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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Creation and annihilation operators in QFT explained

I'm currently self studying QFT from Matthew D. S textbook and David Tongs online notes. I have trouble understanding the creation/annihilation operators in the QFT formalism. I couldn't find any ...
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What is the meaning of the second quantised wave function, actually?

According to what I have read, the second quantisation originally came from the effort to quantise the many body wave function in the Schrodinger equation. We could write down the commutation ...
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Breaking translational invariance on a 1D periodic lattice

I am seeking some clarification on the process of breaking translational symmetry in a bosonic lattice by applying a uniform external magnetic field, which was stated as a fact in this paper: https://...
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71 views

Conservation of quasimomentum in second quantization

I'm trying to work out the expression for the matrix elements in Section VI of this paper (PRA 72, 053604). There is a point I need to expand the contact interaction term $\hat{V} = U \int dr\,\, \...
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Are the creation and annihilation operators time-dependent?

Something that always confused me when first hearing about second quantization were the dependencies of the creation and annihilation operators. On the one hand I have seen expressions such as $$ \...
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628 views

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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Why do we need to embed particles into fields?

In QFT we have the so-called embeding of particles into fields. This is discussed at full generality in Weinberg's book, chapter 5. In summary what one does is: From Wigner's classification, for each ...
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Hamiltonian diagonalisation using quantum Fourier transform [closed]

Here is a problem to solve: diagonalize the following hamiltonian using quantum fourier transform. The hamiltonian reads: $$ \sum_{i,j=1}^N e^{-\theta_{ij}} c_i^\dagger c_j + h.c. $$ Where $c_j$ are ...
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Exponential of ladder operators acting on vacuum state [closed]

How would I solve expressions of the following nature: $$<0|e^{Vt(a+a^\dagger)}|0>$$ and $$<0|e^{\omega aa^\dagger t}|0>~?$$ My intuition is that I have to expand the exponent as a ...
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156 views

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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248 views

Deriving anti-commutation relation between creation/annihilation operators for Dirac fermions

Starting from Dirac fields: $$\Psi(x) = \dfrac{1}{(2\pi)^{3/2}} \int \dfrac{d^3k}{\sqrt{2\omega_k}}\sum_r\left[ c_r(k)u_r(k)e^{-ikx}+d^\dagger_r(k)v_r(k)e^{-ikx} \right]_{k_0=\omega_k}$$ $$\Psi^\...
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Plasmons : doubts about the derivation of the Heisenberg equation of electrons' density

I'm studying plasmons from "Haken-Quantum Field Theory of Solids", and i need some help in the calculation of the equation of motion of eletrons' density \begin{equation} \hat{\rho}_{\overrightarrow{q}...
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How are different sites in a Fock state related?

I'm trying to numerically construct a Hamiltonian of the form $$H = \hat n_1+ \hat n_2 $$ for a two site system in some Fock space which I will truncate to allow a maximum of $N$ particles per site. ...
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Two ways of thinking about the Dirac equation

My impression is that there are two ways of thinking about the Dirac equation: Quantum Mechanically: Here we think of the spinor $\phi$ as a generalization of the Schrodinger wave function which ...
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90 views

Closed formula for $[\hat{H},[\hat{H},…[\hat{H},\hat{c}^\dagger_{\mu,\sigma}]]…]]$

Given the interaction part of a general many-body hamiltonian, $$\hat{H}=\sum_{\alpha, \beta,\gamma,\delta,\sigma,\sigma^\prime}O_{\alpha,\gamma,\sigma}^{\beta,\delta,\sigma^\prime}\hat{c}_{\alpha,\...
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Doubt regarding second quantization

In Schwartz it is stated that second quantization can be viewed as modes having energy given by the relation $E=\hbar\omega$ and then considering each mode as simple harmonic oscillator. So my doubt ...
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One-band 1D tight-binding model: how to find the two-particle eigenstates?

