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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.

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Why are quantum-many body problems difficult to solve? [duplicate]

I am a little confused about which classes of interacting many-problems are considered intractable. Suppose I have some tight-binding system with some nearest-neighbor density-density interactions, ...
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Time evolution of state in first and second quantization

I was trying to work out the time evolution of a single particle state in second quantization and got something apparently contradicting with the first quantized picture. For a system with energy ...
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Momentum space representaion of an electron-phonon coubling Hamiltonian

I am facing a problem transforming the following Hamiltonian into momentum space: \begin{align}\hat{H} = -\gamma \sum_\alpha\sum_{i=1}^2 \hat{X}_{i,\alpha} \hat{c}_{i,\alpha}^+\hat{c}_{i,\alpha} +t\...
elfarhan's user avatar
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Mahan's derivation of energy current of a free-particle system

On p.25 of the 3rd Ed. analogously to the polarisation operator $\textbf{P}$ for particle currents $$\textbf{P}=\int\textbf{r}\rho(\textbf{r})d^3r$$ he defines an operator $$\textbf{R}_E=\frac{1}{2}\...
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Derivation of two-body Coulomb interaction in momentum space

$\newcommand{\vec}{\mathbf}$ In Condensed Matter Field Theory by Altland and Simons, they claim the two-body Coulomb interaction for the nearly-free electron model for a $d$-dimensional cube with side ...
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Superpose two spatially separated single-photons into the same spatically mode

Consider two single photons, and let the states of them be $|H,A; V,B\rangle$. Here, $|H, A; V,B\rangle$ means that a horizontally-polarized single photon state is highly localized in spatial mode $A$ ...
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Math in Hamiltonian of the hyperquantization of EM field

1. Background: I encounter this when looking into the hyperquantization of EM field. We have the secondly quantized field as below: $$\hat{E}^{(+)}(t)=\mathscr{E} e^{-iwt+i\vec{k}\cdot\vec{r}}\hat{a}=\...
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Two-particle Green's function, possible typo in the book referred?

I'm trying to follow a computation in some QFT book, p64. The goal is to derive the equation of motion for the lesser Green's function $G^<$ defined as $$ G^< = \mp i {\rm Tr}\left(\rho \Psi^\...
user2820579's user avatar
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Two-body operators in second-quantized form

I'm currently reading through Altland and Simons' book "Condensed Matter Field Theory" (2nd ed.), where in chapter 2, page 49 they derive the form of two-body second-quantized operators. ...
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(anti)commutation relation between two different fermion fields

This question is relevant to the one enter link description here As one answered in that question, different fermions species also follow anti-commutation. But why in transverse Ising field, spins on ...
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Does the Fermi surface drift opposite to electrons in an electric field?

I have encountered somewhat of a paradox when it comes to electrons in an electromagnetic field. I hope someone here knows where the problem lies. Upon minimal coupling of a free classical electron ...
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Definition of the single-particle density matrix

The single-particle density matrix is defined as $$\rho_{ij } = \langle \psi | a_j^\dagger a_i |\psi \rangle . $$ I am curious about the order of the indices. Why is it not $$ \rho_{ij } = \langle \...
poisson's user avatar
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Ladder operators and creation & annihilation operators - different between $a$, $b$ and $c$ [closed]

Usually, the ladder operator denoted by $a$ and $a^\dagger$. In some case, people talk about the creation operator and denote it by $c$ and $c^\dagger$. Recently I see another notation, $b$ and $b^\...
Yohay Halfon's user avatar
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Hartree-Fock Hamiltonian and higher-order terms

I'm diving into Hartree-Fock methods, and I'm confused on why the Hartree-Fock Hamiltonian reduces into a single particle Hamiltonian. When applying Wick's theorem to the Fermi Sea vacuum, we use the ...
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Ground state of an harmonic chain and Klein-Gordon vacuum

Consider the lagrangian of a system of classical coupled harmonic oscillators of mass $M$, connected with springs with elastic constant $\chi$ and connected to the background with springs of elastic ...
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4 answers
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How can a QFT field act on particle states in Fock space?

Recently I asked a question that was considered a duplicate. However I felt that the related question didn't answer my doubts. After a bit of pondering I have realized the core of my discomfort with ...
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How to interpret QFT fields (in relation with QM)? [duplicate]

In QM we deal with the Schrödinger equation:1 $$i\frac{\partial}{\partial t}\psi = H \psi$$ the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the ...
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On the boundary conditions of the Casimir effect and quantization of the wave vector

I'm reviewing the famous Casimir effect. I'm uploading an image with the starting setup and frame of reference. The electric field field operator is: where $\textbf{e}$ is the polarization vector, $\...
Giuliano Artale's user avatar
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Does a normal ordered Hamiltonian admit the same solutions as the non-normal ordered Hamiltonian

