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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Canonical quantization - can't form creation or annilhation operators?

This question is related to this PSE question I asked yesterday. From this question and what I have read, the general procedure for canonical quantization is as follows (correct me if I am wrong): ...
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Diagonalization of a Hamiltonian in Second Quantization

I have the following Hamiltonian of a fermionic two particle system. $H =2 \epsilon_m f^\dagger f + \epsilon_d d^\dagger d + t df +t f^\dagger d^\dagger + t f^\dagger d + td^\dagger f $ t $\in I R$ ...
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Why must the Bogoliubov transform preserve anticommutation relations?

$\mathbf{Background}$: In my research I am studying the Ising model, expressed in terms of Jordan-Wigner fermions: $$ H = \sum_{j=1}^n(c_j - c_j^\dagger)(c_{j+1} + c_{j+1}^\dagger) + \lambda c_jc_j^\...
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How exactly is “normal-ordering an operator” defined?

(In this question, I'm only talking about the second-quantization version of normal ordering, not the CFT version.) Most sources (e.g. Wikipedia) very quickly define normal-ordering as "reordering ...
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Origin of potential term in non-relativistic QFT?

The Hamiltonian of the free real scalar field is $\hat H = \int d^3k (\hbar \omega_k) \hat a^\dagger_k \hat a_k$ where $\omega_k = c \sqrt{k^2 + \mu^2}$. In the limit where $|k| \ll \mu$, $\omega_k ...
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Second Quantization in Condensed Matter and Quantum Field Theory

There appears to be an apparent dichotomy between the interpretation of second quantized operators in condensed matter and quantum field theory proper. For example, if we look at Peskin and Schroeder, ...
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Diagonalizing a fermionic Hamiltonian

My question somewhat builds off of this answer. For a fermionic Hamiltonian and diagonilzaition of the form $$H = \sum_{i,j} G_{ij}a_i^\dagger a_j = A^\dagger G A = A^\dagger U^\dagger D U A = F^\...
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Question about second quantization

I have a question about second quantization. For two fermions, one in state $\alpha_1$ and the other in state $\alpha_2$, using occupation number representation, one can express them as $|n_{\alpha_1}=...
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362 views

Usage of $ \hat{\psi}^{\dagger}$ operator

When I was reading the original paper about Runge-Gross theorem (Phys. Rev. Lett. 52, 997 (1984)), I saw $\hat{\psi} $ operator notation. I've never seen these notations from my QM text, and I'm ...
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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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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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Doi's second quantization: expected value of Hamiltonian

I am currently trying to understand section 3.3 from this article, about how to use second quantization techniques in statistical mechanics. Here I have the creation and annihilation operators defined ...
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Exact diagonalisation by Bogoliubov-Valatin Rotation in the 'real -particle' space

I have a question about the random ising model, where the model is no longer translationally invariant, and one has to resort to diagonalizing the full hamiltonian rather than solve for decoupled $2X2$...
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Showing a relation for green's function in single site impurity

The Hamiltonian is given as $$H = \sum_k \epsilon_k c_k^{\dagger} c_k + V \sum_k (c_k^{\dagger}d + d^{\dagger}c_k) + \epsilon_d d^{\dagger} d$$. Here, the $d$ operator denotes the annihilation and ...
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Relation between the Fourier transform and the Fock Space

First consider the classical Klein-Gordon field. The equation is $(\Box +m^2)\phi = 0$ which upon using the Fourier transform becomes (here I denote the Fourier transform with the $\hat{}$ as usual. $...
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Scalar product in Fock space and Coulomb interaction in second quantization

This is a doubt into which I've run trying to compute the form of the Coulombian interaction in second quantization in a basis of plane waves. Let $k$ denote the momentum and $r$ the position, and ...
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Intuition about this derivation on QFT

I've found on nLab this post on Wightman axioms which in particular contains a nice example about the quantization of the Klein-Gordon Field. This is a remarkably clean approach from the point of view ...
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How to write exchange interaction as Spin operators

