Page 69 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

60 Schro Èdinger wave mechanics

dhri
hpiˆ m
dt
dhpi
ˆÿh=ViˆhFi
dt
where F is the vector force acting on the particle. The Heisenberg uncertainty
principle becomes
" " "
ÄxÄp x > , ÄyÄp y > , ÄzÄp z >
2 2 2

Multi-particle system
For a system of N distinguishable particles in three-dimensional space, the
classical Hamiltonian is
p 2 p 2 p 2
H(p 1 , p 2 , ... , p N , r 1 , r 2 , ... , r N ) ˆ 1 ‡ 2 ‡ ‡ N
2m 1 2m 2 2m N
‡ V(r 1 , r 2 , ... , r N )
where r k and p k are the position and momentum vectors of particle k. Thus,
the quantum-mechanical Hamiltonian operator is
ÿ" 2 1 1 1
^ 2 2 2
H ˆ = ‡ = ‡ ‡ = ‡ V(r 1 , r 2 , ... , r N ) (2:73)
2 m 1 1 m 2 2 m N N
2
where = is the laplacian with respect to the position of particle k.
k
The wave function for this system is a function of the N position vectors:
Ø(r 1 , r 2 , ... , r N , t). Thus, although the N particles are moving in three-
dimensional space, the wave function is 3N-dimensional. The physical inter-
pretation of the wave function is analogous to that for the three-dimensional
case. The quantity

Ø (r 1 , r 2 , ... , r N , t)Ø(r 1 , r 2 , ... , r N , t)dr 1 dr 2 ... dr N

ˆ Ø (x 1 , y 1 , z 1 , x 2 , ... , z N )Ø(x 1 , y 1 , z 1 , x 2 , ... , z N )dx 1 dy 1 dz 1 dx 2 ... dz N

is the probability at time t that, simultaneously, particle 1 is between x 1 , y 1 , z 1
and x 1 ‡ dx 1, y 1 ‡ dy 1 , z 1 ‡ dz 1 , particle 2 is between x 2 , y 2 , z 2 and x 2 ‡ dx 2,
y 2 ‡ dy 2 , z 2 ‡ dz 2 , ... , and particle N is between x N , y N , z N and x N ‡ dx N ,
y N ‡ dy N , z N ‡ dz N . The normalization condition is

Ø (r 1 , r 2 , ... , r N , t)Ø(r 1 , r 2 , ... , r N , t)dr 1 dr 2 ... dr N ˆ 1 (2:74)
This discussion applies only to systems with distinguishable particles; for
example, systems where each particle has a different mass. The treatment of
wave functions for systems with indistinguishable particles is more compli-