Essentials of Computational Chemistry

m
−b
r eq
−b
3.3 MOLECULAR DYNAMICS 73
p
m
r eq
m
r eq
b √mk
−b √mk
Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described
in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system
evolves clockwise until it returns to the original point, with the period depending on the mass of the
ball and the force constant of the spring
a position b length units displaced from equilibrium. The frictionless spring, characterized
by force constant k, begins to contract, so that the position coordinate decreases.
The momentum coordinate, which was 0 at t0, also decreases (momentum is a vector
quantity, and we here define negative momentum as movement towards the wall). As the
spring passes through coordinate position 0 (the equilibrium length), the magnitude of the
momentum reaches a maximum, and then decreases as the spring begins resisting further
motion of the ball. Ultimately, the momentum drops to zero as the ball reaches position
−b, and then grows increasingly positive as the ball moves back towards the coordinate
origin. Again, after passing through the equilibrium length, the magnitude of the momentum
begins to decrease, until the ball returns to the same point in phase space from which it
began.
Let us consider the phase space trajectory traced out by this behavior beginning with the
position vector. Over any arbitrary time interval, the relationship between two positions is
t2
q(t2) = q(t1) +
t1
b
q
r eq
p(t)
dt (3.10)
m
b
m