# 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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### Klein-Gordon quantization and SHO analogy

I understand that the procedure to quantize Klein-Gordon's field is to manipulate in a such a way to bring up the simple harmonic oscillator behavior of the field. This is done by Fourier transforming ...
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### Canonical quantisation: How to find the scalar product?

I am trying to understand the canonical quantisation procedure. I understood that one takes the classical field equation and replaces the field by an operator Φ which solves the field equations. ...
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### Is the annihilator appearing in linear algebra books the same as the one of second quantization?

I have seen in some linear algebra textbooks such as Hoffman & Kunze, Friedberg & Insel & Spence, or Advanced Linear algebra by Roman the definition of annihilator. Here I take the ...
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### What is the physical meaning of the Fourier transformed Coulomb potential $1/q^2$?

$V(r)=\frac{1}{r}$ means for any two electrons at position $r_1$ and $r_2$, the electric potential is given by $\frac{1}{|r_1-r_2|}$ The Fourier transform of $\frac{1}{r}$ is $\frac{1}{q^2}$. How ...
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### How to interpret a wavepacket in quantum field theory: is it one particle or a superposition of many?

In 'classical' quantum mechanics, a wave packet is a (more or less) localized particle. The wave packet can be expanded in a superposition of plane waves, each with a defined momentum and energy. This ...
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### Physics of a Second quantized Hamiltonian?

It is frequently seen that the (Bosonic) Hamiltonian $H=e a^{\dagger}a+f( a^{\dagger}a^{\dagger}+a a)$ is discussed and diagonalized using Bogoliubov transformation. My question is that what is ...
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### Creation/Annihilation Operators and Positive/Negative Exponentials

One of the principal concepts in QFT is to consider the expasion of the field $$\phi(x)=\int{\frac{d^3 \vec{p}}{2(2\pi)^3\omega_\vec{p}}}(a(\vec{p})e^{-ipx}+b^{\dagger}(\vec{p})e^{+ipx}),$$ with ...
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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 \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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Given the interaction part of a general many-body hamiltonian, $$\hat{H}=\sum_{\alpha, \beta,\gamma,\delta,\sigma,\sigma^\prime}O_{\alpha,\gamma,\sigma}^{\beta,\delta,\sigma^\prime}\hat{c}_{\alpha,\... 2answers 61 views ### Doubt regarding second quantization In Schwartz it is stated that second quantization can be viewed as modes having energy given by the relation E=\hbar\omega and then considering each mode as simple harmonic oscillator. So my doubt ... 0answers 232 views ### One-band 1D tight-binding model: how to find the two-particle eigenstates? Consider the simple hopping model in second quantization, \hat{H} = -J \sum_{i,j=1}^\infty \left(\hat{c}_i \hat{c}^\dagger_j + h.c.\right) where J is real and \hat{c}_i are annihilation ... 1answer 60 views ### Second quantisation of interaction potential (Fermions) If we start with an interaction Hamiltonian for fermions in second quantised form:$$ H_\text{int} = \frac{1}{2} \int d^3r \int d^3r' V(|r-r'|) \hat{n}(r)\hat{n}(r') $$where \hat{n}(r)=c^\dagger(r)... 0answers 209 views ### 2-nd quantized TQFT formalism? Suppose that we have a certain TQFT in the Atiyah-Singer sense. It is given by a functor Z which associates: To connected oriented n-1-manifolds a, b, \dots (in what follows called compact ... 1answer 201 views ### How to diagonalize the BCS Hubbard Hamiltonian using the Bogoliubov transformation? How do I diagonalize the following BCS (Bardeen-Cooper-Schrieffer) Hubbard Hamiltonian: H= \sum\limits_{k \in [-\frac{π}{2}, +\frac{π}{2}[} \begin{bmatrix}c^\dagger_k & c^\... 1answer 62 views ### Dimensions in the Second Quantization of an Operator Consider the one-particle operator \hat A_{1p}. As given in e.g. (Altland and Simons, 2nd ed, 2010; pg47) the second quantized version of this is given by:$$\hat A=\sum_{\mu,\nu} \left< \mu \...
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 ...
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{...