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0
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1answer
175 views

What is the form of the kinetic energy operator on a one-dimensional (real space) lattice? (In second quantization)

I'm trying to figure out how one would write down the Hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...
4
votes
3answers
356 views

Schroedinger field operators and their commutation relations

I've got several questions regarding the so called second quantization of the Schroedinger equation. My professor introduced the field operators for the Schroedinger field by simply stating them as ...
1
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2answers
57 views

Commutation Relations in Second Quantization

I understand that if I have the field operators $\psi(r)$ and $\psi^\dagger(r)$, then I have the canonical commutation relation (in the boson case) $$[ \psi(r) , \psi^\dagger(r')]=\delta(r-r').$$ My ...
1
vote
2answers
80 views

Approximate expression for the ground state of hopping Hamiltonian

In second quantization, the Hamiltonian describing the hopping process between two neighboring sites is given ($N$ - number of particles and $M$ - number of sites) by: $$\hat{\mathcal H} = ...
0
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0answers
13 views

Derive Fermion Particle Density Operator in q-space

I am reading Bernevig's Topological Insulators and Topological Superconductors. In section 3.1.1, it gives an expression for the Fourier transformation of electron density operator, which I cannot ...
8
votes
2answers
319 views

The Origins of the Second Quantization

I've been studying quantum theory for a while now and have a number of closely related questions that are not giving me any peace. I am not sure if such a long format is appropriate here, but I'd like ...
0
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0answers
14 views

Creation operators that differ by a reciprocal lattice vector

In very general terms, if you have an infinite lattice of atoms you can describe the physics in terms of creation (and annihilation) operators $\hat{a}_{\mathbf{R}}$, that create (and annihilate) an ...
7
votes
3answers
182 views

What do the wave functions associated to the Fock states of each mode of a bound state system mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ Consider a string of length $L$ under tension and clamped on each end. This system is described by the wave equation and has a set of modes. ...
18
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2answers
1k views

What is the physical interpretation of second quantization?

One way that second quantization is motivated in an introductory text (QFT, Schwartz) is: The general solution to a Lorentz-invariant field equation is an integral over plane waves (Fourier ...
1
vote
0answers
61 views

reduced single particle density matrix (1-RDM) in second quantization

Is it possible to approach one body density matrix without using field operators ? For example for a double well potential, the reduced single particle density matrix is defined as: $$ ...
1
vote
1answer
37 views

many-body state in second quantization

The ground state of a system of N particles is represented as $$ \mid \Psi \rangle = \frac{1}{\sqrt{2^NN!}}\big( \hat{a}_1^{\dagger} + \hat{a}_2^{\dagger} \big)^{N} \mid 0\rangle $$ or similarly $$ ...
2
votes
1answer
320 views

Derivation of Rashba spin-orbit coupling in tight-binding model

Rashba spin-orbit coupling Hamiltonian in free space can be written as: $H_{\text{so}}=\int d^3r \Psi^{\dagger}(\mathbf{r}) \gamma (p_{x}\sigma _{y}-p_{y}\sigma _{x})\Psi(\mathbf{r})$. I expand ...
3
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0answers
53 views

How to write the second quantization form of spin-orbit coupling(Dzyaloshinskii-Moriya interaction)?

Spin orbit coupling is the single particle term, so the second quantization form can be written like:$\langle \alpha\sigma|s\cdot(\nabla V\times P)|\beta\sigma'\rangle ...
2
votes
1answer
130 views

One body harmonic oscillator states expressed in terms of creation operators

I am reading trough chapter one of Moshinsky's "The harmonic Oscillator in Modern Physics". However i am having some trouble with the mathematics in section 8 of chapter 1. I will sketch a summary of ...
2
votes
1answer
56 views

Bogoliubov-de-Gennes (BdG) formalism

Suppose you treat the mean-field BCS superconductor Hamiltonian $H$ in "BdG style" by re-writing it as $H = \frac{1}{2} \sum_k \psi_k^{\dagger} H_{BdG} \psi_k$ where, in terms of original ...
0
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0answers
53 views

Bogoliubov coefficients and eigenvectors

Suppose you have diagonalized Hamiltonian $H$ in second quantization using Bogoliubov transformation. If Hamiltonian is $N \times N$ matrix, the Bogoliubov transformation will have $N$ coefficients. ...
2
votes
1answer
57 views

Canonical field momentum in quantum field theory

In the context of the second quantization and the use of fields in the canonical quantization, the canonical momentum of the field is defined as the derivative of the field by the time coordinate. But ...
2
votes
1answer
91 views

Can mixed states be treated in the second quantization formalism? [closed]

