412
B Simple Models
Fig. B.5 Square-well
system: x versus V (x)
p = ±
√
2m
h 2
κ
sn
2K(κ))
π
, κ
dn
2K(κ))
π
, κ
.
(B.33)
B.3 Infinite Square-Well Potential
The motion of a particle of mass m with kinetic energy p 2 /2m in an infinite squarewell potential has some special properties that make it especially useful for studying
many aspects of chaotic behavior, both classical and quantum mechanical. The
infinite square-well potential is shown in Fig. B.5. The momentum, p, and position,
x, as a function of time can be obtained by inspection. A plot of the momentum and
position as a function of time is given in Fig. B.6. Analytic expressions are given in
terms of the Fourier series
p(t) =
(2mE o ) sign
sin
2πt
τ
=
(2mE o )
4
π
∞
n = 1
odd
1
n
sin
2πnt
τ
(B.34)
and
x(t) = −a +
4a
τ
|t| for
−
τ
2
< t <
τ
2
,
(B.35)
or
x(t) = −
4a
π 2
∞
n = 1
odd
1
n 2 cos
2πnt
τ
,
(B.36)
where τ =
4ma
√
2mE o
is the period of the motion.