Page 82 - 71 the abc of dft_opt

164 CHAPTER 21. TIME-DEPENDENCE
This is the potential due to an external time-dependent electric field in the dipole approxi-
mation.
Exercise 87 Matrix elements of time-dependent potentials
Calculate the matrix elements V jk (t) for (a) a harmonic oscillator and (b) a particle in a box,
Chapter 21
when the perturbation is an electric field.
We choose
Time-dependence E(t) = A sin(ωt) (1 − exp(−ωt)) (21.8)
This form allows a slow turn-on of a periodic electric field of frequency ω and amplitude
A. Our first example is taken with generic values, A = 10, ω = 4, and the time run up to
21.1 Schr¨odinger equation t = 4. In Fig. 21.1, we show how the system responds. The density barely changes from its
ground-state value, but simply jiggles to left and right. The mean position begins to oscillate
We learn in kindergarten that the evolution of the wavefunction is governed by the time- immediately, not with the frequency of the driving field, but rather with a period of about
dependent Schr¨odinger equation: 2
1/3. This corresponds to the lowest transition, since & 1 = π /2 = 4.93, and & 2 = 4& 1 , so
˙
ˆ
iΨ(t) = H(t) Ψ(t), Ψ(0) given (21.1) that ω 21 = 14.8, and the period is 2π/ω ∼ 0.4. Thus the time-dependent evolution of the
ˆ
ˆ
ˆ
where H(t) = T + V (t), and a dot implies differentiation with respect to time. We will
generally be concerned with a time-dependence of the form
10 4
ˆ
ˆ
ˆ
H(t) = H o + V (t) (21.2) 8 3
2
6
4 1
2 0
where the unperturbed Hamiltonian, H o has eigenstates E(t) 0 100 <x(t)> -1
ˆ -2 -2
-3
-4
H o |j& = E j |j& (21.3)
-6 -4
-8 -5
ˆ
and V (t) is zero for t < 0. We may then expand the time-dependent wavefunction in terms -10 -6
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4
of these unperturbed eigenstates:
t t
-
Ψ(t) = C j (t) exp(−iE j t) |j& (21.4)
j Figure 21.1: Electric field and expectation value of x for a particle in a box of length 1 subjected to the time-dependent field
shown and given in the text.
where the phase factor for undisturbed evolution has been extracted. Insertion into Eq. (21.1)
yields expectation value of operators contains information about transition frequencies.
˙
i C j (t) = - C k exp(iω jk t) V jk (t), (21.5) Next, we illustrate the concept of resonance. We run our calculation with a = 1, but
k
sweep the driving frequency through ω = ω 12 , as shown in Fig. 21.2. We see that, the nearer
ˆ
a coupled set of first-order differential equations, with V jk = %j|V |k&, ω jk = E j − E k and
the driving frequency is to the transition frequency, the greater the effect of the driving field
with initial values on the particle. As ω → ω 21 , the perturbation grows indefinitely and pushes more and more
C j (0) = %Ψ(0)|j&. (21.6)
energy into the particle.
We usually begin in the ground-state, so that C j (0) = δ j0 . Another interesting regime of time-dependent problems is when the time variation is very
To have a feeling for what time-dependent quantum mechanics really means, we immedi- slow. This is the adiabatic limit. For our particle in a box in an external electric field, if the
ately show some examples for a simple case. We take our particle in a box and subject it to external frequency ω 0 ω 12 , then the chances of exciting out of the ground-state in any small
a particular driving potential, interval are exponentially small. Over long times, as the potential changes significantly from
V (xt) = x E(t) (21.7) its original, these small chances add up, and the wavefunction does evolve. But if, instead
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