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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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,...
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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 ...
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178 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}\...
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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 \...
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81 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, ...,...
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Does string theory need operator formalism to quantize?

Can we really use path integral approach to quantize for (first-quantized) string theory? This question is motivated from the following fact: even though we can establish exact correspondence between ...
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Solution to the Klein-Gordon Equation

Most textbooks solve the Klein-Gordon equation with the ansatz $$\varphi(\mathbf{x},t) = \int \frac{\mathrm{d}^3\mathbf{k}}{f(k)}\left(a(\mathbf{k})\, \mathrm{e}^{i k x} + b(\mathbf{k})\,\mathrm{e}^{-...
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Apparent problem in using Wick's theorem to calculate matrix elements of two body operators

In the second quantized notation, a two body operator $\hat{O}$ can be written as $$\hat{O} = \sum\limits_{x_1,x_2,x_3,x_4} O_{x_1,x_2,x_3,x_4} a^\dagger_{x_1}a^\dagger_{x_2}a_{x_4}a_{x_3} ,$$ where ...
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120 views

Symmetry transformation of the second quantization operator

As we know, under the symmetry operation $U$, the operator $\hat A$ and the state $|\alpha \rangle $act as $$\hat A \longrightarrow U\hat A U^{\dagger}$$ $$|\alpha \rangle \longrightarrow U|\alpha\...
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Can the mass term be responsible for creation and destruction of particles?

In an interacting quantum field theory, for example, QED, the Dirac mass $m\bar{\psi}\psi$ is a piece of the free Dirac Lagrangian. On the other hand, the interaction term $j^\mu A_\mu=e\bar{\psi}\...
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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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Bose gas Hamiltonian in second quantization with indefinite parity potential

In the book Bose-Einstein Condensation by Pitaevski, Lev; Petrovitch, and Sandro Stringari (Oxford University Press), the Hamiltonian for weakly interacting Bose gas reads as, $$H=\sum\dfrac{p^2}{2m}\...
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What is a quantum number in a quantum field theory?

In non-relativistic quantum mechanics, quantum numbers are associated with eigenvalues of an operator. For example, $\ell$ is a quantum number associated with the eigenvalue $\ell(\ell+1)\hbar^2$ ...
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Does the order of operators in the the hamiltonian in second quantised form matter?

For a particles that not interact (free particles) we can write the Hamiltonian in second quantized form as $$\hat{H} = -\frac{\hbar^2}{2m} \int \psi^{\dagger}(\vec{x}) \nabla^2 \psi(\vec{x}) d^3x \,...
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Kanamori hamiltonian, rotational invariance and isospin

The Kanamori hamiltonian, if the coefficients satisfy a certain relationship, can be seen to be rotationally invariant. Its symmetry is $U(1)_C\times SU(2)_S\times SO(3)_O$ (I add the subscripts $C,S,...
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Second quantization canonical commutation relation : $\{c_\alpha,c_\beta^\dagger\}=\delta_{\alpha,\beta}$ a counter example?

Suppose two different states $\alpha$ and $\beta$ of some system of fermions such that each state only allows zero or one particle. The canonical commutation relation $\{c_\alpha,c_\beta^\dagger\}=\...
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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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A naive question on the eigenvalues of fermionic operators?

Let $A$ be a fermionic operator which is a product of odd number of fermion operators or a summation of them, say $A=C_{i_1}^{\dagger}\cdot \cdot\cdot C_{i_m}^{\dagger}C_{j_1}\cdot \cdot\cdot C_{j_n}...
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Phase space representations and coherent state representation of $\rho$, W, P and Q functions in quantum optics and related questions

What is the difference between phase space representations (with real x & p as variables) and coherent state representation (with complex $\alpha$ as a variable) of $\rho$, W, P and Q functions in ...
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U(1) Dirac string moved to the SU(2) or SO(3) gauge theory

Dirac string describes the string connecting the U(1) magnetic monopole to the U(1) anti-magnetic monopole in the U(1) gauge theory. Since U(1) is a subgroup of SU(2) and SO(3), we may embed the U(1) ...
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Ghost in quantum many body systems

Since we know the gauge theory can be emergent from local tensor product Hilbert space of quantum many body systems, such as solid state or condensed matter, etc. How do we understand the ghosts in ...
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Delta potential in terms of annihilation/creation operators

Let the Hamiltonian of a system on a discrete lattice be given by $$ \mathcal{H} = \gamma \sum_\vec{x} c^\dagger_\vec{x}c^\vphantom{\dagger}_{\vec{x}+\vec{y}} + \text{h.c.}, $$ where $\gamma$ is ...
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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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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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528 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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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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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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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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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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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: \begin{equation} H= \sum\limits_{k \in [-\frac{π}{2}, +\frac{π}{2}[} \begin{bmatrix}c^\dagger_k & c^\...
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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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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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412 views

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{...