imported>AnomieBOT: Substing templates: {{Format ISBN}}. See User:AnomieBOT/docs/TemplateSubster for info.

2025-10-04T15:45:09Z

Substing templates: {{Format ISBN}}. See User:AnomieBOT/docs/TemplateSubster for info.

← Previous revision		Revision as of 15:45, 4 October 2025
Line 53:		Line 53:
	\| location = San Diego		\| location = San Diego
	\| year = 1975		\| year = 1975
	\| ~~pages~~ = 328		\| page = 328
	\| isbn = ~~9780080925370~~		\| isbn = 978-0-08-092537-0
	}}</ref>		}}</ref>
	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.<ref name="Becchi2010">{{cite journal		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.<ref name="Becchi2010">{{cite journal
Line 63:		Line 63:
	\| volume = 5		\| volume = 5
	\| number = 6		\| number = 6
	\| ~~pages~~ = 7902		\| page = 7902
	\| doi = 10.4249/scholarpedia.7902		\| doi = 10.4249/scholarpedia.7902
	\| year = 2010		\| year = 2010
Line 80:		Line 80:
	Consider a complete set of single-particle wave functions <math>\psi_{\alpha}(\mathbf{r})</math> labeled by <math>\alpha</math> (which may be a combined index of a number of quantum numbers). The following wave function		Consider a complete set of single-particle wave functions <math>\psi_{\alpha}(\mathbf{r})</math> labeled by <math>\alpha</math> (which may be a combined index of a number of quantum numbers). The following wave function
	:<math>\Psi[\mathbf{r}_i]=\prod_{i=1}^{N}\psi_{\alpha_i}(\mathbf{r}_i)\equiv \psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots\otimes\psi_{\alpha_N}</math>		:<math>\Psi[\mathbf{r}_i]=\prod_{i=1}^{N}\psi_{\alpha_i}(\mathbf{r}_i)\equiv \psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots\otimes\psi_{\alpha_N}</math>
	represents an ''N''-particle state with the ''i''th particle occupying the single-particle state <math>\|{\alpha_i}\rangle</math>. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ''i''th single-particle wave function describes the state of the ''i''th particle. The wave function <math>\Psi</math> has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by operators <math>\mathcal{S}</math> for symmetrizer, and <math>\mathcal{A}</math> for [[antisymmetrizer]].		represents an ''N''-particle state with the ''i''-th particle occupying the single-particle state <math>\|{\alpha_i}\rangle</math>. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ''i''-th single-particle wave function describes the state of the ''i''-th particle. The wave function <math>\Psi</math> has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by operators <math>\mathcal{S}</math> for symmetrizer, and <math>\mathcal{A}</math> for [[antisymmetrizer]].

	For bosons, the many-body wave function must be symmetrized,		For bosons, the many-body wave function must be symmetrized,
Line 246:		Line 246:
	:<math>c_\alpha^\dagger\|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle=(-1)^{\sum_{\beta<\alpha}n_\beta} \sqrt{1-n_\alpha}\|\cdots,n_\beta,1+n_\alpha,n_\gamma,\cdots\rangle.</math><ref>Book "Nuclear Models" of Greiner and Maruhn p53 equation 3.47 : http://xn--webducation-dbb.com/wp-content/uploads/2019/02/Walter-Greiner-Joachim-A.-Maruhn-D.A.-Bromley-Nuclear-Models-Springer-Verlag-1996.pdf </ref>		:<math>c_\alpha^\dagger\|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle=(-1)^{\sum_{\beta<\alpha}n_\beta} \sqrt{1-n_\alpha}\|\cdots,n_\beta,1+n_\alpha,n_\gamma,\cdots\rangle.</math><ref>Book "Nuclear Models" of Greiner and Maruhn p53 equation 3.47 : http://xn--webducation-dbb.com/wp-content/uploads/2019/02/Walter-Greiner-Joachim-A.-Maruhn-D.A.-Bromley-Nuclear-Models-Springer-Verlag-1996.pdf </ref>
	:<math>c_\alpha\|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= (-1)^{\sum_{\beta<\alpha}n_\beta} \sqrt{n_\alpha}\|\cdots,n_\beta,1-n_\alpha,n_\gamma,\cdots\rangle.</math>		:<math>c_\alpha\|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= (-1)^{\sum_{\beta<\alpha}n_\beta} \sqrt{n_\alpha}\|\cdots,n_\beta,1-n_\alpha,n_\gamma,\cdots\rangle.</math>
	Recall that the occupation number <math>n_\alpha</math> can only take 0 or 1 for fermions. These two equations can be considered as the defining properties of fermion creation and annihilation operators in the second quantization formalism. Note that the fermion sign structure <math>(-1)^{\sum_{\beta<\alpha}n_\beta} </math>, also known as the [[Jordan–Wigner transformation\|Jordan-Wigner string]], requires there to exist a predefined ordering of the single-particle states (the [[spin structure]]){{clarify\|reason=There is no mention of an ordering of single-particle states in the link, or it is difficult to find. Could it be some spin quantum number instead of spin structure?\|date=April 2015}} and involves a counting of the fermion occupation numbers of all the preceding states; therefore the fermion creation and annihilation operators are considered non-local in some sense. This observation leads to the idea that fermions are emergent particles in the long-range entangled local [[qubit]] system.<ref>{{Cite journal \| doi = 10.1103/PhysRevB.67.245316\| title = Fermions, strings, and gauge fields in lattice spin models\| journal = Physical Review B\| volume = 67\| issue = 24\| year = 2003\| last1 = Levin \| first1 = M. \| last2 = Wen \| first2 = X. G. \| ~~page~~ = 245316\| arxiv = cond-mat/0302460\| bibcode = 2003PhRvB..67x5316L\| s2cid = 29180411}}</ref>		Recall that the occupation number <math>n_\alpha</math> can only take 0 or 1 for fermions. These two equations can be considered as the defining properties of fermion creation and annihilation operators in the second quantization formalism. Note that the fermion sign structure <math>(-1)^{\sum_{\beta<\alpha}n_\beta} </math>, also known as the [[Jordan–Wigner transformation\|Jordan-Wigner string]], requires there to exist a predefined ordering of the single-particle states (the [[spin structure]]){{clarify\|reason=There is no mention of an ordering of single-particle states in the link, or it is difficult to find. Could it be some spin quantum number instead of spin structure?\|date=April 2015}} and involves a counting of the fermion occupation numbers of all the preceding states; therefore the fermion creation and annihilation operators are considered non-local in some sense. This observation leads to the idea that fermions are emergent particles in the long-range entangled local [[qubit]] system.<ref>{{Cite journal \| doi = 10.1103/PhysRevB.67.245316\| title = Fermions, strings, and gauge fields in lattice spin models\| journal = Physical Review B\| volume = 67\| issue = 24\| year = 2003\| last1 = Levin \| first1 = M. \| last2 = Wen \| first2 = X. G. \| article-number = 245316\| arxiv = cond-mat/0302460\| bibcode = 2003PhRvB..67x5316L\| s2cid = 29180411}}</ref>

	==== Operator identities ====		==== Operator identities ====

imported>Flying.elf11: /* Second-quantized Fock states */ The normalization should be different such that one divides by the product

2025-01-28T12:28:03Z

Second-quantized Fock states: The normalization should be different such that one divides by the product

Show changes

Second quantization - Revision history

imported>AnomieBOT: Substing templates: {{Format ISBN}}. See User:AnomieBOT/docs/TemplateSubster for info.

imported>Flying.elf11: /* Second-quantized Fock states */ The normalization should be different such that one divides by the product