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

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4
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3answers
914 views

Direct Sum of Hilbert spaces

I am a physicist who is not that well-versed in mathematical rigour (a shame, I know! But I'm working on it.) In Wald's book on QFT in Curved spacetimes, I found the following definitions of the ...
5
votes
1answer
2k views

Differences between pure/mixed/entangled/separable/superposed states

I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum ...
5
votes
1answer
207 views

Operator norm of creation and annihilation operators

Are the creation and the annihilation operators $a(f)$ and $a^{\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked ...
5
votes
2answers
2k views

Differences between symmetric, Hermitian, self-adjoint, and essentially self-adjoint operators

I am a physicist. I always heard physicists used the terminology "symmetric", "Hermitian", "self-adjoint", and "essentially self-adjoint" operators interchangeably. Actually what is the difference ...
7
votes
2answers
411 views

Proof for the completeness of eigenfunctions of a self-adjoint operator

I always heard the eigenfunctions of a self-adjoint operator form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.
4
votes
2answers
454 views

Schrödinger equation in position representation

$$ \DeclareMathOperator{\dif}{d \!} \newcommand{\ramuno}{\mathrm{i}} \newcommand{\exponent}{\mathrm{e}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} ...
3
votes
1answer
553 views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
5
votes
1answer
436 views

What do up-left orthogonality has in common with up-down and what is their relationship?

I am familiar with the true (or general) notion of orthogonality, given in the Linear Algebra and Pythagoras theorem derived from the $\vec x \cdot \vec y = 0$. I have also recently got to know that ...
9
votes
3answers
1k views

Meaning of inner product $\langle \vec{r} | \psi(t)\rangle $

I have come across the equation which comes out of the nothing in Zettili's book Quantum mechanics concepts and applications p. 167: $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$ ...
4
votes
1answer
274 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 ...
5
votes
1answer
390 views

Hilbert space of a free particle: Countable or Uncountable?

This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but ...
0
votes
1answer
81 views

Eigenvalue $a_n$

Q1: In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
5
votes
2answers
222 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 ...
2
votes
2answers
150 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 ...
21
votes
3answers
956 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
394 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 ...
7
votes
2answers
274 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 ...
2
votes
2answers
351 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
116 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 $\langle 0 | \alpha | 0 \rangle = \alpha$, ...
10
votes
3answers
550 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 ...
-2
votes
1answer
122 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?
3
votes
2answers
956 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
229 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 ...
6
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1answer
479 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. ...
11
votes
3answers
2k 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
481 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 ...
4
votes
5answers
551 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
vote
1answer
114 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 ...
3
votes
1answer
253 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 = ...
3
votes
2answers
211 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 ...
10
votes
3answers
926 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
138 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 ...
13
votes
1answer
546 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 ...
5
votes
1answer
255 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
votes
1answer
175 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
110 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
votes
2answers
281 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 ...
-1
votes
1answer
555 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
240 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
1k 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
208 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
384 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
682 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
vote
2answers
2k 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 ...
2
votes
1answer
411 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
485 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
197 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?
15
votes
1answer
748 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
289 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 ...
7
votes
3answers
4k 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 ...