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. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete or closed. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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Confusion about the Wigner-Eckart theorem

Background This will be a lengthy thread, but I made sure that all 3 questions are related to each other and related to the same topic. I currently encountered the W.E-theorem and while I do ...
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-4 votes
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Whats the meaning of the 1 Ket? [closed]

I am talking this one: $|1\rangle$. If I have 2 orthonormal states $|1\rangle$ and $|2\rangle$ in the 2D Hilbert space, does that imply the vector $\vec{\psi_n}=(1,2)$, if I would like to solve the ...
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5 votes
1 answer
379 views

Question about the kinetic energy operator

The Kinetic Energy Operator is essentially self-adjoint. Under what circumstances does it have a unique extension?
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3 votes
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+50

String theory hilbert space - Gas of free gravitons

I am trying to understand the arguments given in MAGOO in chapter 3.4.1(Hilbert Space of String Theory). The authors give descriptions of the Hilbert space of String Theory when we consider our theory ...
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2 answers
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What is the role of Hermitian Hamiltonians in relativistic QFT?

In single-particle quantum mechanics, the probability of finding the particle in all space is conserved due to the hermiticity of the Hamiltonians (and remains equal to unity for all times, if ...
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1 vote
0 answers
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Hilbert space of a diatomic molecule

In molecular quantum mechanics, it is very common to model a diatomic molecule as a two-level harmonic oscillator with vibrational levels lying within the electronic states: In most of the textbooks ...
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9 votes
1 answer
271 views

Resolution of the identity of operator with mixed spectrum

In most quantum mechanics text books, the resolution of the identity or completeness relation is stated in the following (or similar) form $$ \mathbb I_\mathcal H = \sum\limits_n |\lambda_n\rangle \...
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2 votes
1 answer
47 views

Why does applying the kinetic energy operator to a free particle result in a divergent integral?

The wavefunction of a free particle is just $$\psi = Ae^{i(kx-\omega t)}$$ and when you plug this into the Schrodinger equation you get the dispersion relation $$E = \frac{\hbar^2 k^2}{2m}$$ However, ...
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0 votes
1 answer
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System of two objects (Quantum Mechanics) [duplicate]

If we consider a system made out of two subsystems (i.e particles etc) and we do not consider interaction between the two subsystems. Then we have: $H_1=-\frac {\hbar^2}{2m_1}\Delta_1 + V(r_1)$ (...
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2 votes
1 answer
129 views

Problem understanding expectation value of operators defined with density operator in quantum mechanics

I have a problem in understanding why we can write the expectation value of an operator $\hat{O}$ as the trace of $\hat{\rho}\hat{O}$ where $\hat{\rho}$ is the density matrix defined for pure state. ...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

In this post, I claimed in my answer that $\partial_{\mu}^{\dagger}=\partial_{\mu}$. The reason I claimed that is because $D^{\dagger}_{\mu}=\partial_{\mu}-iqA_{\mu}$ and I assumed that $$(\partial_{\...
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Why sum of squares of the magnitudes of Fourier coefficients in Infinite Square Well equals one but it is not so in regular Fourier analysis?

My question is basically this.. In regular math, Fourier Coefficients give the "amount" a particular frequency is available in any periodic signal. The squares of sum of coefficients is not ...
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1 vote
1 answer
53 views

Product rule for bras and kets

For the time evolution of expectation value of an operator $\Omega$, we can write $$\frac{d}{dt}\langle\psi|\Omega |\psi\rangle=\langle\dot\psi|\Omega|\psi\rangle+\langle\psi|\dot\Omega|\psi\rangle+\...
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0 answers
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Derivation of the path integral in QM [duplicate]

So I'm going through the path integral derivation for a general quantum system where the generalized coordinates and momentum are given as $q^i$ and $p^i$ respectively. The transition amplitude we ...
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1 answer
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Radial position operator

While trying to find the expectation value of the radial distance $r$ of an electron in hydrogen atom in ground state the expression is: $$\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \...
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Spherical harmonics integral

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi $$ I've tried to use the definition of the ...
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Degeneracy of wavefunction in 1 dimension

Suppose we have a one-dimensional bound state, with the degenerate eigenstates given by $\psi(x)$ and $\phi(x)$. Using the Wronskian, we can show that there is no degeneracy, as the two functions are ...
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1 vote
1 answer
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Definition of a wave packet

In Shankar's QM book page 168, the author stated a wave packet is any wave function with reasonably well-defined position and momentum. What does he mean by resonably well-defined position and ...
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2 votes
1 answer
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Fock Space and Coherent state

