What is Second quantization: Definition and 66 Discussions

Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were developed, most notably, by Vladimir Fock and Pascual Jordan later.In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

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  1. joneall

    I Who is Ballentine and why is he important in the world of quantum mechanics?

    People ask me what quantum fields are. (Really...) Quantum fields are the basic stuff of physical (and all) reality, right? But, after second quantization, they are operators! I cannot get my head around this.
  2. T

    A Anti-commutation relation for quantized fields

    Could somebody elaborate following statement from wikipedia in detail on interplay between the "choice" of anti- or commutation relation for quantized fields and the the associated statistics which the field satisfies before get quantized: Very roughly the story with second quantization is one...
  3. T

    A Quantization of real Klein-Gordon field (sign issues)

    I have a pretty naive question about quantization of real Klein-Gordon (so scalar) field ##\hat{\phi}(x,t)##. The most conventional form (see eg in this one ; but there are myriad scripts) is given by ##\hat{\phi}(x,t)= \int d^3p \dfrac{1}{(2\pi)^3} N_p (a_p \cdot e^{i(\omega_pt - p \cdot...
  4. mcas

    Show that an expected value of a vacuum state is equal to 1

    \langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| - \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow} \hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| \hat{c}_{k...
  5. hello_world30

    A Hamiltonian in second quantization

    Hello ! I require some guidance on this prove :I normally derive the Hamiltonian for a SHO in Hilbert space with a term of 1/2 hbar omega included. However, I am unsure of how one derives this from Hilbert space to Fock space. I have attached my attempt at it as an image below. Any input will be...
  6. sophiatev

    A Second Quantization in QFT

    In Quantum Field Theory and the Standard Model by Schwartz, he defines the Hamiltonian for the free electromagnetic field as (page 20, here's a link to the book). This follows (in my understanding) from the fact that the amplitude of the field at a given point in space oscillates as a simple...
  7. sakh1012

    A Dirac Field quantization and anti-commutator relation

    Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to $$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$ My question is how the above mentioned terms can be written as...
  8. LarryC

    Second Quantization - Quasiparticles

    (Simplified version of Baym, Chapter 19, Problem 2) Calculate, to first order in the inter-particle interaction V(r-r'), the energy of an N+1 particle system of spin-1/2 fermions with on particle of momentum p outside an N-particle Fermi sea (quasiparticle state). The answer should be expressed...
  9. pallab

    Book recommendations for second quantization and Jellium model

    please refer me a good book for the detail step by step study on the second quantization. and also where can I find the jellium model for the metal?
  10. CMJ96

    Help with second quantization practice problem

    Hi, to help further my understanding of the second quantization for one of my modules I would like to show that the following expressions $$ \hat{H} = \Sigma_{ij} \langle i| \hat{T} | j \rangle \hat{a_i }^{\dagger} \hat{a_j} $$ $$\hat{\psi}(r,t)= \Sigma_k \psi_k(r) \hat{a}_k(t)$$ Obey the...
  11. RicardoMP

    Bosonic annihilation and creation operators commutators

    Homework Statement After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin. 2. The attempt at a solution I tried to apply the...
  12. RicardoMP

    Bosonic operator eigenvalues in second quantization

    Homework Statement Following from \hat{b}^\dagger_j\hat{b}_j(\hat{b}_j \mid \Psi \rangle )=(|B_-^j|^2-1)\hat{b}_j \mid \Psi \rangle , I want to prove that if I keep applying ##\hat{b}_j##, ## n_j##times, I'll get: (|B_-^j|^2-n_j)\hat{b}_j\hat{b}_j\hat{b}_j ... \mid \Psi \rangle . Homework...
  13. A

    I Understanding second quantization

    Hi, I was reading a book about second quantization and there were a few things that I didn't quite understand entirely. This is what I understood so far: Given an operator ##\mathcal A## and two orthonormal bases ##|\alpha_i\rangle## and ##|\beta_i\rangle## for the Hilbert space, ##\mathcal...
  14. F

    A What is the difference between second quantization and QFT?

    Please teach me the difference between second quantization and QFT?
  15. GiovanniNunziante

    A Derivation of the Heisenberg equation for electron 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}} = \frac{1}{\sqrt{V}} \sum_{\overrightarrow{k}}...
  16. MichPod

    I Why the second quantization Hamiltonian works?

