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1
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1answer
21 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 ...
0
votes
0answers
59 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
45 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
24 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
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0answers
68 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
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0answers
100 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 ...
0
votes
1answer
121 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 ...
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
144 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
53 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
37 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
37 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
26 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
42 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
43 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
votes
0answers
25 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
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0answers
62 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
73 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
94 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
490 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
397 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
46 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
105 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
74 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
30 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
458 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
167 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
votes
0answers
184 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
54 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
52 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: ...
5
votes
1answer
84 views

Non-hermiticity of Dirac Lagrangian: null momentum?

The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are $$ \pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\ \bar \pi=\frac{\partial ...
2
votes
1answer
76 views

What is a single-phonon?

From what I understood from wikipedia, as well as some other resources, each phonon corresponds to a normal mode oscillation, and the creation operator to create a phonon of wavevector $k$ is: $$ ...
2
votes
1answer
49 views

How is intensity defined for quantized EM fields?

Classically intensity is defined as $$ I \equiv \frac{1}{2} c \epsilon_0 E^2, $$ but when you perform a second quantization this definition becomes a bit ambiguous since the $E^2$ could be ...
0
votes
1answer
92 views

Commutation relations in second quantization

I know that for operators $a(\chi_1), a(\chi_2)$ of the same type (fermionic or bosonic) $$ [a(\chi_1), a(\chi_2)]_{-\xi} = [a^\dagger (\chi_1), a^\dagger (\chi_2)]_{-\xi} = 0 \tag{1}$$ where $$\xi ...
1
vote
3answers
224 views

Is Hamiltonian a differential operator in second quantization?

Normally, a free particle Hamiltonian is written $$ \hat{H} = - \frac{\hbar^2}{2m} \Delta $$ which is a differential operator because Laplacian $\Delta$ is. On the other hand, in second ...
2
votes
1answer
73 views

Minus sign in the time ordering operator

The time ordering operator is usually defined as $$\mathcal{T} \left\{A(\tau) B(\tau')\right\} := \begin{cases} A(\tau) B(\tau') & \text{if } \tau > \tau', \\ \pm B(\tau')A(\tau) & \text{if ...
0
votes
1answer
78 views

second quantization lost particle number information?

In first quantization, we can read the particle number from the Hamiltonian. $$ H=\sum_{i=1}^N \hat{T}(x_i) $$ Converting this to second quantization form, the particle number of the system is lost: ...
1
vote
1answer
132 views

Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
1
vote
2answers
119 views

Two particle operator

Why is the two-particle (fermionic, cause for bosonic operators it is immediately clear that both representations are the same) Hamiltonian given by $$ H = \sum_{a,b,c,d} \langle ab|V|cd \rangle ...
4
votes
2answers
210 views

Spin zero photons

As I understand it, the reason why there is no Spin 0 Photon is because the polarisation of an EM field lives in two dimension. Hence we only have two basis vectors, yielding two pairs of ladder ...
2
votes
1answer
83 views

Why are photons bosonic?

I am studying the quantization of the electromagnetic field. My text quantizes by changing amplitudes to ladder operators, by putting in an action and by imposing bosonic commutation relations upon ...
15
votes
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 ...
3
votes
1answer
72 views

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 ...
1
vote
1answer
46 views

Fourier transform of a set of L fermions operators

I have a set of L fermion creation and annihilation operators: $\lbrace{\hat{C}^+_1,...,\hat{C}^+_L\rbrace}$ and $\lbrace{\hat{C}^-_1,...,\hat{C}^-_L\rbrace}$. Every $\hat{C}^+_l,\hat{C}^-_l$ ...
1
vote
0answers
32 views

Creation and annihilation form of hamiltonian to derive a relation between the ac current applied to the crystal and the oscillations of the crystal

in the book "many-particle physics" by G.Mahan in piezoelectric subsection, it uses the second quantization formalism to derive the relation for hamiltonian of the electron-phonon interaction. so ...
0
votes
3answers
159 views

What is the general theory that describes the interactions between strings?

What is the general theory that describes the interactions between strings? I mean the basic object in the theory is (closed) string and they have interactions among them. The string theory, as I ...
1
vote
1answer
56 views

Multiply creation operator by a phase factor

A basic question, but I'm not completely confident what I'm doing is legit. I can multiply a creation operator by an arbitrary phase factor and it doesn't change any physics. True? I have a ...
0
votes
1answer
71 views

Fourier transform of random variables

My question is concerning Fourier transforms of random variables. So if the question itself is too heavy a task but you know of any good resources to learn this topic that would also be very much ...
0
votes
1answer
33 views

Why does trying to remove a non-existing electron from a state give zero?

Setup Creating an electron that is already in a basis set is zero (Pauli's principle): \begin{equation} a_i^+ | \chi_i \cdots \chi_k \cdots \chi_l \rangle = | \chi_i \chi_i \cdots \chi_k \cdots ...
2
votes
1answer
50 views

Slater-Determinant: When is this appopriate?

Imagine we have a N-particle Hamiltonian without any interaction between the electron particles $$ H = \sum_{i=1}^{N} \frac{p_i^2}{2m} + V(r_i)$$ then the solution to this equation $H\Psi = E \Psi$ ...