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Questions tagged [hilbert-space]

This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles, and the space is complete. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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How do operators on kets and wavefunctions correspond?

Let $\hat{A}$ be an operator on Hilbert space vectors. How does one show that there always exists a corresponding operator $\hat{a}$ on wave functions? i.e. $\exists \hat{a}:L^2\rightarrow L^2$ s.t. $$...
Y G's user avatar
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Some confusion about understanding the relativistic quantum mechanics

S. Weinberg in his book "The quantum theory of fields" chapter 2 introduced the notion of symmetry in quantum mechanic as follows: Physical states are represented by rays in Hilbert space. ...
Mahtab's user avatar
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Dirac's Bracket Notation

I have a question on Dirac's bracket notation. In particular, according to this notation, vectors and covectors are represented by $|\psi\rangle$ and $\langle\psi|$ respectively. Moreover, these two ...
Falcy87's user avatar
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What kind of law can accurately describe atomic nuclei? [closed]

I understand that atomic nuclei are much too dense to behave as an ideal gas. Are they degenerate? I would assume so (similar to neutron stars), but couldn't find any laws that would accurately ...
Joe Peters's user avatar
3 votes
0 answers
51 views

Do optimal Lieb-Thirring constants have physical meaning?

In their proof of stability of matter Lieb and Thirring used a particular set of inequalities. Namely, if $H=-\Delta+V(x)$ is a Schrödinger operator, then the sum of (powers of the absolute value of) ...
Severin Schraven's user avatar
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48 views

Kramer's degeneracy and ambiguity in time-reversal operator

To my understanding, time reversal symmetry can be represented by an anti-linear operator $T=U\mathcal{K}$, where $U$ is a unitary operator and $\mathcal{K}$ represents complex conjugation. This ...
TopoLynch's user avatar
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-4 votes
2 answers
75 views

Operator's definition in Dirac picture [closed]

I have a question about the definition of quantum operators in the Dirac picture. The definition is: $$A=\sum_i \sum_j \vert i \rangle A_{ij} \langle j \vert.\tag{1}$$ By deplacing the ket vector I ...
Dayane's user avatar
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2 answers
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The eigenvectors associated to the continuous spectrum in Dirac formalism

I am comfused about the definition of an observable, eigenvectors and the spectrum in the physics litterature. All what I did understand from Dirac's monograph is that the state space is a complex ...
user536450's user avatar
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1 answer
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The meaning of a representation in one-dimensional quantum mechanics

In many places, one reads about chosing a representation for studying a particular one-dimensional quantum system. Usual representations are the position representation, the momentum representation or ...
user536450's user avatar
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0 answers
40 views

About momentum states covariant normalization

I'm following QFT of Schwrtz and I have a doubt about Eq. (2.72). In particular, from Eq. (2.69): $$[a_k,a_p^\dagger]=(2\pi)^3\delta^3(\vec{p}-\vec{k}),\tag{2.69}$$ and Eq. (2.70): $$a_p^\dagger|0\...
Albus Black's user avatar
-1 votes
1 answer
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What is the dimension considered in the Schmidt Decomposition?

In the Schmidt decomposition, is the dimension considered of each Hilbert space the complex or real one? Meaning the complex dimension of $\mathbb C^2$ has dimension $2$, not $4$. If so when you ...
jujumumu's user avatar
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3 answers
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Quantum: why linear combination of vectors (superposition) is described as "both at the same time"?

I want to get a better understanding of quantum phenomena and out world in general. Before long I've thought of Schrödinger cat as being both alive and dead (or spin both up and down). Now after some ...
Martian2020's user avatar
2 votes
0 answers
60 views

Asymptotic states and physical states in QFT scattering theory

Context In the scattering theory of QFT, one may impose the asymptotic conditions on the field: \begin{align} \lim_{t\to\pm\infty} \langle \alpha | \hat{\phi}(t,\mathbf{x}) | \beta \rangle = \sqrt{Z} \...
Steven Chang's user avatar
1 vote
1 answer
108 views

Can the Parity Operator in polar coordinates be defined as $\hat\Pi\psi(r,\theta,\phi) = \psi(r,\theta+\pi,\phi).$?

