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Fock state
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==Bosonic Fock state== [[Boson]]s, which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric<ref name="TIFR">{{Cite web | url=http://theory.tifr.res.in/~sgupta/courses/qm2013/hand13.pdf | title= Quantum Mechanics 1 Lecture Notes on Identical Particles, TIFR, Mumbai}}</ref> under operation by an [[exchange operator]]. For example, in a two particle system in the tensor product representation we have <math>\hat{P}\left|x_1, x_2\right\rangle = \left|x_2, x_1\right\rangle</math> . ===Boson creation and annihilation operators=== We should be able to express the same symmetric property in this new Fock space representation. For this we introduce non-Hermitian bosonic [[creation and annihilation operators]],<ref name="TIFR"/> denoted by <math>b^{\dagger}</math> and <math>b</math> respectively. The action of these operators on a Fock state are given by the following two equations: * Creation operator <math display="inline">b^{\dagger}_{{\mathbf{k}}_l} </math>: *: <math>b^{\dagger}_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}} +1 } |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}+1 ,...\rangle </math><ref name="TIFR" /> * Annihilation operator <math display="inline">b_{{\mathbf{k}}_l} </math>: *: <math>b_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1 ,...\rangle </math><ref name="TIFR" /> [[File:Action of operator on bosonic fock state.jpg|center|''The Operation of creation and annihilation operators on Bosonic Fock states.'']] ====Non-Hermiticity of creation and annihilation operators==== The bosonic Fock state creation and annihilation operators are not [[Self-adjoint operator|Hermitian operators]].<ref name="TIFR"/> {{math proof|title= Proof that creation and annihilation operators are not Hermitian. |proof= For a Fock state, <math>|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \rangle</math>, <math display="block">\begin{align} \left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots \left| b_{\mathbf{k}_l} \right| n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \right\rangle &= \sqrt{n_{\mathbf{k}_l}}\left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots | n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots \right\rangle \\[6pt] \left(\left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \left| b_{\mathbf{k}_l} \right| n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots \right\rangle\right)^* &= \left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1 \dots \left| b_{\mathbf{k}_l}^\dagger \right| n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \right\rangle \\ &= \sqrt{n_{\mathbf{k}_l} + 1}\left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1 \dots | n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} + 1 \dots \right\rangle \end{align}</math> Therefore, it is clear that adjoint of creation (annihilation) operator doesn't go into itself. Hence, they are not Hermitian operators. But adjoint of creation (annihilation) operator is annihilation (creation) operator.<ref name="Altland">{{Cite book | last1 = Altland | first1 = Alexander | last2 = Simons | first2 = Ben | title = Condensed Matter Field Theory | publisher = Cambridge University Press | date = 2006 | url = https://books.google.com/books?id=0KMkfAMe3JkC&pg=PA39 | isbn = 0521769752 }}</ref>{{rp|45}} }} ====Operator identities==== The commutation relations of creation and annihilation operators in a [[Boson|bosonic system]] are : <math>\left[b^{\,}_i, b^\dagger_j\right] \equiv b^{\,}_i b^\dagger_j - b^\dagger_jb^{\,}_i = \delta_{i j},</math><ref name="TIFR"/> : <math>\left[b^\dagger_i, b^\dagger_j\right] = \left[b^{\,}_i, b^{\,}_j\right] = 0,</math><ref name="TIFR"/> where <math>[\ \ , \ \ ]</math> is the [[commutator]] and <math>\delta_{i j}</math> is the [[Kronecker delta]]. ===N bosonic basis states === {| class="wikitable" |- ! Number of particles (N) ! Bosonic basis states<ref name="Bruus">{{Cite book | last = Bruus | first = Flensberg | title = Many-Body Quantum Theory in Condensed Matter Physics: An Introduction | publisher = OUP Oxford | date = 2003 | isbn = 0198566336 }}</ref>{{rp|11}} |- | 0 || <math>|0,0,0...