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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Hubbard Model Hamiltonian in matrix form using basis

I am reading material on Hubbard Model (please see this link) "The limits of Hubbard model" by Grabovski, and I have difficulty deriving/calculating hamiltonian in chapter 8. eq.8.2. How ...
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How to second-quantize an operator if the field operator is a spinor

In non-relativistic QM, one normally second-quantizes an operator using $$ \hat O=\int d^3r~\hat\psi^\dagger(r)O~\hat\psi(r),\qquad(1)$$ where the field operator $\hat\psi$ is given by $$\hat\psi(r)=\...
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Conditions for Bogoliubov-de Gennes Hamiltonian representation

The $H_{BdG}$ hamiltonian is described in topocondmat.org as follows: here we can see that the submatrices along the diagonal are related as negative of complex conjugate of each other, I feel that ...
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Momentum and position representation of BCS Hamiltonian

I am struggling to relate the BCS Hamiltonians in momentum and position representations. The free part of the BCS Hamiltonian is often written $$ H_{\text{pos}} = \int d^3 r ~ \psi^\dagger(r) \left(\...
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Nonzero expectation value of boson creation operator in ground state of a Bose-Einstein condensate

I was following along with these notes, and just above equation (32) on page 3, the author makes the claim that, "for a Bose condensate, the ground state boson creation operator acquires a finite ...
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Help with path integral formalism for quantum field theory [duplicate]

Recently i was doing some work with quantization of fields. I learned to quantize using the canonical method, writing in terms of the ladder operators. Then I saw that there was a more powerful method,...
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Confusion with the definition of propagator in canonical quantization and path integral formalism

Imagine the following $1$-loop diagram with two vertices in interacting $\lambda \phi^4$ theory: In momentum space the way to write down the correlator is: Drawing incoming and outgoing legs, ...
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Explaining the DHR superselection theory through specific examples

The DHR superselection theory is an important result in the framework of algebraic quantum field theory that categorizes the set of all physically admissable superselection sectors of an observable ...
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Sign in Peierls substitution

A method often used to couple a lattice tight binding model to a magnetic field is the Peierls substitution, whereby one changes all hopping elements (schematically) as $t_{ij}\mapsto t_{ij}\exp(\...
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Confusion about ladder operators

Let´s consider a system, that consists out of $N$ bosonic particles, that are not interacting with each other. The Hamiltonian of this system would be given as $$H = \sum_{i=1}^N \frac{\hbar^2}{2m}\...
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States of the Fock space and the Hilbert space

The Fock space is defined as the direct sum of all $n$-particle Hilbert spaces. Are Hilbert space vectors also Fock space vectors or are they just isomorphic to Fock space vectors?
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Increasing the dimensions of a Hamiltonian coefficient matrix to make it diagonalizeable

We consider the following Hamiltonian in the second quantization formalism $$\mathcal{H}=\sum_{k}\begin{pmatrix}c_{k\uparrow}^\dagger & c_{k\downarrow}^\dagger \end{pmatrix} \begin{bmatrix}\...
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Why do we not use the Schrödinger equation/standard QM to describe phonons (or other quasi particles)?

We were just taught about phonons in solid state physics class. Last year we did QM and now we are starting QFT as well. Phonons are excitations of a condensed matter field. I thought that we would be ...
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On diagonalizing a $6\times 6$ Hubbard mean-field hamiltonian

I am struggeling with how to tackle a specific Hamiltonian. I am working with a mean-field Hubbard model and after the introduction of a specific order parameter and transform to momentum space, it is ...
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Can an electron in a given band have a well-defined position?

In a book I am using (Quantum Theory of the Optical and Electronic Properties of Semiconductors, chapter on Quantum Dots), they define separate position space creation operators for an electron and ...
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Motivation for Quantum Field Operators

I am currently learning quantum field theory and trying to wrap my head around a couple of concepts that are still eluding my understanding: What is the reason for transitioning from wavefunctions to ...
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What physical observables are the creation/annihilation operators for the EM field made from?

