Physical Chemistry Third Edition

1272 E Classical Mechanics

As with the angular momentum, the vector sum of the momenta of a set of interacting

particles is conserved if no external forces act on the particles.

TheHamiltonian function, also calledHamilton’s principal functionor theclassical

Hamiltonian, is defined by

H 

∑n

i 1

pi

.

qi−L (E-22)

The Hamiltonian function must be expressed as a function of coordinates and conjugate

momenta. It is equal to the total energy of the system (kinetic plus potential).^7

HK +V (E-23)

The Hamiltonian equations of motion are

.

qi

∂H

∂pi

,

.

pi−

∂H

∂qi

(i1, 2,...,n) (E-24)

There is one pair of equations for each value ofi, as indicated.

E.5 The Two-Body Problem

Consider a two-particle system with a potential energy that depends only on the distance

between the particles. This case applies to the hydrogen atom and to the nuclei of a

rotating diatomic molecule in the Born–Oppenheimer approximation. The equation of

motion of such a system can be separated into two separate equations. We first treat the

case in which there is motion only in thexdirection and apply the Lagrangian method.

The Lagrangian of the system is

LK−V 

1

2

m 1

.

x 1

2

+

1

2

m 2

.

x 2

2

−V(x 2 −x 1 ) (E-25)

We now change to a different set of coordinates:

xc

m 1 x 1 +m 2 x 2

m 1 +m 2

(E-26)

xx 2 −x 1 (E-27)

The coordinatexcis thecenter of mass coordinate, and the coordinatexis therelative

coordinate. We now show that these two coordinates obey separate equations of motion.

We solve Eqs. (E-26) and (E-27) forx 1 andx 2 :

x 1 xc−

m 2 x

M

(E-28)

x 2 xc+

m 1 x

M

(E-29)

whereMm 1 +m 2.

(^7) J. C. Slater and N. H. Frank,op. cit., p. 74ff (note 1).