\rangle</math> |- | 1 || <math>|1,0,0...\rangle</math>, <math>|0,1,0...\rangle</math>, <math>|0,0,1...\rangle</math>,... |- | 2 || <math>|2,0,0...\rangle</math>, <math>|1,1,0...\rangle</math>, <math>|0,2,0...\rangle</math>,... |- | <math>n</math>|| <math>|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle</math> |} ===Action on some specific Fock states=== {{unordered list | For a vacuum state—no particle is in any state— expressed as <math> |0_{{\mathbf{k}}_{1}}, 0_{{\mathbf{k}}_{2}}, 0_{{\mathbf{k}}_{3}}...0_{{\mathbf{k}}_{l}}, ...\rangle</math>, we have: : <math>b^{\dagger}_{{\mathbf{k}}_l}|0_{{\mathbf{k}}_{1}}, 0_{{\mathbf{k}}_{2}}, 0_{{\mathbf{k}}_{3}}...0_{{\mathbf{k}}_{l}}, ...\rangle = |0_{{\mathbf{k}}_{1}}, 0_{{\mathbf{k}}_{2}}, 0_{{\mathbf{k}}_{3}}...1_{{\mathbf{k}}_{l}}, ...\rangle </math> and, <math>b_{\mathbf{k}_l}|0_{\mathbf{k}_1}, 0_{\mathbf{k}_2}, 0_{\mathbf{k}_3}...0_{\mathbf{k}_l}, ...\rangle = 0</math>.<ref name="TIFR"/> That is, the ''l''-th creation operator creates a particle in the ''l''-th state '''k'''<sub>l</sub>, and the vacuum state is a fixed point of annihilation operators as there are no particles to annihilate. | We can generate any Fock state by operating on the vacuum state with an appropriate number of '''creation operators''': : <math>|n_{\mathbf{k}_1}, n_{\mathbf{k}_2} ...\rangle = \frac{\left(b^\dagger_{\mathbf{k}_1}\right)^{n_{\mathbf{k}_1}}}{\sqrt{n_{\mathbf{k}_1}!}} \frac{\left(b^\dagger_{\mathbf{k}_2}\right)^{n_{\mathbf{k}_2}}}{\sqrt{n_{\mathbf{k}_2}!}}...|0_{\mathbf{k}_{1}}, 0_{\mathbf{k}_{2}}, ...\rangle </math> | For a single mode Fock state, expressed as, <math>|n_\mathbf{k}\rangle</math>, : <math>b^\dagger_\mathbf{k}|n_\mathbf{k}\rangle = \sqrt{n_\mathbf{k} + 1} |n_\mathbf{k} + 1\rangle</math> and, : <math>b_\mathbf{k}|n_\mathbf{k}\rangle = \sqrt{n_\mathbf{k}} |n_\mathbf{k} - 1\rangle</math> }} ===Action of number operators=== The number operators <math display="inline">\widehat{N_{{\mathbf{k}}_l}}</math> for a bosonic system are given by <math>\widehat{N_{{\mathbf{k}}_l}}=b^{\dagger}_{{\mathbf{k}}_l}b_{{\mathbf{k}}_l}</math>, where <math>\widehat{N_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle=n_{{\mathbf{k}}_{l}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle </math><ref name="TIFR"/> Number operators are Hermitian operators. ===Symmetric behaviour of bosonic Fock states=== The commutation relations of the creation and annihilation operators ensure that the bosonic Fock states have the appropriate symmetric behaviour under particle exchange. Here, exchange of particles between two states (say, ''l'' and ''m'') is done by annihilating a particle in state ''l'' and creating one in state ''m''. If we start with a Fock state <math>|\psi\rangle = \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2} , .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle</math>, and want to shift a particle from state <math>k_l</math> to state <math>k_m</math>, then we operate the Fock state by <math>b_{\mathbf{k}_m}^\dagger b_{\mathbf{k}_l}</math> in the following way: Using the commutation relation we have, <math>b_{\mathbf{k}_m}^\dagger.b_{\mathbf{k}_l} = b_{\mathbf{k}_l}.b_{\mathbf{k}_m}^\dagger</math> : <math>\begin{align} b_{\mathbf{k}_m}^\dagger.b_{\mathbf{k}_l} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle &= b_{\mathbf{k}_l}.b_{\mathbf{k}_m}^\dagger \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle \\ &= \sqrt{n_{\mathbf{k}_m} + 1}\sqrt{n_{\mathbf{k}_l}} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_{m}} + 1 ... n_{\mathbf{k}_l} - 1 ...\right\rangle \end{align}</math> So, the Bosonic Fock state behaves to be symmetric under operation by Exchange operator. <gallery widths="160" heights="100"> Wignerfunction fock 0.png|Wigner function of <math>|0\rangle</math> Wignerfunction fock 1.png|Wigner function of <math>|1\rangle</math> Wignerfunction fock 2.png|Wigner function of <math>|2\rangle</math> Wignerfunction fock 3.png|Wigner function of <math>|3\rangle</math> Wignerfunction fock 4.png|Wigner function of <math>|4\rangle</math> </gallery>
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