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. D

    Express arbitrary state in second quantization

    How do I express an arbitrary 2-particle state in second quantization? I could write this |\psi\rangle=\sum_{mn}c_{mn} a_m^\dagger a_n^\dagger |0\rangle where c_{mn} is a constant, a_n^\dagger is the creation operator and |0\rangle is the vacuum state. The only problem is that I want to...
  2. N

    Second quantization operators

    Hi guys The fermionic creating and annihiliations operators: Do they satisfy c_{i,\sigma }^\dag c_{i,\sigma }^{} = - c_{i,\sigma }^{} c_{i,\sigma }^\dag for some quantum number i and spin σ, i.e. do they commute?
  3. N

    Interpreting operators in second quantization

    Hi guys When working with operators in second quantization, I always imagine c^\dagger_ic_j as denoting the "good old" matrix element \left\langle {i} \mathrel{\left | {\vphantom {i j}} \right. \kern-\nulldelimiterspace} {j} \right\rangle . But how should I interpret an...
  4. N

    Free particles in second quantization

    Hi guys Today my lecturer talked about second quantization, and at the end he talked about free fermions in second quantization. He said that free electrons in second quantization satisfy that their Hamiltonian is only written as a linear combination in terms of c^\dagger c (the creation and...
  5. N

    Second Quantization: Creation & Annihilation Operators

    Hi all I am reading about second quantization. The kinetic energy operator T we write as \hat T = \sum\limits_{i,j} {\left\langle i \right|T\left| j \right\rangle } \,c_i^\dag c_j^{}. Now, the creation and annihilation operators really seem to be analogous (in some sense) to the...
  6. A

    Operator in second quantization

    Hello, I'm struggling with the second quantization formalism. I'd like to derive the hamiltonian of a system with non-interacting particles \hat{H}=\int dx\,a(x)^\dagger \left[\frac{\hat{P}}{2m}+V(x)\right]a(x), where a(x) = \hat{\Psi}(x). I know the second quantized representation of a...
  7. D

    Second Quantization: Wave Function & Creation/Annihilation Operators

    Hi. In second quantization (not QFT or anything advanced like that) we have the particle density \hat n(x)=\Psi^{\dagger}(x)\Psi(x) using the usual field creation/annihilation operators. For a single particle we obtain for the expectation value in the state |\psi\rangle: \langle \psi |...
  8. P

    Representation of second quantization

    In coordinate representation in QM probality density is: \rho(\vec{r})=\psi^*(\vec{r})\psi(\vec{r}) in RSQ representation operator of density of particles is \hat{n}(\vec{r})=\hat{\psi}^{\dagger}(\vec{r})\hat{\psi}(\vec{r}) Is this some relation between this operator and density...
  9. G

    Second quantization of field operators

    Homework Statement (from "Advanced Quantum Mechanics", by Franz Schwabl) Show, by verifying the relation \[n(\bold{x})|\phi\rangle = \delta(\bold{x}-\bold{x'})|\phi\rangle\], that the state \[|\phi\rangle = \psi^\dagger(\bold{x'})|0\rangle\] (\[|0\rangle =\]vacuum state) describes a...
  10. L

    Second quantization and partial traces

    Hi! Is there a common way to write a fermionic Fock space (finite dimensional) as a tensor product such that it is possible to do a partial trace over one particle type? Sorry, if this is an obvious question, but I just can't see it. Thanks!
  11. L

    Second quantization in ashtekar variable

    I am searching material in second quantization in ashtekar-like formulation gravity. Somebody knows something like that? thanks
  12. A

    Books that cover Second Quantization?

    So I actually decided to make an effort to study for my quantum final ahead of time, and I'm trying to find books that cover second quantization. If possible I'd like to find a book that gives a decent explanation (with examples, maybe?) of the Bogoliubov transformation. Does anyone have any...
  13. M

    Second Quantization and Field Operators

    When defining a field operator, textbooks usually say that one can define an operator which destroys (or creates) a particle at position r. What does this really mean? Are they actually referring to destroying (or creating) a state who has specific quantum numbers associated with the geometry...
  14. E

    Second Quantization: Momentum, Kinetic & Potential Energies + 2 Particles

    Write momentum, kinetic and potential energy, and two particle interaction in second quantization. That is the question that I need to answer for my exam, but I don't have any idea what second quantization is, except that you can solve harmonic oscilator by using ladder operators. I can't find...
  15. DaTario

    Why the name Second Quantization ?

    Why the name "Second Quantization"? Hi all, The title said it all. My question is: How is one to interpret the name second quantization ? Which specifically is quantized twice ? Best Wishes DaTario
  16. Antonio Lao

    What is the Connection Between the Graviton and the Higgs Boson?

    The quantum theory of field is sometimes known as second quantization. There are two distinct types of field in physics: 1. Scalar field. 2. Vector field. For each field, a force is supposed to be associated with it. But for the scalar field, this force is zero. This zero-force...
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