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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2k views

First quantization vs second quantization

What is the difference between first quantization and second quantization and where does the name second quantization come from?
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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 $\...
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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}} + ...
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Where can I find a detailed derivation of the form of two body operators in the second quantization?

I've been looking around online for a couple hours now and I can't find a very informative derivation of the form for two body operators in the second quantization. Is there a resource online (...
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214 views

How to diagonalize the BCS Hubbard Hamiltonian using the Bogoliubov transformation?

How do I diagonalize the following BCS (Bardeen-Cooper-Schrieffer) Hubbard Hamiltonian: \begin{equation} H= \sum\limits_{k \in [-\frac{π}{2}, +\frac{π}{2}[} \begin{bmatrix}c^\dagger_k & c^\...
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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 ...
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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,\...
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In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...
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Real Majorana wavefunction / field: What is the big deal?

It is known that there is a set of gamma matrices that can be purely imaginary (called Majorana basis), thus one can solve the 1st quantized Majorana wave function in terms of real wave function. ...
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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^{\...
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1answer
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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^{\...
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166 views

Delta potential in terms of annihilation/creation operators

Let the Hamiltonian of a system on a discrete lattice be given by $$ \mathcal{H} = \gamma \sum_\vec{x} c^\dagger_\vec{x}c^\vphantom{\dagger}_{\vec{x}+\vec{y}} + \text{h.c.}, $$ where $\gamma$ is ...
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Is it possible to construct a pure Bosonic creation operator in scalar QFT?

In a Klein-Gordon field with Lagrangian density $$\mathcal{L} = \frac{1}{2} \left[\dot{\phi}^2 - (\nabla\phi)^2 - m^2\phi^2\right]$$ the Hamiltonian is given by \begin{align} H &= \int\...
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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_{...
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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 }...
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Basis transformation of creation and annihilation operators

I am reading through the chapter on second quantization in Advanced Quantum Mechanics by Schwabl. In §1.5.1, the book suggests that because the two orthonormal basis $\{|\lambda\rangle\}_\lambda$ and $...
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1answer
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How to understand a Hamiltonian of the form $c^\dagger \sigma^x c$

In a 2-dimensional lattice Dirac model (a discretized Hamiltonian on a lattice, the model could be found in this dissertation, equation (2.19)), I found a Hamiltonian with terms like: $$ H = \sum_{m,n}...
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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{...
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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}$...
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Schrödinger wavefunctional quantum-field eigenstates

The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time. The problem is to construct the eigenstates of a quantum ...
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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(...
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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 ...
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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. ...
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Book recommendations for second quantization

I am trying to familiarize myself with the ideas of second quantization. However, the literature that I can find online seems only to outline the tools of this formalism of quantum mechanics. ...
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Solving the BCS Hamiltonian via the Bogoliubov Transformation

I was doing a calculation in Giamarchi's Introduction to Many Body Physics, chapter 3, on BCS theory and second quantization, and ran into some confusion with the BCS Hamiltonian. The pdf is here for ...
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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{\...
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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 ...
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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 ...
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1answer
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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....
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Hedin's equations and the ground state energy

Hedin's equations are an iterative scheme to calculate the Green's function $G$, the self-energy $\Sigma$, the vertex $\Gamma$, the polarizability $\chi$, and the screened interaction $W$. However, ...
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How to write the output state of a beam splitter?

I refer to this pdf. On Page 41 (Quantum-state transformation of number states) the output state on a beam splitter is derived, based on $$b_1^{+} = T^{*} a_1^{+} + R^{*} a_2^{+}$$ $$b_2^{+} = -R ...
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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 (...
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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 $\...
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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^\...
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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 ...
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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 ...
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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 $...
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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. ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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Initial values of creation/annihilation operators

I have a question about creation/annihilation operators. For example, if I have an evolution equation for annihilation operator of photon $$ \frac{da_k}{dt} = -i \omega_k a_k$$ I obviously obtain $$...
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Annihilation and Creation Operators in QFT

I have following question about creation and annihilation operators in QFT: The Klein-Gordon field is introduced as continuous interference of plane waves $\mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})...
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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 ...
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1answer
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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 ...
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1answer
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Connection between the 'spin' and 'polarization' of relativistic and non-relativistic particles

Context 1 The spin $s$ of a relativistic particle of mass $m$ can be read off from the eigenvalue $s(s+1)$ of the operator $- \frac{W_\mu W^\mu}{m^2}$ in the rest frame of the particle where $W^\mu=\...
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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 "...
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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*} \...