Can a coherent photon state also belong to the Fock space? If yes, under what conditions? For example I read that $$\exp\bigg\{-\frac{1}{2}\sum_i|\alpha_i|^2\bigg\}\exp\bigg\{-\sum_i\alpha_ia_i^{\...
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  • 537
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0 answers
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QFT field being operators

Why is it necessary for fields to be operator valued in quantum field theory instead of just having scalar amplitudes?
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1 vote
1 answer
47 views

Diagonalizing a given Hamiltonian

The following Hamiltonian, which has to be diagonalized, is given: $H = \epsilon(f^{\dagger}_1f_1 + f_2^{\dagger}f_2)+\lambda(f_1^{\dagger}f_2^{\dagger}+f_1f_2)$ $f_i^{\dagger}$ and $f_i$ represent ...
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-1 votes
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Unitarity and Vector Space relation [migrated]

Can an operator be a unitary operator in a vector space and not be a unitary operator in another? If so, can I have a simple example or can anyone tell me when does that happen?
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1 answer
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Reconciling special relativity and quantum mechanics

I have been referring to QFT for the gifted amateur, p. 75. To evaluate whether a particle can exist beyond its forward light cone, we check if it has a non-zero amplitude. The amplitude being ...
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8 votes
2 answers
250 views

How are we able to use quantum field theory to study systems?

I've been trying to understand the concept of locality in QFT, and I was reading this paper by Edward Witten, where he explains (on pg 13) that the state space cannot be factored into a tensor product ...
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1 answer
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Directly get tangent vector of Bloch sphere from quantum state (qubit)?

We know that Bloch sphere is a good way to represent a qubit(two energy quantum systems). Now I want to know the tangent vector in Bloch sphere, e.g. for states $\frac{1}{\sqrt{2}}\left( \begin{array}{...
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2 votes
0 answers
44 views

Why does normal-ordering ensure finiteness?

I will be using Jan von Delft's rigorous construction of bosonization/refermionization as an example, but I will try to explain my question in more general terms. Consider an (countably) infinite-dim (...
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2 votes
0 answers
34 views

$S$-matrix commutation with Hamiltonian

I know from scattering theory that $S$-matrix and the free Hamiltonian $H_{0}$ commute due to energy conservation of incident and outgoing asymptotic states, but can the $S$-matrix and $H = H_{0} + V$,...
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0 votes
2 answers
60 views

Do operators always give a number after operating?

I am having some doubts regarding operators. In QM, when operators work on a wave function, will it always give a number times the wave function? Suppose I applied it on any normal function of x. Will ...
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3 votes
1 answer
57 views

Is there a minimum value different from zero for the uncertainty in momentum for a particle in a box?

Lately, I have been studying QM more deeply and I just discovered how many important subtleties the 'well-known' particle in an infinite potential well hides, which are precious for extending ...
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5 votes
1 answer
135 views

What does Haag's theorem say about the Schrodinger picture?

Suppose there are two interacting fields $\phi _1 $ and $\phi_2 $. Let $\psi [\phi_1, \phi_2]$ be a functional with the two fields as the input functions and complex numbers as the output, such that ...
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2 votes
0 answers
73 views

Wave function of a real scalar field in interacting quantum field theory

In interacting real scalar field theory, if I intuitively define the "wave function" of a state as $$\Psi(x)\equiv\langle\Omega|\hat{\phi}(x)|\Psi\rangle.$$ Does this wave function satisfy ...
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1 vote
2 answers
107 views

Discrete and continuous basis in Quantum Mechanics

In the context of Quantum Mechanics and Hilbert spaces, I understand that a function can be interpreted as $\psi(x) = \langle x \vert \psi \rangle$ in the position basis, and things like $$\int_a^b|\...
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1 vote
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State time-evolution in the Interaction picture

What is the Schrödinger-like equation $$i\frac{d}{dt}|\psi(t)\rangle_I=V_I|\psi(t)\rangle_I$$ telling us for the behavior of the interaction picture state vectors, $|\psi(t)\rangle_I$, at infinity/...
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0 votes
1 answer
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How would I normalize this ket vector? [closed]

So I am given the vector: $$|Ψa⟩ = |x⟩ + |y⟩ − |z⟩$$ And I need to normalize it. I know that I have to take the dot product of the vector with itself (and it needs to equal 1) but how would I do this ...
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1 vote
1 answer
47 views

Common eigenstate of incompatible observables

In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an ...
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0 votes
2 answers
88 views

Are qubits just analog, continuous classical bits?

