By Vasko F.T., Raichev O.E.
ISBN-10: 0387260285
ISBN-13: 9780387260280
ISBN-10: 0387280413
ISBN-13: 9780387280417
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Sample text
Because the matrix Gss odic in q, one should consider q inside the first Brillouin zone so that q is a quasi-wave vector. The set of solutions es (ql) is normalized according to s e∗s (ql)es (ql ) = δll , and e∗s (ql) = es (−ql). Employing the normal coordinates Qql (t), which include the exponential time-dependent factors, we write uns (t) ≡ us (Rn t) as uns (t) = (N Ms )−1/2 Qql (t)es (ql)eiq·Rn , ql (5) 37 Elements of Quantum Dynamics where Q∗ql (t) = Q−ql (t) because the displacements are real.
10) is an example of such operators. 10). ˆ + and Ψ ˆx Instead of a ˆ+ ˆγ¯ , one can use the field operators Ψ γ¯ and a x defined by the relations ∗ + φ(γ) ˆγ , x a ˆ+ Ψ x = γ ˆx = Ψ φ(γ) ˆγ . x a (25) γ ˆ + should be considered as the creation operator for the The operator Ψ x ˆ x is the corresponding annihilation particle with coordinate x, while Ψ operator. The anticommutation rules for these operators are obtained from Eq. (21) according to the condition of completeness for the sets of one-electron wave functions φ(γ) x : ˆ +, Ψ ˆ + ]+ = [Ψ ˆ x, Ψ ˆ x ]+ = 0, [Ψ x x φx(γ) ∗ φ(γ) x = δ(x − x ).
Nγ¯ − 1, . . This operator connects N - and (N − 1)-particle ket-vectors. The definitions of the creation and annihilation operators result in the anticommutation rules ˆγ ]+ = δγγ , [ˆ a+ γ ,a [ˆ a+ ˆ+ aγ , a ˆγ ]+ = 0, γ ,a γ ]+ = [ˆ (21) which can be checked by the action of the pairs of operators from Eq. (21) on an arbitrary ket-vector |{nγ } . The anticommutator in Eq. (21) is introduced as [A, B]+ = AB + B A, where A and B are arbitrary operators. The operator of the occupation number of the state γ is introduced as n ˆγ = a ˆ+ ˆγ .
Quantum Kinetic Theory and Applications - Electrons, Photons, Phonons by Vasko F.T., Raichev O.E.
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