Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
58 views

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{\...
0
votes
0answers
35 views

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 ...
2
votes
1answer
55 views

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 ...
1
vote
1answer
39 views

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....
0
votes
0answers
21 views

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 (...
0
votes
0answers
13 views

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^\...
4
votes
0answers
122 views

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 $\...
1
vote
1answer
45 views

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^{\...
2
votes
1answer
38 views

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 ...
0
votes
1answer
43 views

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 $...
0
votes
1answer
91 views

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 ...
0
votes
0answers
24 views

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. ...
2
votes
2answers
68 views

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 ...
0
votes
1answer
44 views

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 ...
2
votes
1answer
76 views

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 ...
-2
votes
1answer
43 views

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 ...
2
votes
2answers
67 views

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 ...
1
vote
0answers
34 views

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 ...
2
votes
1answer
114 views

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 ...
0
votes
1answer
101 views

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 ...
0
votes
0answers
23 views

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 "...
0
votes
0answers
26 views

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*} \...
0
votes
1answer
39 views

Expanding the interaction term in Hamiltonian for weakly interacting Bose gas

Let the second quantization Hamiltonian and the Bogoliubov prescription Could you explain me how to obtain the following expression I know that the second term in this expression is equivalent to ...
0
votes
1answer
46 views

Dirac delta normalization of electromagnetic fields

Usually, to quantize electromagnetic fields we use box normalization and therefore the normalization constant contains the dimensions of Volume V of the box. But if we perform the Dirac-delta ...
1
vote
1answer
108 views

Bogoliubov transformation for fermionic Hamiltonian

I have the Hamiltonian $H=\sum\limits_k [Ab^{\dagger}_{k}b_{k} + B(b^{\dagger}_kb^{\dagger}_{-k}+b_{k}b_{-k})]$, where $b^{\dagger}_k$ and $b_k$ are fermionic creation and annihilation operators. ...
1
vote
2answers
134 views

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 $$...
1
vote
1answer
84 views

Quick questions about second quantization?

What were the historical problems that the second quantization solved? My current understanding is that in re-normalisation one splits the result into a finite and a divergent part and only keeps ...
1
vote
1answer
68 views

Expressing the vacuum projection operator in terms of number operator

I've been reading this book, in which the author expresses the vacuum projection operator $\vert 0\rangle\langle 0\vert$ in terms of the number operator $\hat{N}=\hat{a}^{\dagger}\hat{a}$, where $\hat{...
1
vote
0answers
57 views

Product of Fermionic annihilation and creation operators

I have a bunch of fermions with annihilation $c_i$ and creation $c_i^\dagger$ operators. The index $i$ corresponds to different fermions. I'm interested in calculating the product $c_1^{\dagger} \Pi_{...
1
vote
0answers
23 views

Quantization of the massless neutrino field

If a massless neutrino or anti-neutrino is considered (in the whole post I consider neutrinos res. anti-neutrinos as mass-less), it is described by the Weyl-equation: $$\overline{\sigma}^{\mu}\...
0
votes
1answer
82 views

Momentum Space Representation of the Tight Binding Hamiltonian

I am trying to represent the tight-binding Hamiltonian \begin{equation} \hat{H}_{TB} = \sum_{\sigma} \sum_{\alpha,\beta} \sum_{\mathbf{R}_1,\mathbf{R}_2} t^{\alpha,\beta}_{\mathbf{R}_1,\mathbf{R}_2} \...
0
votes
0answers
52 views

Identity when Diagonalising Single-Particle Hamiltonian

Sorry the title is not precise; wasn't sure how to make it so (this is perhaps a straightforward question). The following is an identity I see quite often when reading lecture notes about ...
1
vote
1answer
65 views

Hamiltonian of a quantum heat bath

I have seen the Hamiltonian for a heat bath written as: $$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$ I was hoping to understand this equation better. This suggests that ...
1
vote
0answers
54 views

Reference request on Bogoliubov de Gennes (BdG) formalism

I have tried to gain an understanding about the BdG formalism by just following the calculations I found here and there of people bringing their superconducting Hamiltonians into matrix forms but I am ...
0
votes
0answers
41 views

Quantization of EM field

Usually the first step to quantization of EM field is the Fourier expansion of vector potential: $$ A = \sum_k A_k e^{jkr} .$$ For example, in book "The classical theory of fields" by Landau, ...
0
votes
1answer
19 views

How to write a wave-fuction of bose einstein condesate bosons in second quantization representation?