Topologically, classical bits (cbits) are essentially special cases of qubits restricted to the poles of the Bloch sphere. However, this restriction doesn't seem to be classical per se, but is simply ...
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0 votes
0 answers
20 views

Differential equations for spontaneous emission for a three-level system

what is the differential equation describing the spontaneous emission of a quantum mechanical three-level system ? Let $|C_i|^2$ ($i=1,2,3$) be the probabilities of the atom being in state $i$. My ...
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0 votes
1 answer
44 views

$y$ Pauli Operators Eigenvectors - How are they orthogonal?

I am struggling to obtain that the eigenvectors of the Pauli $y$ operator are orthogonal, and would appreciate guidance on where I am going wrong. I have calculated the eigenvalues as: 1, -1 And ...
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5 votes
1 answer
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What is the Hilbert space in quantum field theory?

My understanding is that in classical field theory, we study a classical field $\phi(x,t)$ where for each $x\in\mathbb{R}^3$, $t\in\mathbb{R}$, $\phi(x,t)$ is a scalar. In quantum field theory, we ...
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3 votes
1 answer
73 views

Recommendations for Algebraic quantum mechanics book

I am familiar with quantum mechanics and quantum information at the level of Sakurai and Preskill's lecture notes / Nielsen and Chuang. I want to study the $C^*$ algebraic formulation of quantum ...
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1 answer
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Why is the state of a quantum system called "Density $\textbf{Operator}$"?

In quantum mechanics, a $d$-dimensional pure state is represented by a vector belonging to a $d$-dimensional Hilbert space $\mathcal{H}^d$. A mixed state is represented by a density matrix $\rho \in \...
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0 votes
0 answers
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Bra ket notation in spherical coordinates

A position eigenket can be written using a tensor product of individual Cartesian eigenkets as $\mathbf x=|x\rangle \otimes|y\rangle \otimes |z\rangle$ Can I also using spherical coordinates write the ...
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0 votes
1 answer
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Particle number conservation in matrix product state

I've been trying to understand how particle number conservation is enforced in matrix product state algorithms. As far as I understand, if the Hamiltonian commutes with the number operator, you can ...
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4 votes
1 answer
132 views

Is the Hilbert-space description of quantum many-body physics misleading and unphysical?

It is well known that any quantum time-evolution of local, time-dependent Hamiltonians can be described using a poly-depth (in number of qubits) quantum circuit (DOI:10.1103/PhysRevLett.106.170501; ...
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3 votes
1 answer
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Irreducible representations and Hilbert spaces

I am reading Howard Georgi's book "Lie Algebras in Particle Physics" where he writes the following (chapter 1.14:eigenstates): "... if some irreducible representation appears only once ...
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1 vote
1 answer
94 views

2 fermions in a box (infinite potential well)

I have 2 fermions in a box. I know that they are in the state: $$|\psi\rangle = {1 \over \sqrt2}\, (|1\rangle |2\rangle -|2\rangle|1\rangle)\,|+,+\rangle$$ If I hadn't spin, I could find wave ...
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0 votes
0 answers
32 views

Are the 'wrong' states eigenstates of perturbed Hamiltonian?

Townsend quantum mechanics In our earlier derivation we assumed that each unperturbed eigenstate $\left|\varphi_{n}^{(0)}\right\rangle$ turns smoothly into the exact eigenstate $\left|\psi_{n}\right\...
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Inner product evaluation in QM

On wikipedia on the page for inner product it states that for any two $x,y$ in a vector space $V$ the inner product $(\cdot , \cdot)$ satisfies $(ax, y) = a(x,y)$ where $a\in\mathbb{C}$. The inner ...
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2 votes
1 answer
40 views

Coupled and uncoupled qubits - Hilbert space representation

Suppose I have two situations: one where two qubits, $q_A$ and $q_B$, exist independently (on separate sides of the quantum chip, maybe), and one where they exist with some coupling between them. And ...
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  • 1,335
0 votes
2 answers
51 views

Does the location of the Hilbert space of momentum eigenstates in QFT change under time translations and boosts?

I have two questions concerning Wigner's transformation law for irreps of the Poincare group: \begin{equation} U[\Lambda,\vec{a}]\vert p,\sigma\rangle=e^{ip\cdot a}D_{\sigma'\sigma}[\Lambda;p]\vert \...
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