The hilbert-space tag has no wiki summary.
5
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1answer
273 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.
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2answers
958 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 = ...
12
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1answer
376 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
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2answers
301 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 ...
4
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3answers
717 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
$$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right).$$
...
10
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2answers
890 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 ...
7
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1answer
253 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
$$\left| \vec{p}_1 \cdots \vec{p}_n \; \text{out} \right\rangle
=
S^{-1}
...
5
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3answers
224 views
What is a dual / cotangent space?
Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
13
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3answers
216 views
Hilbert space of harmonic oscillator: Countable vs uncountable?
Hm, this just occurred to me while answering another question:
If I write the Hamiltonian for a harmonic oscillator as
$$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$
then wouldn't one set of ...
3
votes
1answer
83 views
State space of QFT, CCR and quantization, and the spectrum of a field operator?
In the canonical quantization of fields, CCR is postulated as (for scalar boson field ):
$$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$
in analogy with the ordinary QM commutation relation:
...
6
votes
1answer
177 views
Entangled or unentangled?
I got a little puzzled when thinking about two entangled fermions.
Say that we have a Hilbert space in which we have two fermionic orbitals $a$ and $b$. Then the Hilbert space $H$'s dimension is just ...
10
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3answers
329 views
How to tackle 'dot' product for spin matrices
I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
3
votes
3answers
307 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
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
votes
2answers
401 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 ...
6
votes
2answers
698 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
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
193 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
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, ...
0
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0answers
103 views
Can experiment distinguish the basis in which a singlet state is represented?
Let $\left(|\uparrow\rangle,|\downarrow\rangle\right)$ and $\left(|\nearrow\rangle,|\swarrow\rangle\right)$ be two bases of the $2$-dimensional Hilbert space $H$.
Can an experiment distinguish ...
