By Tim Spiller, Hoi-Kwong Lo

This ebook relies on a lecture sequence held at Hewlett-Packard Labs, simple study Institute within the Mathematical Sciences (BRIMS), Bristol from November 1996 to April 1997, and in addition comprises different contributions.

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X | = exp |p = exp p ⇒0 p ⇒0 1 p · (−i|∇p ) C exp −i| i p · Q |p = 0 . 28) 9 Here, the concept of a vacuum state does not have a special meaning since |0 represent arbitrary reference position. It is introduced simply to bring analogy with zero-eigenvalue of non-Hermitian operators in later chapters, there the state |0 has a distinguished position. 5in swp0002 Quantum Mechanics: Perspectives 17 We also note from Eq. 12) that ∂ i exp p·Q ∂Q | i p·Q . 29) Therefore, P exp i p·Q | = exp i p · Q P + p exp | i exp − p · Q P exp | i p·Q | i p·Q , | = P + q.

5in 38 swp0002 Nonequilibrium Quantum Transport Physics in Nanosystems Upon substitution in the Hamiltonian, Eq. , Φll ,κκ ,αβ = Φκκ ,αβ (Xl ,κ − Xl,κ ) . We can now write the summation over l as summation over h = Xl ,κ − Xl,κ , thus lh,αβ Φκκ ,αβ (h) exp {−i ([q + q ] · Xl,κ + q · h)} = l,αβ = exp {−i ([q + q ] · Xl,κ )} NV δ (q + q ) αβ h h Φκκ ,αβ (h) exp {−iq · h} Φκκ ,αβ (h) exp {−iq · h} . 5in swp0002 Lattice Vibrations in Crystalline Solids: Phonons 39 Thus the Hamiltonian can be written as H= Pα (q,κ)Pα (−q,κ) mκ κ,α,q + Uα (q, κ) 1 αβ κκ qq 2NV Φκκ ,αβ (h) exp {−iq · h} × NV δ (q + q ) h ×Uβ (q , κ ) H= 1 Pα (q,κ)Pα (−q,κ) mκ κ,α,q + Uα (q, κ) αβ 2 × h κκ q Φκκ ,αβ (h) exp {iq · h} ×Uβ (−q, κ ) Using Eqs.

2 Deterministic Schrödinger Wave Equation The particle Hamiltonian operator, H, acting on the state |Ψ now reads, using the position eigenfunction expansion of |Ψ , i| ∂ |Ψ = H |Ψ = C ∂t dq − |2 2 ∇ + V (q) ψ (q, t) 2m q |q . Since H is Hermitian, the presence of i| renders the time evolution as a unitary evolution of the quantum states. 15) where V (q) is the external potential seen by the particle. 16) or Hψ (q) = Eψ (q) , where i Φ (t) = e− | Et . Similarly, in the momentum representation the Schrödinger equation is given by i| ∂ ψ (p, t) = Hψ (p, t) ∂t p2 + V (i|∇p ) ψ (p, t) .