113 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 ...
4
votes
2answers
130 views
Uniqueness of eigenvector representation in a complete set of compatible observables [duplicate]
Possible Duplicate:
Uniqueness of eigenvector representation in a complete set of compatible observables
Sakurai states that if we have a complete, maximal set of compatible observables, ...
4
votes
1answer
141 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' ...
4
votes
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$ ...
4
votes
1answer
266 views
Born-Oppenheimer Approximation equivalent to Tensor-product ?
If you have a wave function $\Psi$ of a system consisting of an electron and the vibrational modes of the crystal, THEN we represent the wavefunction $\Psi%$ to be in the Hilbert Space formed by the ...
4
votes
1answer
191 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
2answers
147 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 ...
4
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0answers
67 views
Shape of the state space under different tensor products
I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).
Recall: In a ...
3
votes
1answer
105 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 ...
3
votes
2answers
372 views
Bra-ket notation and linear operators
Let $H$ be a hilbert space and let $\hat{A}$ be a linear operator on $H$.
My textbook states that $|\hat{A} \psi\rangle = \hat{A} |\psi\rangle$. My understanding of bra-kets is that $|\psi\rangle$ is ...
3
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5answers
230 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 ...
3
votes
3answers
306 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 ...
3
votes
3answers
376 views
Existence of creation and annihilation operators
In a multiple particle Hilbert space (any space of any multi-particle system), is it sufficient to define creation and annihilation operators by their action (e.g. mapping an n-particle state to an ...
3
votes
4answers
201 views
How to apply an algebraic operator expression to a ket found in Dirac's QM book?
I've been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac's book. In the fourth edition, 33rd page, starting from this:$$\xi|\xi'\rangle=\xi'|\xi'\rangle$$
(Where ...
3
votes
2answers
300 views
Uniqueness of eigenvector representation in a complete set of compatible observables
Sakurai states that if we have a complete, maximal set of compatible observables, say $A,B,C...$ Then, an eigenvector represented by $|a,b,c....>$, where $a,b,c...$ are respective eigenvalues, is ...
3
votes
2answers
197 views
QM formalism is one big confusion - lack of geometrical explaination with images
I have been trying to learn QM and it went well (all untill harmonic oscilator) until i had to face the formalism:
Hilbert space- As a novice to QM i am very sad that in none of the books i have ...
3
votes
2answers
399 views
Observing the exponential growth of Hilbert space?
One of the weirdest things about quantum mechanics (QM) is the exponential growth of the dimensions of Hilbert space with increasing number of particles.
This was already discussed by Born and ...
3
votes
1answer
50 views
Quantum Field Theory and Hilbert space dimensionality
Much (All?) of quantum theory can be done in separable Hilbert spaces with a countable basis.
How about quantum field theory? Is it “quite happy” (mathematically consistent) if everything is ...
3
votes
1answer
131 views
How does a state in quantum mechanics evolve?
I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
$$
I am ...
3
votes
1answer
83 views
State space of QFT, CCR and quantization, and the spectrum of a field operator?
In the canonical quantization of fields, CCR is postulated as (for scalar boson field ):
$$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$
in analogy with the ordinary QM commutation relation:
...
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 ...
3
votes
3answers
355 views
Can we have discontinuous wavefunctions in the Infinite Square well?
The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an ...
3
votes
1answer
315 views
Quantum mechanic newbie: why complex amplitudes, why Hilbert space?
I'm just starting learning quantum mechanics by myself (2 "lectures" so far) and I was wondering
why we need to define quantum states in a complex vector space rater than a real one?
Also I was ...
3
votes
1answer
345 views
Where does the wave function of the universe live? Please describe its home
Where does the wave function of the universe live?
Please describe its home.
I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.)
Or maybe ...
2
votes
3answers
250 views
If I go to the church of the greater Hilbert space, can I have Unitary Collapse?
Actually, unitary pseudo-collapse?
Von Neuman said quantum mechanics proceeds by two processes: unitary evolution and nonunitary reduction, also now called projection, collapse and splitting.
...