    I am puzzled by the fact that a "single-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a multi-particle case (non-interacting particles) or that (only) a "two-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a...
  17. K

    Gamma matrices in higher (even) dimensions

    Homework Statement I define the gamma matrices in this following representation: \begin{align*} \gamma^{0}=\begin{pmatrix} \,\,0 & \mathbb{1}_{2}\,\,\\ \,\,\mathbb{1}_{2} & 0\,\, \end{pmatrix},\qquad \gamma^{i}=\begin{pmatrix} \,\,0 &\sigma^{i}\,\,\\ \,\,-\sigma^{i}...
  18. DeathbyGreen

    I Creation operator and Wavefunction relationship

    Hello, I've noticed that some professors will jump between second quantized creation/annihilation operators and wavefunctions rather easily. For instance \Psi_k \propto c_k + ac_k^{\dagger} with "a" some constant (complex possibly). I'm fairly familiar with the second quantized notation, and...
  19. L

    Hamiltonian in terms of creation/annihilation operators

    Homework Statement Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators. Homework Equations Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
  20. V

    I Is the Fermion number operator squared equal to itself?

    What the title says. Acting on a fermionic state with the number operator to a power is like acting with the fermionic operator itself. Does this allow us to define ## \hat{n}^k=\hat{n} ##? Or is there any picky mathematical reason not to do so?
  21. L

    I Understanding the scalar field quantization

    I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field. The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...
  22. N

    I Eigenvalues of Fermionic field operator

    Hello, I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...
  23. N

    I Symmetrized Lagrangian (second quantization)

    Hello, I need some help to find the correct symmetrized Lagrangian for the field operators. After some work I guess that $$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$ should be the correct Lagrangian but I'm not sure with this. I'm...
  24. T

    I Dispersion relation in tight binding model

    Hamiltonian of tight binding model in second quantization is given as H = -t \sum_{<i,j>} a_i^{\dagger} a_j After changing basis it is H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}} where E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}} where \vec{b} is a nearest...
  25. N

    A Fetter & Walecka's derivation of second-quantised kinetic term....

    On page 9 of *Quantum theory of many-particle systems* by Alexander L. Fetter and John Dirk Walecka, during the derivation of the second-quantised kinetic term, there is an equality equation below: >\begin{align} \sum_{k=1}^{N} \sum_{W} & \langle E_k|T|W\rangle C(E_1, ..., E_{k-1}, W...
  26. V

    Solid State Books for second quantization and condensed matter

    Hi. I'll be doing a master's degree in nanophysics and working on electron transport in arrays of qubits. I don't know anything (or barely) about the second quantization and would like a book which covers it, and on condensed matter overall. So far I've been told about Bruus&Flensberg's...
  27. S

    I Second quantization and creation/annih. operators

    I'd like some help with something in this introduction to second quantization ... http://yclept.ucdavis.edu/course/242/Class.html They start by considering two operators ##a## and ##a^\dagger## such that ##[a,a^\dagger]=1## and they are adjoint to each other. Also introduced is a state...
  28. stevendaryl

    Second Quantization vs Many-Particle QM

    Apparently, there are two different routes to get to quantum field theory from single-particle quantum mechanics: (I'm going to use nonrelativistic quantum mechanics for this discussion. I think the same issues apply in relativistic quantum mechanics.) Route 1: Many-particle quantum mechanics...
  29. T

    Second Quantization Density Matrix

    Homework Statement Homework Equations and attempt at solution I think I got the ground state, which can be expressed as |\Psi \rangle = \prod_{k}^{N}\hat{a}_{k}^{\dagger} |0 \rangle . Then for the density matrix I used: \langle...
  30. G

    What is the first and second quantization?

    I believe the title says it all. What is the first and second quantization? What is the difference? Thank you.
  31. Einj

    Second quantization operators

    I have a doubt on the second quantization formalism. Suppose that we have two spin-1/2 fermions which can have just two possible quantum number, 1 and 2. Consider the wave function: $$ \psi(r_1,r_2)=\frac{1}{\sqrt{2}}\left(\psi_1(r_1)\psi_2(r_2)-\psi_1(r_2)\psi_2(r_1)\right). $$ The second...
  32. A

    Second Quantization: Single Particle Basis States

    This is just a simple question. I am reading about the second quantization but every text I read it starts with something like: suppose we have a set of single particle basis states {la1>,la2>,...,lan>}, which are used to label the wavefunction etc. I just need to make sure I understand what...
  33. A

    Second quantization of the Schrodinger fields

    Hi, I'm reading www.phys.ethz.ch/~babis/Teaching/QFTI/qft1.pdf and trying to understand the canonical quantization of the Schrodinger field. In particular, the Lagrangian: \begin{equation} \mathcal{L} = \frac{i}{2}\psi^* \partial_0 \psi - \frac{i}{2}\psi \partial_0 \psi^* +...
  34. T

