Questions tagged [second-quantization]

Second quantization or canonical quantization in quantum field theory and many-body systems is the collective organizing and accounting of an infinity of quantum excitations and their interactions through quantum field operators.

Filter by
Sorted by
Tagged with
0
votes
0answers
10 views

Why is the Coulomb repulsion between particles minimized if the orbital two-particle wave function is anti-symmetric?

In exchange coupling, we obtain the interaction term \begin{equation} \sum_{i\ne j} J_{ij} a_{i\sigma}^* a_{j\sigma'}^* a_{i\sigma'} a_{j\sigma} = --2 \sum_{i\ne j} J_{ij} (S_i\cdot S_j +\frac{1}{4} ...
2
votes
0answers
35 views

Mathematically rigorours formulation of the Bogoliubov transform for bosons

Let $\mathfrak{H}$ denote the Hilbert space describing the single-particle states and $|k\rangle$ denote an orthonormal basis of $\mathfrak{H}$. Let $c_k$ denote the corresponding annihilation ...
1
vote
0answers
20 views

A mathematical definition of quasi-particle based on second quantization

Is there a mathematical definition of a quasi-particle? So far, from what I see in textbooks, a quasi-particle is that created by the creation operator $d^*$ which was obtained by a Bogoliubov ...
0
votes
0answers
19 views

A relation for a scattering state using second quantization

Consider the scattering of a particle with momentum $\textbf{k}$ and energy $\varepsilon$ by any target. I saw in an article that, if $\Psi_{\textbf{k}}^{(+)}$ is the scattering state with outgoing ...
1
vote
0answers
45 views

Quantization of field with other complete orthogonal system

I've learned the quantization of Klein-Gordon field using Fourier expansion. I understand that this process is kind of exchanging complex fourier coefficients to operator and makes it satisfying the ...
5
votes
1answer
83 views

Is a state with no fluctuations in particle density necessarily a stationary state of the Hamiltonian?

Consider a system of identical particles (bosons or fermions) with field operator $\hat{\psi}(x)$. The particle density operator is $\hat{\psi}^\dagger(x)\hat{\psi}(x)$. Suppose that the particle ...
2
votes
1answer
88 views

Motivation for introducing quantum field theory in particle physics

Why is it so that because particles can be destroyed and recreated we introduce QFT? I read at the begining of some textbook that this is so. My main problem is not the rest of the book but the first ...
2
votes
0answers
22 views

Collective, emergent motion, solitons etc. in multiparticle system described by second quantization?

Multiparticle systems are described by vectors of creation and annihilation operators $a_i^\dagger$, $a_i$ for each particle $I_i$. The atate is described by $b_1 b_2\dots b_n$ applied to the ground ...
2
votes
1answer
35 views

Confusion about interaction in 2nd quantization

Interaction terms in second quantization is written as $$ \sum_{ijkl}c_{i}^{\dagger}c_{j}^{\dagger}V_{ijkl}c_{k}c_{l} $$ Now, is spin is there then this term is written like $$ \sum_{ijkl\sigma\sigma'...
0
votes
1answer
33 views

Ambiguity in second quantized current operator

In Mahan eq. 1.193 I see an expression for the second quantized current operator of the form: $$j(r)=\frac{e}{2mi}[\psi^\dagger(r)\nabla\psi(r)-\psi(r)\nabla\psi^\dagger(r)]$$ However, in other ...
0
votes
1answer
31 views

Is the Hubbard 2-body potential non diagonal in both direct and momentum space?

I was looking at the following table from these lecture notes: http://www.lassp.cornell.edu/clh/Book-sample/1.1.pdf And was wondering if the 2-body potential is always non-diagonal, or if there is a ...
0
votes
0answers
25 views

Particle hole symmetry in 2nd quantization

In second quantization one the Particle hole trasnformation is defined as \begin{align} \hat{\mathcal{C}} \hat{\psi}_A \hat{\mathcal{C}}^{-1} &= \sum_B U^{*\dagger}_{A,B} \hat{\psi}^{\dagger}_B \\...
0
votes
1answer
46 views

Fermion parity operator

Fermion parity operator is defined as $$ \hat{\mathcal{Q}}=\exp(i\pi\sum_j \hat{n}_j) = (-1)^{\sum_j \hat{n}_j} $$ And also if $\sum_j \hat{n}_j = \sum_j c^{\dagger}_{j}c_j=N $ is constant then it ...
0
votes
1answer
53 views

How Creation and Annihilation operator transform under an unitary transformation?

\begin{align} \hat{\mathcal H}= \sum_{i,j} \hat{\psi}^{\dagger}_i H_{i,j}\hat{\psi}_j \end{align} The $\mathcal H$ is the full second quantized Hamiltonian for a system and $H$ is the single particle ...
0
votes
3answers
85 views

From second quantization to first quantized Hamiltonian

In second quantization the Hamiltonian can be written as $$ \hat{H} = \sum_{ij} \psi_i^{\dagger} H_{ij} \psi_j = \psi^{\dagger} H\psi $$ Where, $\psi, \psi^{\dagger}$ are the annihilation and creation ...
0
votes
1answer
36 views

How does the Hubbard hamiltonian change when considering a Peierls distortion (bipartite lattice)?