I have been reading about quantum magnetism from Assa Auerbach's book, and Equation (2.9) reads $$J^F \sum_{s s'} c^{\dagger}_{is} c^{\dagger}_{i's'}c_{is'}c_{i's} = -2 J^F\left (\mathbf{S}_i \cdot \...
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Second quantization and Klein Gordon equation

This is what I understood from Klein Gordon equation : We start from $$E^2=p^2+m^2.$$ We quantize it replacing $E \rightarrow \partial_t$, $p \rightarrow -ih\nabla$, $m \rightarrow m$ Thus, we get ...
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How to fill the gaps in this QFT construction?

I've seem in several books and lecture notes the quantization of the free KG field, and perhaps because I'm a kind of person that feels umconfortable with "hand waving" constructions, I still feel the ...
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Methods for nonquadratic Hamiltonian in second quantizaion

Quadratic Hamiltonian ($A, B, C$ constants) $H_Q = A a^{\dagger} a + B a^{\dagger} a^{\dagger} + C a a $ may be diagonalized by Bogoliubov transformation. Consider other terms $H_{NQ} = D a^{\...
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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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Expansion coefficients in the solution of the Dirac equation for a free particle

So my question is why do we need to write the coefficients $b$ (that after the second quantization are going to be promoted as the antiparticle creation operators) as complex conjugate? I mean, why ...
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Normal order of 1

Say in my Hamiltonian I had a term $$Q[a_j,a^\dagger_j]_\pm$$ where $Q$ is a constant. Suppose I didn't realise that this quantity equals 1 and calculated a normal order. Of course you get 0 However,...
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Spinor quantization: contradiction between covariant anticommutator and canonical rules?

Starting from the free lagrangian $$\mathscr L = \bar\Psi(i\displaystyle{\not}\partial - m)\Psi$$ I compute the canonical momenta $$\Pi =\frac{\partial \mathscr L}{\partial\dot{\Psi}}=i\Psi^\dagger ...
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What's the right Fourier transformation definition of creation or annihilation operators in lattice model?

Very often one can view the solid as lattice model and then use the the language of second quantization,namely taking the occupation number representation, to express the Hamiltonian of the system. In ...
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Classical and quantum field in beam-splitter

Why is the annihilation operator $ \hat a $ the quantum analog of a classical field amplitude $ \mathcal{E} $? Is the quantized electromagnetic field not a sum of the annihilation and creator operator?...
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Second quantization using matlab [closed]

I am attempting to create creation and annihilation operator used in 2nd quantisation in quantum mechanics. I want to do this both numerically and symbolically. Here is the idea: $n= {c^\dagger}c$ ...
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Terms $\hat{c}^{\dagger}_{i\uparrow} \hat{c}^\phantom{\dagger}_{i+1\downarrow}+\text{h.c.}$ in tight-binding hamiltonians

The basic tight-binding hamiltonian consists of terms of the form $$\hat{c}^{\dagger}_{i\uparrow} \hat{c}^\phantom{\dagger}_{i+1\uparrow}+\text{h.c.}$$ (where $\text{h.c.}$ denotes the adjoint of the ...
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Klein Gordon Field Quantization: why this is the correct way to express the field?

I'm reading a book in QFT and the first thing tackled is the quantization of the Klein Gordon Field. The classical Klein Gordon field satisfies the partial differential equation $$(\partial^\mu\...
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Eigenvalues of fermionic field operators

Consider the fermionic field operators $\psi_a(x), \psi^{\dagger}_b(y)$ with the canonical anti-commutation relations $$\{\psi_a(x),\psi_b(y)\} = 0 $$ and $$\{\psi^{\dagger}_b(t,\vec{x}),\psi_a(t,\vec{...
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Majorana Operators

My question concerns a comment made on this post: Because one needs two Majorana modes $\gamma_1$ and $\gamma_2$ to each regular fermionic $c^\dagger c$ one, any two associated Majorana modes ...
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Second quantization field derivative commutator

