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(r)≡〈r| ¦ ¥¥r′〉 - Condensed Matter Theory

(r)≡〈r| ¦ ¥¥r′〉 - Condensed Matter Theory

(r)≡〈r| ¦ ¥¥r′〉 - Condensed Matter

¡ ¢ ¢ ¡ ¢ ¡ ¥ Postgraduate Lectures 1. Time Independent Green’s functions a) Formalism Green’s functions are defined as the solutions of the inhomogeneous equation: where z = λ + is is a complex variable and φ n (r) and [z − (r)] G(r, r′; z) = δ(r − r′) (1.1) (r) is a time–independent, linear, Hermitian operator with eigenfunctions (r)φ n (r) = λ n φ n (r) (1.2) φ n (r) and G(r, r′; z) satisfy the same boundary conditions on the surface of the domain Ω. considered orthonormal and complete i.e. φ ∗ n(r)φ m (r) dr = δ nm Ω n φ n (r)φ ∗ n(r′) = δ(r − r′) The set {φ n } can be (1.3a) (1.3b) Note that the £ sum and ¤ integral operations are more or less interchangeable depending on the nature of the operator , continuum or tight binding, and its spectrum, continuous or discrete. It is useful to consider an abstract vector space and use Dirac’s notation φ n (r) ≡ 〈r|φ n 〉 (1.4a) δ(r − r′) (r) ≡ 〈r| ¦¥ ¥ r′〉 (1.4b) G(r, r′; z) ≡ 〈r| § (z) ¥ ¥ r′〉 (1.4c) 〈r|r′〉 = δ(r − r′) (1.4d) dr |r〉 〈r| = 1 (1.4e) (z − )§ (z) = 1 (1.4f ) ¥ ¥ ¦¥ φn 〉 = λ n ¥ φn 〉 (1.4g) n 〈φ n |φ m 〉 = δ nm (1.4h) ¥¥φ n 〉 〈φ n ¥ We can revert to the r–representation by taking the 〈r|, ¥ ¥ r′〉 matrix element, thus ¥ = 1 (1.4i) 〈r| (z )§ − (z) ¥ r′〉 = 〈r| 1 ¥ ¥ r′〉 = 〈r|r′〉 = δ(r − r′) (1.5a) ¥ If z ≠ {λ n } then we can write Multiplying (1.6) by (1.4i) and using F( ) ¥ zG(r, r′; z) − 〈r| § (z) ¥ ¥ r′〉 = zG(r, r′; z) − dr′′ 〈r| ¥ r′′〉 〈r′′ ¥ ¥ § (z) ¥ ¥ r′〉 ¦¥ = zG(r, r′; z) − (r)G(r, r′; z) = δ(r − r′) (1.5b) § (z) = 1 z − ¥ φ n 〉 = F(λ n ) ¥ ¥ φ n 〉 gives (1.6) § (z) = 1 z − = ¢ n = ¢ n G(r, r′; z) = ¢ n 1 1 z − n ¥¥φ n 〉 〈φ n ¥ ¥ ¥¥φ n 〉 〈φ n ¥ ¥¥φ ¥ n 〉 〈φ n ¥ (1.7) z − λ n φ n (r)φ ∗ n(r′) z − λ n

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