Can we write wave function of bose einstein condensation like this $|\psi \rangle=\frac{{c^{\dagger}}^N}{ \sqrt{N!}} |0\rangle$. Or it will be different?
0
votes
1answer
28 views

Different ways to write t-V model Hamiltonian

The $t-V$ model is usually written as: $$H = -t\sum_{i}^{L}c_i^\dagger c_{i+1} + c_{i+1}^\dagger c_i + V\sum_i n_i n_{i+1}$$ where $c_i^\dagger(c_i)$ are creation (annahilation) operators and $n_i$ is ...
-1
votes
1answer
75 views

Slater determinant in second quantization using the creation operators help [closed]

$$ \left\langle 0\left|\hat{\Psi}\left(x_{1}\right) c_{\alpha_{1}}^{\dagger}\right| 0\right\rangle=\left\langle 0\left|\varphi_{\alpha_{1}}\left(x_{1}\right)-c_{\alpha_{1}}^{\dagger} \hat{\Psi}\left(...
1
vote
1answer
86 views

The role of the harmonic oscillator eigenfunctions in quantum optics

In quantum optics we quantize the electromagnetic field and describe it using the harmonic oscillator model and the formalism of annihilation and creation operators. For the electric field operator we ...
1
vote
0answers
109 views

Proving the collapse of a many body system (Fetter and Walecka problem 1.2)

I was trying to solve the problem 1.2 from Quantum theory of many-body systems by A. Fetter and J. D. Walecka. I succeeded in the first part, obtaining the suggested formulation for the expectation ...
3
votes
1answer
126 views

How to derive the retarded Green's function matrix for a quadratic Hamiltonian?

Start with the quadratic Hamiltonian for fermion: $$\hat{H}=\sum_{ij}H_{ij}\hat{c}_i^\dagger \hat{c}_j$$ and the definition of retarded Green's functon in time domain: $$G_{i,j}^r(t_1,t_2)=-i\theta(...
0
votes
1answer
87 views

Can a general many-body Hamiltonian with quadratic and biquadratic terms be diagonalized?

Can an arbitrary many-body hamiltonian in second quantization form with quadratic and biquadratic terms $$H=\sum_{v_1,v_2} \alpha_{v_1 v_2}\ c_{v_1}^{\dagger}c_{v_2}+ \sum_{v_1,v_2,v_3,v_4}\beta_{v_1 ...
2
votes
2answers
106 views

Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization

It is well known that in elementary QM the so-called destruction/creation operators $$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$ are introduced when ...
0
votes
1answer
50 views

Proving equivalence of first and second quantisation (Pathria's way)

I'm trying to solve problem 11.1 form Pathria R. K. & Beale P. D. - Statistical mechanics book (the hyperlink will get you straight to the page of the problem). The point (b) is to show the ...
0
votes
1answer
89 views

Tricks to evaluate expectation values for operator strings in second quantisation

I am taking a course in many-body quantum mechanics. Often, I have to evaluate expectation values on strings of creation/annihilation operators. I was told that to evaluate these, I should use the (...
0
votes
1answer
47 views

Operator algebra for momentum and potential vector in second quantisation

While reading about the interaction of matter with the quantised electromagnetic field I found that, after applying the minimal coupling $\hat{p_i}\rightarrow \hat{p_i}-\frac{e_i}{c}\hat{A}(\vec{r}_i,...
1
vote
1answer
40 views

Energy per unit time of spontaneous emission within second quantisation problem

I'm studying second quantisation and I have the following problem concerning the spontaneous emission that corresponds to the decay of an atom from the level $ \lvert2\rangle$ to $ \lvert1\rangle$ and ...
1
vote
1answer
161 views

Density operator in second quantization form [closed]

In first quantization the particle density operator is $$n(x)=\sum_{\alpha}\delta^{3}(\vec{x}-\vec{x}_{\alpha})$$ In second quantization I have: $$ n(\vec{x})=\sum_{\alpha,i,j}\langle i|_{\alpha}\...
1
vote
1answer
96 views

Spin and many particle systems

I learned about many particle systems and second quantization recently. The Fock space of distinct particles with single particle Hilbert space $\mathcal H$ was defined to be $\bigoplus_{N=0}^\infty \...
1
vote
1answer
78 views

Recipe to determine symmetries of quadratic fermionic Hamiltonian in second quantisation

Consider an arbitrary 1D chain (of length $N$) of fermions with an arbitrary quadratic Hamiltonian of the form $$\mathcal{H}=\hat{\Psi}^\dagger H \hat{\Psi}$$ with $$\hat{\Psi}=\left(a_1, a_2, ...,...