The explanations of quantizing the Electric/Magnetic (E/M) fields that I've read have all basically worked by using the Coulomb gauge in free space to define the vector potential in some volume as $$ \...
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Fourier transform of the Hamiltonian of harmonic oscillator from $k$-space to $x$-space

Conventionally, I have the sum of an infinite number of harmonic oscillators with the free Hamiltonian $H_{0}$ to be $$ \begin{equation} H_{0} = \sum_{k}\hbar\omega_{k}\left(a_{k}^{\dagger}a_{k} + \...
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Canonical quantization and the parallels between QFT and QM [closed]

Questions similar to this have been asked multiple times on stack exchange about quantum field theory and the role of operators and the similarity of the theory to regular quantum mechanics. However, ...
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Second quantization and composite systems

I'm considering a one-dimensional Hamiltonian of the form $$ H=\sum_i a^\dagger_i a_i b^\dagger_i b_i, $$ where the different operators correspond to different species of bosons. This is an ...
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Ground state of homogeneous electron gas in mean field approximation

For the homogeneous electron gas in the Hartree-Fock approximation the Hamiltonian is approximated by: $\hat{H} = \sum_{\bf{k}}\epsilon_{\bf{k}}\hat{c}_{\bf{k}}^\dagger \hat{c}_{\bf{k}} - \frac{1}{2}\...
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Why does the density operator in second quantization represent a density?

Usually, in second quantization, the number operator counts the number of particles occupying a given state. For example, for an occupation number state $\psi$, we have that $\hat{n_k}|\psi> = n_k|\...
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Terminology referring to the term "quantization" in Schrödinger and Dirac equation

When people write "Quantization of Dirac equation" is the word "Quantization" the same as "second quantization"? As I understand it, both Schrödinger and Dirac equations ...
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Second quantization in QFT

My first introduction to second quantization was in the context of condensed matter physics. The idea is if we have a system of $N$ indistinguishable particles then the $N$ fold tensor product of 1 - ...
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Why do we get extra symmetric terms/factors when applying annihilation operators to multiparticle states of definite momentum?

Sorry for the wordy title, but I wasn't sure how else to express what I want to ask succinctly. My question is best illustrated with an example. Suppose we are calculating a scattering amplitude in ...
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Verify that a field operator creates a particle

In example 4.1 of Lancaster and Blundell's "Quantum field theory for the gifted amateur", we verify that a field operator creates a particle as follow: Let $|\Psi\rangle=\hat{\psi}^{\dagger}(...
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Specific commutator calculation using the path integral

Consider the path integral quantisation of a scalar field $\phi$ on flat spacetime. Let the Lagrangian be $\mathcal{L}$. I would like to prove the following equal time commutation relation: \begin{...
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D-branes mechanics and D-branes field theory

Following those two questions: The different frameworks around the point particle (for $p=0$), The different frameworks around the string (for $p=1$), I was wondering if the discussion could ...
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The different frameworks around the string

I am studying string theory and I realize that the relations between the different frameworks are not clear to me. Following this question, one could repeat the discussion but now taking $p=1$. We get ...
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The different frameworks around the point particle

I am studying string theory and I realize that the relations between the different frameworks are not clear to me. Starting from (reativistic) classical mechanics, the "state" of a point ...
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Intuition & physical interpretation behind asymmetry in applying field operators in second quantization

Equivalently to the simple quantum harmonic oscillator, we define the creation and annihilation operators in second quantization for identical particles with the commutator $[a_i, a_j^\dagger] = \...
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Why we use fields instead of wave functions?

I'm starting studying QFT and I have a problem about a concept that maybe it's stupid but I don't understand exactly. Why we use fields instead of wave functions? I tried to answer myself, and I found ...
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Density matrix element in Jaynes-Cummings model

In the Jaynes-Cummings model, when using the density matrix to describe mixed states for the atom-field system, after some calculations I got to this matrix element: $$ \rho_{ee}^A = \sum_{n=0}^{\...
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What is $\varepsilon_i$ in second quantization Hamitonian?