I was reading about Symmetries & Conservation Laws from Introduction to Quantum Mechanics, David J. Griffiths when I came across this question about the parity operator in three dimensions: ...
Kapil's user avatar
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3 answers
104 views

Can I use any linearly independent, orthogonal, eigenkets as starting basis to construct $S_x$, $S_y$ and $S_z$? [closed]

I know how to construct $S_z$ using $|\uparrow\rangle$=$\left(\begin{matrix}1\\0\end{matrix}\right)$ and $|\downarrow\rangle$=$\left(\begin{matrix}0\\1\end{matrix}\right)$ as starting basis. And I can ...
Siddaram's user avatar
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0 answers
25 views

Completeness meaning (complete basis vs complete metric space) [migrated]

Today my professor started talking about the formalism of QM. We talked about that that eigenvectors of a Hermitian operator (over Hilbert space) is a "complete set". He also mentioned ...
R24698's user avatar
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1 answer
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Alternative way to compute expectation value of momentum? [closed]

This might be ridiculously incorrect, but is it possible to find the expectation value of momentum like this? In the position space: $$\langle x | \psi \rangle = \psi(x)$$ $$\langle \hat{A} \rangle_{x\...
Aryan MP's user avatar
-2 votes
2 answers
134 views

How can rotation about Z be a superposition of rotations in X ($|\uparrow \rangle_z = |\uparrow\rangle_x + |\downarrow\rangle _x $) [closed]

I am fully aware of the mathematical form that relates spinstates when measured from different axes. For example the unnormalized relation: $$|\uparrow_z \rangle = |\uparrow_x \rangle + | \downarrow_x ...
Steven Sagona's user avatar
2 votes
2 answers
217 views

Why we use trace-class operators and bounded operators in quantum mechanics?

The set of trace-class operators $\mathcal{B_1(H)}$ on the Hilbert space $\mathcal{H}$ is like the Banach space $l^1$, while the set of bounded operators $\mathcal{B_\infty(H)}$ is like the Banach ...
Godfly666's user avatar
3 votes
2 answers
538 views

Does this double well potential contradict the fact that there is no degeneracy for one-dimensional bound states?

This example is taken from Modern Quantum Mechanics by Sakurai. Consider a symmetric double well potential in one-dimension with a barrier of height $V_0$ and width $a$ at the middle. The eigenstates ...
Solidification's user avatar
4 votes
3 answers
859 views

Is the zero vector necessary to do quantum mechanics?

Textbook quantum mechanics describes systems as Hilbert spaces $\mathcal{H}$, states as unit vectors $\psi \in \mathcal{H}$, and observables as operators $O: \mathcal{H} \to \mathcal{H}$. Ultimately, ...
Silly Goose's user avatar
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Why the Slavnov operator is self-adjoint? [duplicate]

In the context of BRST we can define the Slavnov operator $\Delta_{BRST}$ which generates BRST transformations. My lecture notes claim that $\Delta_{BRST}$ is self-adjoint, but I don't see why.
Alex's user avatar
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4 votes
0 answers
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Why order matters when combining angular momentum

This seems like the answer should be trivial but when decomposing the direct product of 4 spin-$\frac{1}{2}$ states into a direct sum, one gets two singlets, namely $$\frac{1}{\sqrt{2}} \left(\mid{\...
lionelbrits's user avatar
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0 votes
2 answers
116 views

Do different bases of Fock space commute?

$\newcommand\dag\dagger$ Suppose we have a Fock space $\mathcal{F}$ with two different bases of creation and annihilation operators $\{a_\lambda, a^\dag_\lambda\}$ and $\{a_{\tilde \lambda}, a^\dag_{\...
zeroknowledgeprover's user avatar
1 vote
0 answers
33 views

Why Is There No Oscillator Representation for Operators in Planar ${\cal N}=4$ SYM Theory?