The following is the Hubbard contribution to the hamiltonian in the Hubbard-Tight Binding model. $$H_{hubbard}=U \sum_i n_{i \uparrow}n_{i\downarrow}$$ where $n_{i \sigma}=c_{i\sigma}^\dagger c_{i\...
0
votes
0answers
24 views

Why interaction operator in 2nd quantization form is $V=\int dxdx' V(x-x') \rho(x)\rho(x')$?

The question is as above, where $V$ is a two-particle operator whose value depends only on relative coordinate. I am asking this question because I think the result should be not there. My claim: It ...
0
votes
1answer
75 views

What's the difference between canonical quantization and second quantization?

I am wondering the difference between the canonical quantization and the second quantization in quantum field theory. For example, a harmonic chain, one can write down its lagrangian density $\...
1
vote
0answers
47 views

Trouble following canonical quantization of massive scalar field

For the massive scalar field, the mode expansion is given by $$\hat{\phi}(x) = \int \frac{d^3 p}{(2\pi)^{3/2}} \frac{1}{(2E_\mathbf{p})^{1/2}} (\hat{a}_\mathbf{p} e^{-i\mathbf{p} \cdot \mathbf{x}} + ...
2
votes
1answer
84 views

Question about the “wave function” on Relativistic Quantum Mechanics (RQM) and Quantum Field Theory (QFT)

I'm enrolled on a short and conceptual couse on RQM and QFT and the professor made a distinction about the Klein-Gordon (K-G) equation on RQM and the K-G equation on QFT. Roughly speaking, he said ...
2
votes
0answers
67 views

Tunneling elements in the Hubbard model

Consider the tunneling Hamiltonian in the Hubbard model for a 1D lattice of quantum dots. $$\begin{align}\hat{H}_t=t\displaystyle\sum_{i,j,\sigma}c_{i,\sigma}^{\dagger}c_{j\sigma}+c^{\dagger}_{j,\...
1
vote
0answers
33 views

Fermion commutation with two quantum numbers

I am trying to understand the second quantization formalism. Let's say we have a system of fermions (e.g. electrons) with spin in an array of quantum dots. The creation and annihilation operators $c^{\...
0
votes
1answer
41 views

Nabla Operator in Kinetic Energy Hamiltonian in 2nd Quantization

Why can I, in the 2nd quantisation representation of a kinetic energy Hamiltonian $$ H=\frac { -\hbar ^ { 2 } } { 2 m } \nabla^2 $$ write the Laplace (=Nabla$^2$) operator out like this? $$ \hat { T }...
0
votes
1answer
63 views

Completeness relations in propagators

Let's consider a system of $N$ identical fermions with a time-independent Hamiltonian $H$. We define the Green's function or propagator as $$G(k_{1}, k_{2}, t, t') = -i \langle \Psi_{0}^{N} | T[c_{k_{...
1
vote
1answer
37 views

Interpretation of creation and annihilation operators acting in the state of a interacting system

If I have a system of $N$ non-interacting fermions, I can write the wave function of the ground-state of the system using a Slater determinant $$ \Phi_{0}(\textbf{r}_{1}, ..., \textbf{r}_{N}) = \frac{...
0
votes
2answers
89 views

Why has the free boson a charge $c=1$ in 2D CFT?

In the free scalar field theory in 2D conformal field theory, we consider the correlation functions of the derivatives of the fields, i.e. $$\langle \partial \phi(z) \partial \phi(w) \rangle, \tag{1}$...
2
votes
3answers
70 views

Appropriate space in second quantization

The appropriate space for the study of a system of identical bosons, for instance, is something like \begin{equation} \tag{1} \mathbb{C}\oplus\mathcal{H}\oplus(\mathcal{H}\otimes\mathcal{H})_S\oplus(...
1
vote
1answer
51 views

Second quantisation for fermions

I am trying to build a model for reactions on a lattice in the Doi-Peliti formalism. Suppose there exists a lattice of $N$ sites indexed by $i$. Each site can be either occupied or unoccupied. ...
1
vote
1answer
71 views

Interpretation of field operators

In the book Field Quantization of Greiner, in section 3.2 he introduces the field operators (for bosons), that are postuleted to satisfy the commutation relations $$[\hat{\psi}(\textbf{x},t), \hat{\...
0
votes
0answers
36 views

Second quantisation for dynamical systems

The paper "Perturbative approach to an $A + B \rightarrow C$ reaction-diffusion system", (Z. Phys. B 96, 137-144 (1994)), by Conrad and Trimper, applies the Fock Space formalism for the master ...
2
votes
1answer
61 views

Calculating wavefunctionals for general QFT states or one-particle states

Is there a standard method for calculating the wavefunctional $\Psi[\phi] = \langle \phi | \psi \rangle$ for a given state $|\psi\rangle$, where $|\phi\rangle$ are field eigenstates? We can take a ...
1
vote
1answer
51 views