I'm studying a solid state physics course I have difficulties with hamiltonian defined $$\hat H = \sum_{i}\varepsilon_i \hat c^\dagger \hat c = \sum_{i} \varepsilon _i \hat n_i .$$ I thought ...
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Showing that the creation operator is a spin tensor operator

In the second quantization formalism of quantum chemistry, according to the book 1 on page 43, the creation operators $a_{p,\beta}^{\dagger},a_{p,\beta}^{\dagger}$ satisfy the relations $[\hat S_\pm,...
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Second Quantisation - Changing Basis

If you have an Interacting Hamiltonian \begin{equation}H_{int}= \sum_{\sigma,\sigma'}\sum_{\mathbf{p},\mathbf{k}} a^\dagger_{\mathbf{p},\sigma}a^\dagger_{\mathbf{k},\sigma'}a_{\mathbf{k},\sigma'}a_{\...
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What does a field represent in QFT? [duplicate]

In quantum mechanics we work using wave functions, which represent (if we take the module squared) the probability of finding a particle in a certain position or momenta. On the other hand in the case ...
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Is BdG transformation unitarily equivalent?

Consider the fermionic Hamiltonian of the form $$H = \sum_{i,j=1}^n\left( \alpha_{ij} c^\dagger_i c_j + \frac12 \gamma_{ij} c^\dagger_i c^\dagger_j + \frac12 \gamma^*_{ji}c_ic_j \right).$$ In BdG ...
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Similarities between occupation number representation/second quantisation and the Hartree-Fock method

I've been reading the Hartree-Fock approximation method to determine ground state wavefunctions for many-electron atoms, but I've become a little confused due to the similarities the approach has to ...
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Pair correlation function for non-interacting spinless bosons

While trying to solve a second quantization exercise regarding a bosonic gas, I've been having trouble trying to understand the $(1-\delta_{\text{pq}})$ term in the decomposition of the following ...
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Two-particle operators in QFT and the factor 1/2

I am learning about QFT through the book Quantum Field Theory for the Gifted Amateur and I am having trouble understanding the factor 1/2 in the definition of two particle field operators. In the book ...
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Many-body Bose-Hubbard Interaction Energy

I am trying to understand the derivation in this link. They start with a slightly modified Hamiltonian (with a gauge field) (I call $H_j$ the tunneling, since my question is regarding the interaction ...
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Deriving a squeezing Hamiltonian in the context of second-quantization

I intend to rederive the Hamiltonian of a two-mode system which describes squeezing/amplification. This is given as (see Gerry, Knight Eq. 7.187) $$ H = \hbar\omega_{s}a_{s}^{\dagger}a_{s}+\hbar\...
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Evaluating kinetic and potential energy of a single-body Hamiltonian in QFT

I am trying to derive the free Hamiltonian of a particle, specifically the kinetic and potential energy, which under second quantization, takes the form of $$ \hat{T} = -\frac{\hbar^{2}}{2m}\int d\...
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Pair correlation function in a 1D tight binding chain

I want to calculate the pair correlation function in a non-interacting Fermi system which is defined in "Quantum Theory of the Electron Liquid" by Gabriele Giuliani and Giovanni Vignale as: $...
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Approximate the two-body density matrix in terms of product of one-body density matrices

Given a set of boson operators $\hat a_\alpha$, i.e. satisfying $[\hat a_\alpha,\hat a^\dagger_\beta]=\delta_{\alpha,\beta}$, the one-body density matrix $\rho^{(1)}_{\alpha\beta}$ is defined via the ...
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Coulomb interaction in 2D crystal

My question is very simple. What is the correct way of modelling a Coulomb interaction on a 2D lattice? Usually for a system that is infinitely big $(N\to\infty)$ and not discrete $(a_0\to 0)$, the ...
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From QM to QFT learning approach

The passage from the undergraduated course of quantum mechanics and the first course of qft was plenty traumatic: usual concepts like the hamiltonian of a system with its energy eigenstates, orbital ...
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Why do the Laplacian and creation operator commute in second quantisation?

I am specifically asking this question in regard to showing that the first and second quantisation hamiltonian coincide. Namely, we want to show that $$H = \int dx\, a^\dagger(x)(-\nabla^2)a(x)$$ is ...
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Anticommutation relation for the Weyl spinors in Minkowski space+time

For $D$-dimensional Minkowski space+time, I suppose that the Dirac spinor has the following anticommutation relation: $$ \{ \psi(x_1), \psi(x_2)\}=\{ \psi^\dagger(x_1), \psi^\dagger(x_2)\}=0 $$ $$ \{ \...
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