10.3 Quantum Nonlinear Resonances
349
10.3.3 Quantum Resonance Overlap
Resonance overlap and destruction of conserved quantities occur in quantum
systems in a manner analogous to classical systems. For the double-resonance
system in Eq. (10.22), the resonance condition and resonance widths are the same
as for the classical case,
¯
n m 0 =
ω 0
2m 0
, ¯
n n 0 =
ω 0
2n 0
, ,n U = 2
2U 0 , ,n V = 2
2V 0 .
(10.29)
To show resonance overlap in the double-resonance system in Eq. (10.23), set ω 0 =
300 and V 0 = U 0 /9. There is a resonance zone centered at ¯
n m 0 = 150 and a
resonance zone centered at ¯
n n 0 = 50. In order to see the effect of these resonances,
we find how the probability, |ψ n (τ )| 2 , spreads after a long time (many periods,
T =
2π
ω ) starting with an initial condition |ψ(0) = |m. We use 250 angular
momentum basis states.
In Fig. 10.3, we plot the probability distribution for the case U 0 = 657 and V 0 =
73. Figure 10.3a, we show the probability distribution at time t = 50T for the
initial. condition m = 55. In Fig. 10.3.b, we show the probability distribution at
Fig. 10.3 |ψ n (t)| 2 versus n for m 0 = 1, n 0 = 3, and ω = 300. (a) U 0 = 637, V 0 = 73, m = 55,
and t = 50T . (b) U 0 = 637, V 0 = 73, m = 130, and t = 50T ; (c) U 0 = 1478, and V 0 = 164,
m = 130 and t = 18.75T ; (d) U 0 = 1478, and V 0 = 164, m = 700 (far from the double-resonance
region) and t = 26.5T (Reichl and Lin 1986)
349
10.3.3 Quantum Resonance Overlap
Resonance overlap and destruction of conserved quantities occur in quantum
systems in a manner analogous to classical systems. For the double-resonance
system in Eq. (10.22), the resonance condition and resonance widths are the same
as for the classical case,
¯
n m 0 =
ω 0
2m 0
, ¯
n n 0 =
ω 0
2n 0
, ,n U = 2
2U 0 , ,n V = 2
2V 0 .
(10.29)
To show resonance overlap in the double-resonance system in Eq. (10.23), set ω 0 =
300 and V 0 = U 0 /9. There is a resonance zone centered at ¯
n m 0 = 150 and a
resonance zone centered at ¯
n n 0 = 50. In order to see the effect of these resonances,
we find how the probability, |ψ n (τ )| 2 , spreads after a long time (many periods,
T =
2π
ω ) starting with an initial condition |ψ(0) = |m. We use 250 angular
momentum basis states.
In Fig. 10.3, we plot the probability distribution for the case U 0 = 657 and V 0 =
73. Figure 10.3a, we show the probability distribution at time t = 50T for the
initial. condition m = 55. In Fig. 10.3.b, we show the probability distribution at
Fig. 10.3 |ψ n (t)| 2 versus n for m 0 = 1, n 0 = 3, and ω = 300. (a) U 0 = 637, V 0 = 73, m = 55,
and t = 50T . (b) U 0 = 637, V 0 = 73, m = 130, and t = 50T ; (c) U 0 = 1478, and V 0 = 164,
m = 130 and t = 18.75T ; (d) U 0 = 1478, and V 0 = 164, m = 700 (far from the double-resonance
region) and t = 26.5T (Reichl and Lin 1986)