Pauli matrices acting on creation operators in the second quantization formalism

I'm looking at some lecture notes for electron scattering taking place at a ferromagnet-superconductor junction. The idea is to start from a tight binding model, and eventually obtain the BdG equation....
0
votes
0answers
30 views

Wave Function of the Tight Binding Model

The Gutzwiller wavefunction, i talked in brief in this other question, is introduced to compute the expectation value of the Hubbard Hamiltonian. It is composed by a uncorrelated Slater determinant (...
0
votes
0answers
13 views

How to know the symmetries of a coupling in the Hamiltonian

Suppose you have an interaction term in your hamiltonian that looks like \begin{equation} H=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \end{equation} where $U$ is the coupling and $c$, $c^\...
4
votes
0answers
160 views

Legal values of quantum field can take? $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

Main issue: What are the legal and possible values of the quantum field can take? Clarify by examples: (1) For example, for the spin-0 Klein Gordon field $\phi$, we may choose it to be: real $\...
1
vote
1answer
75 views

Constructing singlet state in second quantization formalism

I am trying to understand the second quantization formalism. Let's say we have a system of fermions (e.g. electrons) with spin in an array of quantum dots. The creation and annihilation operators $c^{\...
2
votes
1answer
39 views

Symmetric BCS state

The ground state wavefunction for the BCS can be written $$|\Psi_{G}\rangle\equiv\prod_{\textbf{k}}[u_{k}+v_{k}c_{\textbf{k}1}^{+}c_{\textbf{-k}-1}^{+}]|\phi_{0}\rangle,$$ where $|\phi\rangle$ denotes ...
0
votes
1answer
46 views

Superconductivity and phase overlap

Given the following state $$|\Psi^{\phi}\rangle=\prod_{\mathbf{k}}(u_{k}+v_{k}e^{i\phi}c_{k1}^{+}c_{-k-1}^{+})|\phi_{0}\rangle,$$ where $|\phi_{0}>$ is the vacuum, $u_{k}, v_{k}\in\mathbb{R}$, and $...
1
vote
1answer
220 views

Why do we need creation and annihilation operators in QFT?

2. Why do we need creation and annihilation operators? Main point is that a particle can be created by creation operator and destroyed by ...
0
votes
0answers
28 views

How to calculate a one-body reduced density matrix

I calculated eigenvalues and eigenvectors of a many-body problem for a SBEC (spinorial Bose-Einstein Condensate) in the SMA approximation . Then I can calculate the density matrix of this problem. ...
2
votes
2answers
99 views

What are the fermions in the SYK model doing?

The Hamiltonian of the SYK model is \begin{equation} H = \mathcal{N}\sum_{ijkl}^N J^{ijkl} \chi_i \chi_j \chi _k \chi _l \end{equation} where $\mathcal{N}$ is some normalization to make the energy ...
0
votes
1answer
52 views

commutation relations when calculating Hamiltonian

I am reading Topics on Superfulidity of Walter Greiner Book Titled "Quantum:Mechanics Special Chapters" In Exercise page no 200, Hamiltonian has been discussed and derived throughly using commutation ...
2
votes
1answer
124 views

Mean-Field Theory in Second Quantization Formalism

Consider the Ising model in statistical physics $$H=-J\sum_{\left<i,j\right>}s_{i}s_{j}-\mu h\sum_{i}s_{i}$$ In this case mean-field approximation is done by replacing the surrounding spins ...
-2
votes
1answer
49 views

Is the expectation value of creation operator zero?

Let $c^\dagger, c$ be creation and annihilation operators respectively. And we denote expectation value of operator $A$ calculated via Hamiltonian without interaction as $\left< A \right>_0$. In ...
2
votes
2answers
76 views

Tensor product of photon number states

I'm looking to compute the tensor product of photon number states. I suspect this is a fairly simple quantum optics problem, but am having the following problem. Consider a qubit which is in the ...
2
votes
0answers
49 views

Symmetry operators of a Bloch Hamiltonian

Consider a lattice with a 3 atom basis, e.g. the Lieb lattice, and some completely arbitrary on-site energies and hopping energies and phases between the different atoms. In momentum space we can ...
2
votes
1answer
116 views

Expressing the Schrödinger equation in 2nd quantised language

For times sake, I will only write about the non-interacting part of the Hamiltonian, $$H_0=\sum_{j=1}\left(-\frac{\hbar^2}{2m}\frac{\partial}{\partial x_j^2}+U(x_j)\right)$$ where $U(x_j)$ is some ...
0
votes
1answer
121 views

Second quantization notation - Hamiltonian on triplet state

So I'm struggling quite a bit with dirac notation and second quantization and it seems like no one wants to really do calculations step-by-step to at least get the notation right. We were given the ...
0
votes
0answers
25 views

Generic modes of light

Typically in second quantization one employs creation and destruction operators for modes which are eigenstates of the momentum: $a_{k}, a^{\dagger}_{k}$ where $k$ is the momentum eigenvalue. The "...
0
votes
0answers
28 views

Bosonic Pair Distribution Function

In Schwabls Book "Advanced Quantum Mechanics" in the chapter for Bosons he calculates the Bosonic pair distribution function for noninteracting bosons. He said the expectation value of \begin{align*} \...