# Classical Barrier Crossing and Phase Space

Classical Barrier Crossing and Phase Space

This Demonstration illustrates the classical motion of a particle governed by the Hamiltonian . Here and are obtained by solving Hamilton's equations of motion, subject to the initial conditions =x(0) and =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 and . In the upper two panels, the points with minimum momentum (turning points) are labeled with blue 's.

H=+

2

p

2m

-α

2

x

e

x(t)

p(t)

x

0

p

0

x(t)

p(t)

x