Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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1answer
199 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
129 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
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1answer
88 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
198 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
366 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
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1answer
224 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)\\ ...
2
votes
1answer
715 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
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1answer
186 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
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2answers
286 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
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3answers
498 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. $$\int \limits^{}_{V} \left|\psi \right|^2 \, \textrm{d} V = 1\\$$ ...
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2answers
813 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 ...
1
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0answers
224 views

Time-ordering in QFT

In Srednicki QFT page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition ...
8
votes
3answers
398 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
191 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?
14
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1answer
601 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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3answers
230 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 ...
6
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3answers
3k views

Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
2
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2answers
1k 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
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0answers
146 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
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0answers
238 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
369 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
252 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
547 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
333 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 ...
8
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1answer
600 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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3answers
117 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 ...
7
votes
3answers
2k 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 ...
5
votes
2answers
261 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
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2answers
363 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 ...
4
votes
3answers
383 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 ...
9
votes
1answer
453 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} ...
4
votes
3answers
1k views

Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
3
votes
3answers
574 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
342 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. ...
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2answers
293 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
123 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 ...
9
votes
4answers
412 views

Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
3
votes
1answer
488 views

Quantum mechanic newbie: why complex amplitudes, why Hilbert space? [duplicate]

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
522 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
49 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
211 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
187 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 ...
4
votes
2answers
584 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 ...
11
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2answers
2k 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 = ...
6
votes
0answers
100 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
534 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
120 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
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4answers
237 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 ...
11
votes
2answers
577 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 ...
9
votes
4answers
958 views

What is the physical meaning of a “complete” Hilbert space in QM?

What does the word "complete" means from the physical point of view? I do not understand what it physically means to say that a Hilbert space is a complete vector space.