I'm reading "Lectures of Quantum Field Theory" by Ashok Das, where I encountered for the first time the normal ordering of an Hamiltonian (Chapter 5.5). In the book, the Hamiltonian for the ...
Yotam Ohad's user avatar
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Second quantization of hamiltonian of the Klein-Gordon field [closed]

Good day everyone. When I try to do a second quantization on the hamiltonian, I end up with the following equation, $$ H = \int \frac{d^3p}{(2\pi)^3} \omega_{\vec{p}} {a_{\vec{p}}}^{\dagger} {a_{\vec{...
King Meruem's user avatar
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Heisenberg Hamiltonian in terms of spinon operators

In Chapter 9 of Xiao-Gang Wen's book Quantum Field Theory of Many-body Systems, the spin operator $\textbf{S}_i$ is represented by $$\begin{equation}\textbf{S}_i=\frac{1}{2}f^{\dagger}_{i\alpha}\...
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Wick's theorem for an interacting theory in the $n=4$ case

I was working with the following expression related to the Wick's theorem for four fermionic operators. $$ \langle c^\dagger_i c_j c^\dagger_p c_q \rangle = \langle c^\dagger_i c_q \rangle \langle c_j ...
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Second quantization, creation and annihilation operators in space basis

I am working through some basic knowledge of second quantization. At the beginning, everything is neat and clean. We have $|n_{1}, n_{2}, ...\rangle = a_{1}^{\dagger}a_{2}^{\dagger}...|0\rangle$, ...
QFT's user avatar
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Why covariant quantization is not enough for a relativistic quantum mechanics theory?

It is often said in quantum field theory books that a quantum theory of fields is needed because every other attempt to develop a quantum-mechanical theory compatible with the principles of relativity ...
JustWannaKnow's user avatar
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Sign of momentum doesn't affect Bogoliubov coefficients in Bogoliubov transformation for BEC

I'm running into some issues with the deriving Bogoliubov transformation. Specifically, in order to diagonalize the Hamiltonian $$\begin{align}H = \sum_p \frac{p^2}{2m} \hat a^\dagger_p \hat a_p + \...
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Question about Witten 0+1 quantum gravity

I am trying to follow Witten’s article as much as I can. What every physicist should know about string theory. Physics Today 68 (11), 38–43 (2015); my knowledge is very basic GR and QFT. I am really ...
Pato Galmarini's user avatar
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Why does the phonon displacement have an overall factor of $i$?

I'm going through Mahan's book on Many Body Physics, and I'm a bit confused about one of his claims. First he expresses the position of an atom as $$\textbf{R}=\textbf{R}_{i}^{(0)}+\textbf{Q}_i$$ ...
Redcrazyguy's user avatar
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How do you get from the polarisation operator to particle current?

I'm going through Mahan's "Many Particle Physics", and I'm a bit confused about his reasoning. He introduces the polarisation operator as $$\textbf{P}=\int\textbf{r}\rho(\textbf{r})d^3r$$ ...
Redcrazyguy's user avatar
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Given a self-adjoint operator $A$, how does one calculate $d\Gamma(A)$?

Given a self-adjoint operator $A$, I am interested in calculating the generator of the second quantization operator $d\Gamma(A)$. In particular, I am interested in the case where $A=x\partial_x-\...
Ryan Hendricks's user avatar
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Diagonalization of Hamiltonian involving two particle interactions

For a non-interacting Hamiltonian, $H = \sum_{\alpha\beta} H_{\alpha\beta} c_\alpha^\dagger c_\beta$, we can diagonalize the $H_{\alpha\beta}$ matrix to find the eigenstates, which allows us to write ...
Bio's user avatar
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Transition from position as operator in QM to a label in QFT

In David Tong's lecture "Quantum Field Theory" - Lecture 2, he said that "In Quantum mechanics, position is the dynamical degree of the particle which get changed into an operator but ...
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What is a proper symmetrization procedure for polynomial k.p. Hamiltonians in magnetic field?

Suppose we find a polynomial approximation of a Hamiltonian in two-dimensional crystal around some point in momentum space to have the form $H=k_x^n k_y^m$, with $n \neq m$. After adding magnetic ...
D. Oriekhov's user avatar
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2 answers
188 views

Squared spin operators in second quantization

Spin operator in second quantization can be written as: \begin{equation} \hat{\vec{S}}_{i} = \frac{1}{2} \sum_{\sigma \sigma'} \hat{c}^{\dagger}_{i\sigma} \hat{\vec{\sigma}}_{\sigma \sigma'} \hat{c}_{...
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1 answer
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Angular momentum operator in second quantization

I wanted to know what the angular momentum operator in the second quantization would look like in terms of the annihilation and creation operators.
sajad oskouie's user avatar
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Transition current density to generate electromagnetic field

The second quantization quantize the electromagnetic field $\boldsymbol{A}$ and $\Phi$ (vector and scalar potential): they became operators. The vector potential in case of an external current source $...
fefetltl's user avatar
2 votes
1 answer
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Misunderstanding the notion of occupation numbers

In the context of calculating the partition function of a quantum ideal gas of $N$ indistinguishable particles, we introduced the notion of ocupation numbers $n_{p,s_z}$as the number of particles in ...
Lourenco Entrudo's user avatar
2 votes
1 answer
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How is a Fock state from QFT related to the wave function from quantum mechanics?