I'm studying the planar ${\cal N}=4$ Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum ...
iron's user avatar
  • 43
0 votes
1 answer
43 views

Bogoliubov transformation of Bunch-Davies vacuum

Let $\left|0\right>$ be the Bunch-Davies vacuum state of a QFT, for example a free scalar field, in de Sitter space. The creation and annihilation operators w.r.t. this state is a vacuum, i.e. $a^...
Aralian's user avatar
  • 505
2 votes
1 answer
175 views

Problem Deriving "The General Uncertainty Principle" in Section 5.7 of Susskind's "Quantum Mechanics"

I'm having a problem in section 5.7 of Susskind's "Quantum Mechanics, The Theoretical Minimum". Specifically, I'm trying to derive eq. 5.11, $$ 2\sqrt{ \langle \mathbf{A}^2 \rangle \langle \...
BoCoKeith's user avatar
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2 answers
101 views

What exactly does it mean for two bosons to be in the same state?

If I understand QM correctly, it's a fact that two bosons can have the same wave function in principle. What I'm wondering is if the particles governed by the wave functions can also be in the same ...
Francisco Skrobola's user avatar
0 votes
1 answer
67 views

How are quantum states of particles represented in particle processes?

For example, lets say we have an electron-positron annihilation scenario. What will be the density matrix representing the quantum state of the electron and the positron? What will be the density ...
cdebanil's user avatar
0 votes
1 answer
71 views

How is the quantum harmonic oscillator related to Fock states?

The question is basically in the title. From what I understand, in the Fock state there is a certain number of particles in each energy level. The creation/annihilation operators create or destroy a ...
Andris Erglis's user avatar
1 vote
1 answer
129 views

The expectation of $L^2_x$

In some problems, we use $$\langle L^2_x\rangle=\frac{1}{2}(\langle L^2\rangle-\langle L^2_ z\rangle)$$ But in other problems, we use $$\langle L^2_x\rangle=\frac{1}{4}\langle[L^2_+ + L^2_- +2(L^2-L^...
Suhail Sarwar's user avatar
2 votes
1 answer
115 views

How to model evolution of quantum systems in Hilbert space? [closed]

If we let $\mathcal{H}$ be a Hilbert space of all quantum states. Can one use something like $C^1(\mathbb{R},\mathcal{H})$, the space of continuous differentiable functions from the real numbers to H, ...
jujumumu's user avatar
  • 139
1 vote
1 answer
86 views

Symmetrizing of projectors with identical particles

I am dealing with systems of $N$ identical particles in quantum mechanics. The tensor product state space is : $V^{\otimes N}$. (in the question is use the term symmetrized to designate either ...
cmatteo's user avatar
  • 254
6 votes
2 answers
639 views

When is a state entangled?

I have read from What's the difference between an entangled state, a superposed state and a cat state? that an entangled state is one that cannot be expressed as product state. Suppose we have the ...
Daniel Janjani's user avatar
0 votes
0 answers
49 views

Equality of Hilbert subspaces

If $A,B\in \mathscr{L_H}$ in the lattice of subspaces of a Hilbert space $\mathscr{H}$, then is it always true that $$A\sqsubseteq B\ \&\ B\sqsubseteq A \implies A=B\ ~ ?$$ Or is there maybe an ...
eigengrau's user avatar
  • 298
2 votes
1 answer
62 views

Operator systems in functional analysis & quantum mechanics: intuition

I saw this concept of operator systems in here but I am not sure if I want to get deep into it before knowing roughly what it is used for in, say, quantum information or quantum mechanics. My very ...
Evangeline A. K. McDowell's user avatar
8 votes
4 answers
1k views

Examples of systems with infinite dimensional Hilbert space, whose energy is bounded from above

We often encounter (and love to!) deal with systems whose energy is bounded from below, for good reasons like stability, etc. But what about systems whose energy is bounded from above? In finite ...
Sanjana's user avatar
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1 vote
0 answers
35 views

Vacuum expectation of polynomial of bosonic creation and annihilation operators [duplicate]

Let $\hat{a}^\dagger,\hat{a}$ be creation and annihilation operators with commutator $$ [\hat{a},\hat{a}^\dagger] = 1. $$ Let $|0\rangle$ be vacuum state that $$ \hat{a} |0\rangle=0. $$ Let $\beta$ be ...
Luessiaw's user avatar
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1 vote
2 answers
49 views

In degenerate perturbation theory why can we assume that matrix elements above and below the degenerate subspace disappear?