In the first quantization formalism, mixed states can be handled using density matrices. When treating many-body quantum systems however, the second quantization formalism often comes handier, ...
0
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0answers
47 views

Single-particle operator in second quantization

I am new to second quantization and I am still rather uncomfortable with the bra-ket notation. I feel like I am slowly getting the hang of it but when it comes to shifting bra's and ket's around, I ...
1
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0answers
47 views

Eigenvalues of Hamiltonian with on-diagonal coordinate

A bit abstract, but if I take the standard graphene Hamiltonian (around the Dirac point) and introduce an on-diagonal term proportional to the coordinate $\hat{y}$, how would I find the eigenstates ...
2
votes
2answers
61 views

Fetter & Walecka's derivation of second quantised potential term in many-particle TDSE

For the potential term in the Hamiltonian, I understand that we go through the same process as with the kinetic energy term. On the RHS of the TDSE, we get something like ...
1
vote
0answers
23 views

Confused about anti-fermion notation

Classically anti-fields are obtained by charge conjugation, right? But sometimes authors label hermitian conjugated fields as anti-particles (or barred fields in Dirac language). But h.c. and charge ...
0
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0answers
63 views

Fetter & Walecka's derivation of second quantised canonical Schrodinger equation for fermions

On page 18, before the occupation number variables for states i and j are changed $n_i \rightarrow n'_i = n_i - 1$ and $n_j \rightarrow n'_j = n_j + 1$ respectively, could we not have rewritten eq. ...
2
votes
1answer
49 views

Matrix in two boson system

If there are $N$ single-particle states labeled by $1,2,3,\cdots,N$, it is said that the general two-boson state is given by $$|\Psi\rangle=\sum_{i,j=1}^N \omega_{ij}a_i^\dagger a_j^\dagger ...
0
votes
1answer
43 views

why is the chemical potential included in the hamiltonian for a systeme coupeled to a particle reservoir

I am beginning with second quantification language so i saw that if we are in grand canonical ensemble then: $$ H=H_0 - \mu N $$ naturally i thought that this $ \mu $ would be included in the ...
1
vote
0answers
74 views

Definition of partity in quantized Dirac Theory.

I'm studying from the book "An Introduction to Quantum Field Theory" from Michael E. Peskin and Daniel V. Schroeder, and I read the following: "The operator P should reverse momentum of a particle ...
6
votes
0answers
110 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
19
votes
2answers
6k views

Kubo Formula for Quantum Hall Effect

I'm trying to understand the Kubo Formula for the electrical conductivity in the context of the Quantum Hall Effect. My problem is that several papers, for instance the famous TKNN (1982) paper, or ...
3
votes
3answers
264 views

Second quantization and Hamiltonian diagonalization

So I want to diagonalize my Hamiltonian (it is bosonic hamiltonian) which is: $H=(E+\Delta)a^{\dagger}a + 1/2\Delta(a^{\dagger}a^{\dagger} + aa)$ My class didn't cover this material so I don't ...
0
votes
1answer
61 views

Derivation of Hartree-Fock equations using 2nd quantization [closed]

I derived the following effective Hamiltonian: $$ H_{eff} = \sum_k{ \left( \, \epsilon_k + \sum_{k_2}{\left(<k \, |<k_2 \, |\,u\,| \, k_2>|\, k> - ...
0
votes
1answer
56 views

Kinetic energy operator in second quantization formalism

If we want to express a quantum mechanical oeprator $ \hat{A}$ in second quantization formalism, it is $$ \hat{A} = \sum_{\alpha, \beta} \langle \alpha | \hat{A}|\beta \rangle ...
2
votes
1answer
41 views

Problem using spin-restricted form of the second-quantized nonrelativistic Hamiltonian

I have a problem that confuses me a lot. The two-electron part of the electronic nonrelativistic Hamiltonian can be written \begin{equation} \frac{1}{2}\sum_{pqrs} (pq|rs) ...
1
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0answers
31 views

Definition of vacuum and occupation number in expanding Universe

Suppose for simplicity we have theory of free quantum scalar field in expanding Universe (metric plays the role of background field) $g_{\mu \nu} = \text{diag}(1, -a^2,-a^2,-a^2)$, where $a(t) \sim ...
1
vote
1answer
48 views

Second Quantization: The Identity Operator does not Commute?

