Page 168 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.1 Two-particle problem 159
2 2
jp R j jp r j
H ˆ ‡ ‡ V(r) (6:8)
2M 2ì
We see that the kinetic energy contribution to the Hamiltonian is the sum of
two parts, the kinetic energy due to the translational motion of the center of
mass of the system as a whole and the kinetic energy due to the relative motion
of the two particles. Since the potential energy V(r) is assumed to be a function
only of the relative position coordinate r, the motion of the center of mass of
the system is unaffected by the potential energy.
^
The quantum-mechanical Hamiltonian operator H is obtained by replacing
2
2
2
2
2
2
jp R j and jp r j in equation (6.8) by the operators ÿ" = and ÿ" = , respec-
R r
tively, where
@ 2 @ 2 @ 2
2
= ˆ ‡ ‡ (6:9a)
R @X 2 @Y 2 @ Z 2
@ 2 @ 2 @ 2
2
= ˆ ‡ ‡ (6:9b)
r 2 2 2
@x @ y @z
The resulting Schrodinger equation is, then,
È
" #
" 2 2 " 2 2
ÿ = ÿ = ‡ V(r) Ø(R, r) ˆ EØ(R, r) (6:10)
r
R
2M 2ì
This partial differential equation may be readily separated by writing the
wave function Ø(R, r) as the product of two functions, one a function only of
the center of mass variables X, Y, Z and the other a function only of the relative
coordinates x, y, z
Ø(R, r) ˆ ÷(X, Y, Z)ø(x, y, z) ˆ ÷(R)ø(r)
With this substitution, equation (6.10) separates into two independent partial
differential equations
" 2
2
ÿ = ÷(R) ˆ E R ÷(R) (6:11)
R
2M
" 2
2
ÿ = ø(r) ‡ V(r)ø(r) ˆ E r ø(r) (6:12)
2ì r
where
E ˆ E R ‡ E r
È
Equation (6.11) is the Schrodinger equation for the translational motion of a
È
free particle of mass M, while equation (6.12) is the Schrodinger equation for a
hypothetical particle of mass ì moving in a potential ®eld V(r). Since the
energy E R of the translational motion is a positive constant (E R > 0), the
solutions of equation (6.11) are not relevant to the structure of the two-particle
system and we do not consider this equation any further.