Page 83 - 71 the abc of dft_opt

21.1. SCHR ¨ ODINGER EQUATION 165 166 CHAPTER 21. TIME-DEPENDENCE
. For our particle in a box, it is simply
∞
2
-
∗
7 5.4 n(xt) = |Ψ(xt)| = c (t) c k (t) φ j (x) φ k (x) (21.14)
j
6 t=13 t=13 j,k=1
t=14
t=14
5 t=15 5.3 t=15 A new quantity, that will play a central role in the development of TDDFT, is the current
100 <x(t)> 4 t=16 <H(t)> 5.2 t=16 density. The operator is defined as
3
5.1
2
N
1 5 1 -
0 ˆ j(r) = (p i δ(r − r i ) + δ(r − r i )p i ) (21.15)
-1 4.9 2 i=1
0 0.5 1 1.5 2 0 0.5 1 1.5 2
and its expectation value given by evaluation on the time-dependent wavefunction. For our
t t
particle in a box,
j(xt) = 2φ (xt)dφ(xt)/dx (21.16)
∗
Figure 21.2: Expectation value of x and energy of a particle in a box of length 1 subjected to time-dependent fields of
frequencies ω = 13, 14, 15, and 16. Since, in any dx around a point x, the change in the number of electrons per unit time must
equal the difference between the flow of electrons into dx from the left and out from the
of using the unperturbed eigenfunctions as our basis, we used the instantaneous eigenstates, right,
i.e., those satisfying ˙ n(xt) = −dj(xt)/dx (21.17)
ˆ
H(t) Ψ n (t) = E n (t) Ψ(t) (21.9)
This is the equation of continuity. More generally, the time evolution of any (time-independent)
these chances would remain exponentially small, and so the system stays in the adiabatic
ground state. Essentially, the potential is moving so slowly that the wavefunction of the operator is ˆ
˙
ˆ
ˆ
A = i[H(t), A] (21.18)
particle simply rearranges itself to the new ground state at any given time.
In Fig X, I’ve calculated the case where ω = 1, t = 0.1, and a = 2000. Thus ω is much Applying this to the density yields
2
lower than the lowest transition, and ωt 0 1. This yields a final electric field of a(ωt) /2
ˆ
˙ n(rt) = %Ψ(t)|i[T, ˆn(r)]|Ψ(t)& = −∇j(x) (21.19)
or 10. The particle adiabatically follows the ground-state of the potential. This limit can be
found by simply turning on a static external potential, and asking what the ground-state is. where the density commutes with all but the kinetic operator in the Hamiltonian, and its
commutation (with a little algebra) yields the divergence of the current operator.
Exercise 88 Harmonic oscillator in a time-dependent electric field
Consider a harmonic oscillator of frequency ω 0 in an electric field. If it starts out at rest at
the origin, show that its classical position is given by 21.2 Perturbation theory
t E(t )
1 #
#
x cl (t) = − dt # sin(ω 0 (t − t )) (21.10) This concept is best seen in perturbation theory. Imagine integrating the equations of motion
0 mω 0 for the coefficients just a small amount in time, so that the excited states are only infinites-
Then show that imally populated. Then one may insert the initial condition into the right-hand-side of Eq.
φ(xt) = φ 0 (x − x cl (t)) exp(iα(t)) (21.11) X to find the probability of excitation. Next consider the effect of varying the external fre-
is a solution to the time-dependent Schr¨odinger equation. How important is α(t)? quency, and one finds a huge response as it passes through a transition frequency. We find
the transition rate, i.e., the probability of excitation per unit time, to be
In this framework, the time-dependent density is given by the usual formula of integrating
˙
the square of the wavefunction over all coordinates but one. The density operator may be W(k) = P(k) = ... (21.20)
written as
N which is known as Fermi’s golden rule.
ˆ n(r) = - δ(r − r i ) (21.12) The full range of behaviors of time-dependent systems is unnervingly large. In many cases,
i=1
and we are interested only in the response to a weak potential, i.e.,
ˆ
ˆ
ˆ
ˆ
n(rt) = %Ψ(x)|ˆn(r)|Ψ(x)& (21.13) H(t) = T + V + δV (t) (21.21)