Classical Barrier Crossing and Phase Space

initial conditions

x

0

-2.5

p

0

0.5

time

0.001

This Demonstration illustrates the classical motion of a particle governed by the Hamiltonian

H=

2

p

2m

+

-α

2

x

e

. Here

x(t)

and

p(t)

are obtained by solving Hamilton's equations of motion, subject to the initial conditions

x

0

=x(0)

and

p

0

=p(0)

. The three panels animate synchronously: (1) the motion of the particle as it approaches the potential, (2) the phase space trajectory, and (3) the time series of

x(t)

and

p(t)

. In the upper two panels, the points with minimum momentum (turning points) are labeled with blue

x

's.