Consider the simple hopping model in second quantization, $\hat{H} = -J \sum_{i,j=1}^\infty \left(\hat{c}_i \hat{c}^\dagger_j + h.c.\right)$ where $J$ is real and $\hat{c}_i$ are annihilation ...
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61 views

Second quantisation of interaction potential (Fermions)

If we start with an interaction Hamiltonian for fermions in second quantised form: $$ H_\text{int} = \frac{1}{2} \int d^3r \int d^3r' V(|r-r'|) \hat{n}(r)\hat{n}(r') $$ where $\hat{n}(r)=c^\dagger(r)...
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2-nd quantized TQFT formalism?

Suppose that we have a certain TQFT in the Atiyah-Singer sense. It is given by a functor $Z$ which associates: To connected oriented $n-1$-manifolds $a, b, \dots$ (in what follows called compact ...
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226 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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65 views

Dimensions in the Second Quantization of an Operator

Consider the one-particle operator $\hat A_{1p}$. As given in e.g. (Altland and Simons, 2nd ed, 2010; pg47) the second quantized version of this is given by: $$\hat A=\sum_{\mu,\nu} \left< \mu \...
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462 views

Position field and momentum field in QFT

Im new in QFT (quantum field theory) We know tha in QM we have operators: Position $\hat{x} \psi(x) = x\psi(x) $ Momenutum ${\mathbf {\hat {p}}}=-i\hbar \nabla $ How are defined the operator field ...
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Bogoliubov Transformation with Complex Hamiltonian

Consider the following Hamiltonian: $$H=\sum_k \begin{pmatrix}a_k^\dagger & b_k \end{pmatrix} \begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \pm \omega_0\end{pmatrix} \begin{...
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113 views

Dispersion relation in tight binding model with even indices only

Given a tight binding model with Hamiltonian $H= \sum_{i(even)}t[c_{i+1}^\dagger c_i+h.c]$ containing even indices only, how can we find out the dispersion relation? Attempt: My guess is that the ...
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Switching between Dirac notation and creation/annihilation operators?

I am wondering: What is the explicit difference between the two Hamiltonians, $$H_1=\sum_{m=1}^Ntc_m^\dagger c_m+h.c.,$$ $$H_2=\sum_{m=1}^Nt |m\rangle\langle m | + h.c..$$ The Hamiltonians describe ...
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How can I show that Second Quantization Hamiltonian is Hermitian?

How can I show that the non relativistic second quantization hamiltonian \begin{equation} \hat{H}=\lmoustache d^3x \hat{\psi}^\dagger_\alpha(x) T(x)\hat{\psi}^\dagger(x)+\frac{1}{2}\lmoustache d^3x\...
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86 views

Vector spaces in second quantization

Studying about fermionic commutation relations, the convention I'm following is to consider a set of creation (destruction) operators $\hat{a}_{i}^{\dagger}\left(\hat{a}_{i}\right)$ with $i=1,...,n$ ...
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79 views

What do quantum spin hamiltonians describe?

I've learned all particles are either fermions or bosons, obeying their respective operator algebras, and then I've seen Hamiltonians describing models carrying one of these two types of particles. So ...
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63 views

Adding energies vs. adding momenta

Suppose we have a (time and space) translationally invariant system with the Fock space for a Hilbert space. Temporal and spatial translation invariance implies that energy and momentum are good ...
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206 views

What is the precise formal correspondance between an oscillator and a quantum field?

A common route of introduction to quantum field theory is to note a similarity between the mathematical structure of a quantum harmonic oscillator and of a quantum field "at a point". The quantised ...
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Rigorous explanation of second quantization of Hamiltonian

I am having trouble understanding the formalism of second quantization. I haven't found any reference with a rigorous explanation. For simplicity, lets say I want to write the second quantized form of ...
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999 views

The Schrödinger equation as an Euler-Lagrange equation

In the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrodinger equation in the form $$i\hbar\frac{\partial\psi}{\partial t}=\Big[-\frac{\hbar^2\nabla^2}{2m}+U(\textbf{r})\Big]...
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The necessity of ground state in QFT