    Proof of second quantization operators

    Please, can somebody show me why a Hamiltonian like \sum_nh(x_n) can be written as \sum_{i,j}t_{i,j}a^+_ia_j, with t_{i,j}=\int f^*_i(x)h(x)f_j(x)dx? Thank you.
  35. J

    Second Quantization for Fermions: Creation Operator

    So, I'm studying Second Quantization for fermions and came across this equation. I was just wondering why there is a summation needed? And why do we do it with (i≠p).? Please can someone explain this to me? Reply and help is much appreciated.
  36. J

    Second Quantization for Fermions

    Please let me know if I get this right. Second Quantization for Fermions used the definition of its annihilation and creation operators instead of wavefunctions. We use second quantization to express this many body problem in a hamiltonian. Am I right? Can someone please explain this to me in...
  37. L

    Charge conjugation in second quantization

    We know that under charge conjugation the current operator reverses the sign: \hat{C} \hat{\bar{\Psi}} \gamma^{\mu} \hat{\Psi} \hat{C} = - \hat{\bar{\Psi}} \gamma^\mu \hat{\Psi} Here \hat{C} is the unitary charge conjugation operator. I was wondering should we consider gamma matrix...
  38. F

    Density matrix elements, momentum basis, second quantization

    Hello everyone, I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise. Specifically, I'm having a hard time understanding the matrix elements of the density matrix...
  39. M

    Second quantization question: one particle or n particle?

    For the simple harmonic oscillator case, the energy is E=(n+1/2)hw, and N|n>=n|n>. It seems second quantization explain it as there are n bosons with each particle has energy homework plus vacuum 1/2hw. But we know before second quantization, there is only one particle with energy nhw plus...
  40. E

    Second Quantization: Motivation & Field Operators

    Hi,all What is the motivation of using Second Quantization ? What kind of situation does people want to use field operators ? Euphemia
  41. F

    Path Integral in first and second quantization

    Is it true that in first quantization the PI includes the possible trajectories a particle can take, but it does not include how particles can change into other kinds of particles (electrons to photons, etc). And QFT (second quantization) calculates how particles can branch off into other...
  42. T

    Second quantization derivation of spin-orbit coupling.

    Can anyone recommend books/reviews that derives the spin-orbit coupling in second quantization. I am working on a tightbinding model and I should be able to convert the spin-orbit hamiltonian from k-space to atomic representation using Warnier states, but I can't figure out some of the aspects...
  43. Q

    Understanding Second Quantization and Its Application in Quantum Mechanics

    I begin with \int (\bar{\psi}(x) (\mathcal{H} \psi(x)) d^3x This is just \int (\bar{\psi}(x) ({\frac{p^2}{2M} + \frac{1}{2}M \omega^2 (x)} \psi(x)) d^3x If one identified that \bar{\psi}(x) and \psi(x) are creation and annihilation operators, I assume that I can simply restate my...
  44. U

    In the second quantization spin operator, what are Pauli spin vector indices?

    If you look up the second quantization spin operator, you'll notice that there are two indices on the pauli vector for two possible spins. The operator sums over these two indices. Since the pauli vector is an unchanging quantity what do these indices physically correspond to?
  45. N

    Second Quantization: Explaining c^\dagger_ic_j = \delta_{i,j}c_jc^\dagger_i

    Hi Say I have the following two fermionic creation/annihilation operators c^\dagger_ic_j 1) Yesterday, my lecturer said that the following is valid c^\dagger_ic_j = \delta_{i,j}c_jc^\dagger_i Can you guys explain to me, where this formula comes from? I originally thought that it was one...
  46. D

    Commutators in second quantization

    Hi. I've been trying to calculate a couple of commutators, namely [\Psi(r),H] and [\Psi^{\dagger}(r),H] where H is a free particle hamiltonian in second quantization. I have attached my attempts and I would greatly appreciate if anyone could tell me if I am right or if there is a better way to...
  47. H

    Second Quantization: Exploring the Strange Results

    Didnt seem to be many threads about this subject although I don't find it trivial at all.. Lets start with a question: If we now have <N_i - 1|â_i|N_i> = N_i^0.5 but let â operate on our ket it should give: <N_i - 1||N_i - 1> = N_i^0.5 its adjoint however is the creation operator (right?)...
  48. N

    Second Quantization: Explaining Creation/Annihilation Operator Transformation

    Hi I have a question regarding second quantization. In the following link...
  49. D

    Second quantization particle current

    Hi. I'm reading an article which writes the following "... and the well-known expression for the charge current is" j=-\frac{ie}{m}\int dr\psi^\dagger (r)[\nabla-ieA(r)]\psi(r) Why does it have an integral sign? And when you define it this way, you integrate out the r-dependence...