Page 85 - 71 the abc of dft_opt

21.3. OPTICAL RESPONSE 169 170 CHAPTER 21. TIME-DEPENDENCE
When discussing first-order perturbation theory, it is always useful to define response func- 21.4 Questions about time-dependent quantum mechanics
tions of the system at hand. A well-known one is the Green’s function, which, for a one-
electron system, may be written in operator form as 1. If a system is in a linear combination of two eigenstates, does its density change?
ˆ (−1)
ˆ
G(z) = (z − H) (21.37) 2. Calculate the oscillator strengths for a particle in a box, and show that they satisfy the
Thomas-Reich Kuhn sum rule.
where z is a complex number with positive imaginary part, and boundary conditions are 3. Calculate the imaginary part of the frequency-dependent susceptibility of the 1d H-atom.
chosen so that G(x, x ) vanishes at large distances, just like the bound-states. In real space,
#
G satisfies the differential equation: 4. Calculate the transmission and reflection amplitudes of the 1d H-atom, assuming a par-
ticle incident from the left.
< 1 =
(3)
2
− ∇ + v(r) − z G(rr ; z) = δ (r − br ) (21.38)
#
#
2 5. Calculate the retarded green’s function for the 1d H-atom.
Formally, one may write G in terms of sums over the eigenstates 6. Using the first two levels of the particle in a box, calculate the Rabi frequency, and
compare its time evolution with that of Fig X.
#
∗
- φ (r) φ i (r )
i
G(rr ; z) = . (21.39) 7. Show how a harmonic oscillator evolves in a time-dependent electric field.
#
i z − & i
8. Calculate the oscillator strengths for the harmonic oscillator, and show that they satisfy
Note that the Green’s function is ill-defined whenever its argument matches that of an eigen-
value. In such cases, we define the Thomas-Reich Kuhn sum rule.
±
#
#
G (rr , λ) = G(rr , λ ± iη) (21.40)
where + refers to the retarded Green’s function, while − refers to the advanced one. When
Fourier-transformed, the Green’s function yields the time-evolution of any initial state, for a
time-independent Hamiltonian:
1 1
+
# #
#
3 #
#
Ψ(rt) = d r dt G (rr , t − t ) Ψ(r t ) (21.41)
since
ˆ +
ˆ
#
#
G (t − t ) = exp(−iH(t − t )) (21.42)
Simple expression for G in terms of L and R phi.
Relate density matrix to equal-time G
Define χ and discuss meaning
Relate χ and G for non-interacting systems.
Relate pair density to equal-time χ.
21.3 Optical response
Define polarizability, static and dynamic.
Define oscillator strength.
Define optical spectrum.