Why we always want to have a ground state in every physical theory? For example, when we try to quantize Dirac Hamiltonian and encounter a Hamiltonian without a ground state, we take a step back and ...
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206 views

Rotating harmonic oscillator

The Hamiltonian operator $$H=\frac{{\bf p}^2}{2m} +\frac{m\omega^2}{2}{\bf r}^2-\Omega L_z$$ with $L_z=xp_y-yp_x$, can be written as $$H=\hbar\left(\omega+\Omega\right)\alpha^\dagger\alpha+\hbar\...
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Exercise on bosonic vacuum

Consider bosonic canonical transformation, generated by operator $S = e^{\lambda (a^{\dagger})^2}$. Show, that \begin{equation} b \equiv SaS^{-1} = a - 2\lambda a^{\dagger}. \end{equation} ...
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256 views

Hamiltonian operator in terms of field operators

I'm struggling to derive the second-quantised expression of the Hamiltonian operator of a many-particle system. Starting from \begin{equation*} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\ket}[1]{...
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83 views

Variation of Fermionic Field Operator

Suppose we have a Hamiltonian containing some interaction term $$V = \sum _{\sigma \sigma '\sigma ''\sigma '''}\iint d^3rd^3r'\hat{\psi }_{\sigma}^\dagger (\textbf{r})\hat{\psi }_{\sigma'}^\dagger (\...
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190 views

Basic understanding of the Fock space of a quantized real scalar field

The states in quantum mechanics belong to some Hilbert space while the states in quantum field theory belong to a Fock space. For simplicity, let me stick to the Fock space emerging after the ...
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152 views

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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Is there a natural operator that is canonically conjugate to the Hamiltonian?

As is well known, the Heisenberg uncertainty principle states that the position and momentum satisfy an uncertainty relation, which follows from the canonical commutation relation \begin{equation} [\...
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Original derivation of the Josephson effect

In my efforts to understand Josephson's original derivation of the effect that bears his name, I was led to Eqs. 3.68 on pg. 69 from Tinkham's book (Introduction to Superconductivity, McGraw-Hill, 2nd ...
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Reality of field quantization

In the quantization of the electromagnetic field (using the "second quantization" method), one says that the transverse vector potential must be $$A(r,t)=\sum_k (\alpha_k e^{i \vec{k} \vec{r}}+\...
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Feynman rules and second quantization in condensed matter physics

Given a hamiltonian in the form: $$\hat{H}=\hat{H}_0+\hat{\mathcal{O}}=\sum_{\alpha, \beta, \sigma} t_{\alpha,\beta} \hat{c}^\dagger_{\alpha \sigma}\hat{c}_{\beta \sigma}+\frac{1}{2}\sum_{\alpha,\...
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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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Representation of a general unitary operator acting on all electrons in terms of creation and annihilation operators

My goal is to find the representation of a general unitary operator acting on all electrons in terms of creation and annihilation operators. Suppose a set of single-particle basis functions $\{\psi_i:...
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basic mathematical concepts concerning the manipulation of field operators

I am self-studying QFT by following Mark Srednicki and trying to derive some results in the text. I am not sure whether my steps are correct in a precise mathematical sense, so I quote the problem and ...
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Some questions on ensemble of single-fermion systems

Example 3.9 (from Introduction to Many-Body Physics) (a) Enumerate the energy eigenstates of a single fermion Hamiltonian $H = Ec^{\dagger} c$ (b) Calculate the number of fermions at ...
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Hartree-Fock decoupling of Hubbard model

Hartree-Fock approximation requires wavefunctions be as separable as possible. I know the basic idea of Hartree-Fock but having some trouble in formalism of second quantization. I am trying to ...
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1answer
141 views

Question about second quantization?

I've spent awhile trying to understand where this equality I marked with '?' may come from. If $\hat{o}$ is just the $O_1$ operator in a special basis then I've no idea where from $n_\lambda$ could ...
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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 $...