I am currently studying quantum field theory as part of my degree. I'm just lacking intuition or an understanding of some basic concepts. So please don't hesitate to correct me if i got something ...
Benny's user avatar
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How to expand Electromagnetic fields in term of Laguerre–Gaussian (LG) beams?

I was studying canonical quantization for the electromagnetic fields. I know that we can expand our fields in other normal modes which one of them is the Laguerre–Gaussian (LG) wave set but in most ...
amir moghaddam's user avatar
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How to calculate the Green function of 1D Kitaev chain?

After performing Jordan-Wigner transformation, a uniform transverse Ising model becomes a 1D Kitaev chain as $\hat{H}_{p=0,1} = -J\sum_{j=1}^{L}{(\hat{c}_{j}^{\dagger}\hat{c}_{j+1}+\hat{c}_{j}^{\...
Shuohang Wu's user avatar
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Expectation value of non-interacting groundstate

Assume that I have a tight binding model given in second quantized form as follows; \begin{equation} H = \sum_i f_i^{\dagger}f_i + t \sum_{i,j} f_i^{\dagger}f_j \end{equation} In real space, ...
zagor's user avatar
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Most general Fock state of spin-1/2 fermions: a parametrization

I wanted to ask if it is true that a valid parametrization of a generic (unnormalized) state in the Fock space of spin-$1/2$ fermions can be written as: $$|\psi>= \prod_{j=1}^L[\alpha_j +\beta_j \...
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1 answer
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What is the ground state energy of $H = H_{0}-\mu N$?

Suppose we take $\mathscr{H} = L^{2}(\Lambda)$ our one-particle space, with box $\Lambda = [-L/2,L/2]^{d}\subset \mathbb{R}^{d}$ for some $L > 1$. Let $H_{0}$ denote the kinetic energy: $$H_{0,1} = ...
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What is the significance of these lines in the explicit Bose-Hubbard Hamiltonian?

I was doing some 2nd quantization computational physics and as my first system i decided to build up a Bose-Hubbard Hamiltonian $$ H = \sum_{k} \left\{ \tau_k(a^\dagger_{k} a_{k+1} + a_{k} a^\dagger_{...
Mephistopheles Faust's user avatar
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Bloch Hamiltonian and density matrix

The Bloch Hamiltonian is usually defined for periodic condensed matter systems in the absence of interactions. In momentum space, the full Hamiltonian of the system takes the form: $$\mathcal{H} = \...
condmatphysguy's user avatar
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How can i write the matrix representation of the following Hatano - Nelson model Hamiltonian?

I have a $1$D and one band lattice model with hopping constants $J_R $ (to the right) and $J_L$ (to the left) and under open boundary condition. It has the following Hamiltonian : $$H = \sum_{n} (J_R ...
muzbi's user avatar
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2 votes
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Tight binding on a square lattice with three orbitals symmetries

I came across a tricky problem while studying tight binding within the second quantization frame: Consider a square lattice with one atom per unit cell, where each atom has three active hydrogen atom ...
Chris Ze Third's user avatar
3 votes
1 answer
112 views

What justifies the formula for $e^{a^{\dagger}(f)}$?

When studying coherent states, one usually finds the following formula: $$e^{a^{\dagger}(f)}\Omega = \sum_{n=0}^{\infty}\frac{f^{\otimes^{n}}}{n!}$$ where $a^{\dagger}(f)$ is the usual creation ...
MathMath's user avatar
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How to compute the Feynman propagator for the Proca field?

I was repeating each step of the exercise 6.4 of the Greiner's book "Field quantization" when I discovered that there is a passage which I can't reproduce, the calculations are lengthy and ...
Filippo's user avatar
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Why do we need the Dirac's hole picture?

When I have to quantize a Dirac field I have to start by the usual classical Lagrangian and find the associated Lagrange equations, then quantize the solutions promoting them to quantum operators. In ...
Filippo's user avatar
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What is the ground state of a Hamiltonian in $k$-space after Bogoliubov transformation? [duplicate]

Consider the following Hamiltonian in $k$-space, quadratic in terms of the $\gamma$ operators: \begin{equation} \hat{H}_2=\frac{1}{2}\sum_k \begin{pmatrix} \gamma_k^\dagger & \gamma_{-...
Humberto Emiliano's user avatar

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