For the symmetrized quantum field theory Lagrangian of the free dirac field $$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$ the terms are ...
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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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Tight-binding model of diatomic solid

I am working on a problem based on Problem 1.6 in Gerald D. Mahan - Many Particle Physics, 3rd edition. The problem: Consider a tight-binding solid which has alternating atoms of type A and B. The ...
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Many body physics - changing to $k$ space

I have an example in my notes starting with Linear Chain: $$H=-t\sum_{\langle i i' \rangle} c_i^\dagger c_{i'} = -2t \sum_k c_k^\dagger c_k \cos{k}$$ I don't know where the $2 \cos{k}$ comes ...
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Time reversal symmetry of fock states

How does the time reversal symmetry work in the second quantization frame of non-relativistic quantum mechanics? In particular what is the time-reversed of a given Fock-state? As an example let's ...
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Does E commute with B (in second quantization)?

In second quantization, there's a standard procedure where we first find solutions to Maxwell's Equations. After doing so we apply quantum mechanical properties to these solutions. So for some ...
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Diagonalizing Hamiltonian for a ring of particles

I have a ring with $N$ interacting particles (bosons) separated by length $a$ with the hamiltonian $$H = -t \sum_{\alpha=0}^N (b_{\alpha+1}^\dagger b_\alpha + h.c.) + u \sum_{\alpha=0}^N b_\alpha^\...
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If all particles are fields, why does first quantization work for some particles?

After a lot of Google and asking professors about the two quantization methods, I have learned that first quantization is what you use to quantize classical particles, while second quantization is ...
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When do annihilation operators act - second quantisation

I am confused as to when does an annihilation operator annihilate its creation operator counter-part. As an intro to second quantisation in my QFT notes we have: The operators $c_j$ annihilate ...
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Momentum creation operator

I am looking at the Bose-Hubbard model. Specifically the solution in the limit of weak interactions $U = 0$. I understand the proof to show the Hamiltonian is diagonal in momentum space, however I ...
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What are the spins of Goldstone bosons in Condensed matter systems

Acoustic phonons, magnons etc are quanta of the Goldstone modes. What is the spin of these particles? Has it been measured in experiments?
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Is there an exchange of electrons between two Cooper pairs in the BCS state?

The BCS state is $$\left|\psi\right>=\prod_k \left(u_k +v_k c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger\right)\left|0\right>$$ which pairs all the electrons into Cooper pairs. The ...
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An exactly solvable model of 2D Majorana zero modes

The Kitaev's Majorana Model is an exactly solvable model of p-wave superconductor with localized Majorana zero modes in 1D quantum wire. For the 2D case, the general theory of Majorana zero modes near ...
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Quantum Fields as Functionals

In single-particle quantum mechanics, particles are replaced by wave-functions -- which are functions from the space of possible particle positions to the complex numbers. It seems that the most '...
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Notation in second quantization

I got a bit confused about the transition of notation between the first and second quantization. When a state is written as: $\rho =a\vert H \rangle \langle H \vert + b\vert V \rangle \langle V \vert+ ...
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Which book do you recommend for extensive review of tight binding model in second quantization formalism?

I need a book that explains the tight binding model in second quantization formalism. for example by telling the meaning of the eigenfunctions of kernel of Hamiltonian in $k$ space. also explains $2$ ...
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What is the origin of this huge degeneracy in this almost free fermion model?

Consider the following second quantized Hamiltonian: $$H=\int_S \frac{1}{2m}(\partial_x-i\partial_y)\psi^\dagger_{\mathbf{r}}~(\partial_x+i\partial_y)\psi_{\mathbf{r}}~\mathrm{d}^2\mathbf{r},$$ where $...
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Meaning of Fock Space

In a book, it says, Fock space is defined as the direct sum of all $n$-body Hilbert Space: $$F=H^0\bigoplus H^1\bigoplus ... \bigoplus H^N$$ Does it mean that it is just "collecting"/"adding" all ...