Lecture 3: Introduction into second quantization.

Motivation of a large Hilbert space H_{0} ⨁ H_{1} ⨁ ... {α_{1}...α_{N} ⟩ = a_{α_{1}}^{†}...a_{α_{N}}^{†} |0⟩ (F)

\{α_{1}...α_{N} ⟩ = ∏_{λ} \frac{1}{\sqrt{n_{λ}}} a_{α_{1}}^{†}...a_{α_{N}}^{†} |0⟩ (B)

a_{α}^{†}{α_{1}...α_{N} ⟩ = {αα_{1}...α_{N} ⟩ (F) if {α_{1}...α_{N} ⟩_{U} ≡ a_{α_{1}}^{†}...a_{α_{N}}^{†} |0⟩ (B) a_{α}^{†}\{α_{1}...α_{N} ⟩_{U} = \sqrt{n_{λ} + 1} \{αα_{1}...α_{N} ⟩_{U} (B)

\{α_{1}...α_{N} ⟩ = ∑_{p} \frac{signP}{\sqrt{N!}} P |α_{1}⟩ ... |α_{N}⟩ (F)

\{α_{1}...α_{N} ⟩ = ∑_{p} \frac{1}{\sqrt{N!}}P |α_{1}⟩ ... |α_{N}⟩ (B)

changes of basis Q_{β}^{†} = ∑_{α} a_{α}^{†} ⟨α|β⟩

Lecture 4: matrix elements of one and two body.

T = ∑_{αβ} ⟨α|T |β⟩ a_{α}^{†}a_{β}; T = ∑_{i} T (i)

V = \frac{1}{2} ∑_{αβγδ} ⟨αβ|T |γδ⟩ a_{α}^{†}a_{β}^{†}a_{δ}a_{γ}; V = ∑_{i<j} T (ij)