The hilbert-space tag has no wiki summary.
3
votes
3answers
351 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 ...
2
votes
3answers
248 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.
...
1
vote
2answers
176 views
Interference, photon's phase, and the Hilbert space
Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference ...
0
votes
0answers
102 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 ...
3
votes
1answer
310 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
3answers
375 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 ...
1
vote
1answer
47 views
In which way is decoherence not symmetric between the two considered systems?
If a quantum system interacts with a "big" quantum system, you have dephasing.
The models of decoherence all have this atog aproach to them, about what is to understood of the interaction of the ...
2
votes
1answer
149 views
How can Hilbert spaces be used to study the harmonics of vibrating strings?
The overtones of a vibrating string.
These are eigenfunctions of an associated Sturm–Liouville problem.
The eigenvalues 1,1/2,1/3,… form the (musical) harmonic series.
How can Hilbert spaces be ...
7
votes
2answers
156 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 ...
3
votes
1answer
343 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 ...
8
votes
2answers
949 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 = ...
4
votes
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 ...
1
vote
1answer
302 views
Bra space and adjoint vectors
If I'm not wrong, a bra, $ \langle \phi_n | $, can be thought as a linear functional that when applied to a ket vector, $| \phi_m \rangle$, returns a complex number; that is, the inner product it's a ...
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 ...
3
votes
4answers
198 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 ...
8
votes
2answers
336 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 ...
2
votes
4answers
236 views
Books on Hilbert space and phase space?
Can you recommend books or papers that highlight or discuss extensively, or at least more than average, the similarites/differences between phase space and Hilbert space? I am primarily interested in ...
2
votes
2answers
312 views
What is the Hilbert space of a single electron?
Is it the same as the space of all possible descriptions of a single electron?
If not, how do they differ?
Please give the mathematical name or specification of this space or these spaces.
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 ...
2
votes
3answers
401 views
Hilbert space and Lie algebra in quantum mechanics
We are looking for a publication or website that explains the Standard Model in terms of Hilbert space and Lie algebra.
We are reading Debnath's Introduction to Hilbert Spaces and Applications and ...
4
votes
1answer
265 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
2answers
129 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, ...
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 ...
17
votes
2answers
85 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 ...
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 ...
30
votes
2answers
210 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. ...
5
votes
3answers
483 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 ...
10
votes
2answers
885 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 ...
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 ...