Let me take the simplest possible example. Consider the fermonic Fock-space $\Lambda^*(\mathbb{C}^n)$ built out of a finite-dimensional, oriented single-particle Hilbert space $\mathbb{C}^n$ with ...
1
vote
0answers
50 views

Photon absorption and emission in 2nd quantization

I am looking for models which describe the interaction of matter (lets take a 1D chain of atoms) with photons, especially the emission and absorption. I would love to see the derivation of models in ...
0
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0answers
26 views

Rotational invariance of on-site repulsion term in Hubbard model

I'm trying to prove to myself something I assumed that was obvious: that the term $n_{\uparrow} n_{\downarrow} = \widetilde{n}_{\uparrow} \widetilde{n}_{\downarrow}$ where, $n_{\sigma} = ...
0
votes
0answers
73 views

How to Fourier transform creation/annihilation operators?

Zee's QFT in a Nutshell pages 65-66. For a complex scalar QFT $$ \varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[a(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + ...
0
votes
1answer
84 views

Do different creation/annihilation operators always commute?

In a complex (non-hermitian) scalar QFT, is it correct that the creation/annihilation operators $a,a^\dagger$ (particle) and $b,b^\dagger$ (anti-particle) commute, i.e. $[a,b] = [a,b^\dagger] = ...
0
votes
0answers
140 views

How do you fourier transform a tight binding hamiltonian numerically?

The task is to do a fourier transformation of a tight binding hamiltonian of a 1D-chain with unit cell size 2, but even after many tries and googling I still don't have a idea how to do it correctly. ...
8
votes
2answers
513 views

Some questions about anyons?

(1) As we know, we have theories of second quantization for both bosons and fermions. That is, let $W_N$ be the $N$ identical particle Hilbert space of bosons or fermions, then the "many particle" ...
2
votes
1answer
421 views

A missing factor of 2 in the standard Hartree-Fock mean field?

Let's start from a very simple argument: If $A$ and $B$ are some operators, then I can write their product as $$AB = (A-\langle A\rangle)(B - \langle B \rangle) + \langle A \rangle B + A \langle B ...
0
votes
0answers
67 views

Probability current density and Hamiltonian commutator in 2nd quantization

If the current density operator is $$ \hat j(r) = \frac{1}{2i} [ \hat\psi^\dagger(r)\nabla\hat\psi(r) - \nabla\hat\psi^\dagger(r)\hat\psi(r)] $$ then how does it follow that $$ \langle \Psi(t) | [\hat ...
1
vote
2answers
117 views

Gauge covariant derivative of a creation operator

Suppose we define the (gauge) covariant derivative or as $$\tilde{\nabla}=\nabla+ie\textbf{A},$$ where the vector potential $\textbf{A}$ has a matrix structure where only the diagonal has nonzero ...
1
vote
0answers
99 views

Finding Ground State from Hamiltonian in Second Quantization

I am looking at the following mean-field Hamiltonian: $H=-\sum_{i,j,\sigma}t_{ij}c_{i\sigma}^\dagger c_{j\sigma}-\Delta \sum_i (n_{i\uparrow}-n_{i\downarrow})$ If I didn't have the ...
0
votes
0answers
32 views

Fermion gas exercise with annihilation and creation operator

The number states in the ground state of fermion gas are, \begin{align} n_l &= 1, \qquad \epsilon_l < \epsilon_F \quad (\epsilon_{F} \text{ is the Fermi energy.}) \\ n_l &= 0 \qquad ...
7
votes
4answers
493 views

Why do we need $2^\text{nd}$ quantization of the Dirac equation

As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion, about it. So let me start with its definition: The Dirac equation, is given by $ i ...
1
vote
1answer
251 views

How do you prove that the number operator commutes with a general Hamiltonian?

If you have a hamiltonian, in the case of a bosonic system $$ H=\sum_{ij}H_{ij}a_i^\dagger a_j, $$ and the number operator $$ N=\sum_{i}a_i^{\dagger}a_i. $$ How do you show that they commute? I have ...
4
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0answers
195 views

Has Sen quantized superstring fields?

Today I saw a paper by Ashoke Sen titled "BV Master Action for Heterotic and Type II String Field Theories". Is it really the "quantization" of superstring fields for the first time? What can be its ...
0
votes
2answers
55 views

Boson ladder operator $n+1$ factor [closed]

Looking at Boson creation and annihilation operators, I come across that \begin{equation} b_a|n_\alpha\rangle=\sqrt{n_\alpha}|n_\alpha-1\rangle \end{equation} and \begin{equation} ...
0
votes
1answer
60 views

One-electron reduced density matrix: Argument for positive semidefiniteness

I cannot follow an argument for positive-semidefiniteness of the one-electron density matrix given in "Molecular Electronic-Structure Theory" by Helgaker/Jorgensen/Olsen. First some definitions: ...