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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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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 ...
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40 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 ...
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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 ...
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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, ...
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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?
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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 ...
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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(...
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68 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 ...
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88 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 ...
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85 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(...
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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 ...
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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 ...
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44 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 ...
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44 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 (...
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46 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,...
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34 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 ...
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103 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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67 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 \...
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53 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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1answer
64 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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101 views

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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81 views

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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2answers
81 views

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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1answer
30 views

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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104 views

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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51 views

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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understanding electron-photon interaction hamiltonian

My apologies for a somewhat out-of-context question in advance. In equation 26 of the paper here , they have the following hamiltonian $\hat{H_i} = \sum_{p',p, \mathbf{k}} = (g_{p,p',\mathbf{k}})(u^...
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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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200 views

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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1answer
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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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63 views

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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1answer
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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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1answer
52 views

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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1answer
63 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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880 views

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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402 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 ...