The picture shows some original Hamiltonian H which has some degeneracies. Suppose I have some perturbation V to the system and I want to find the new energies and eigenstates of the system. Then from ...
David's user avatar
  • 103
2 votes
1 answer
58 views

For any pure state, can I find a pair of non-commuting observables which saturate the uncertainty bound?

Given some pure state $|\psi\rangle$ we have the following bound on the uncertainty for two non-commuting operators $A$ and $B$ \begin{equation} \sigma_A\sigma_B\geq\left|\frac 1{2i}\langle[A,B]\...
Andrew Forbes's user avatar
5 votes
1 answer
286 views

Are "good" states in perturbation theory eigenstates of both the unperturbed and perturbed Hamiltonian?

In my quantum course, my professor asked us the true/false question: "Are 'good' states in degenerate perturbation theory eigenstates of the perturbed Hamiltonian, $H_0 + H'$?" I was ...
nnn's user avatar
  • 63
0 votes
0 answers
51 views

What are the similarities and differences between the Magnus expansion and the Schrieffer-Wolff transformation?

The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates ...
NikNack's user avatar
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0 votes
2 answers
58 views

How can I construct a trivial product state in the continuum?

When working on the lattice it is easy to define a trivial product state. A state $|\psi\rangle$ is a trivial product state if it admits the following tensor decomposition, \begin{equation} |\psi\...
Truth and Beauty and Hatred's user avatar
1 vote
1 answer
93 views

Confusion on Shankar's Motivation for the Dirac delta Function

I was reading Shankar's Principles of Quantum Mechanics and got confused on page 60, where he motivates the delta function from the normalization problem of the inner product for function spaces. We ...
Han's user avatar
  • 13
0 votes
2 answers
111 views

Normal Base for Hilbert Space of delta Potential Well

I'm interested in the problem of an attractive $\delta$ potential. The Hamiltonian is given by $$ H = - \frac{\partial_x^2}{2m} - V \delta(x). $$ Solving this typically entails looking at scattering ...
Daniel Hauck's user avatar
0 votes
2 answers
79 views

Definition of expectation value for momentum [duplicate]

I think this is probably a stupid question but I'm confused over how the expectation value for momentum is calculated. It is always given as $$⟨𝑝⟩ = ⟨𝜓|\hat{p}𝜓⟩ = −𝑖\hbar∫𝜓^*(𝑥)\frac{d𝜓(𝑥)}{...
user1184477's user avatar
1 vote
0 answers
90 views

In the path integral formulation of QFT, how do we get quantized particles out of a field?

Every QFT textbook starts by basically postulating that we have discrete states connected by creation and annihilation operators. In Quantum Mechanics, we started from a differential equation and ...
A. Kriegman's user avatar
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0 votes
2 answers
67 views

Is the tensor product injective on pure quantum state vectors?

I am reading lecture notes on quantum information/computing, and the tensor product of two pure qubit states $|b_1\rangle\otimes |b_2\rangle\in\mathbb{C}^{2\times2}$ was introduced as the "...
td12345's user avatar
0 votes
0 answers
16 views

Quantum sensing using Ion penning traps[[ODF, Dicke states and pi/2 pulse]]

I am reading the article https://www.science.org/doi/full/10.1126/science.abi5226 It's about Quantum sensing in ion penning traps. Question 1. Does the axial mode of the penning trap have nothing to ...
lalala's user avatar
  • 39
0 votes
1 answer
49 views

Seperable Quantum States

Some similar questions have been ask before, but I still don't really get the definition of seperable states in quantum mechanics. Consider a bell state of a two qubit system. \begin{align} \left|\Psi\...
Aralian's user avatar
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