Page 118 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4.2 Quantum treatment 109
P(y)

21 y 1
Figure 4.1 Classical probability density for an oscillating particle.

4.2 Quantum treatment
The classical Hamiltonian H(x, p) for the harmonic oscillator is

p 2 p 2 2 2
1
H(x, p) ˆ ‡ V(x) ˆ ‡ mù x (4:11)
2m 2m 2
^
The Hamiltonian operator H(x, ^ p) is obtained by replacing the momentum p
in equation (4.11) with the momentum operator ^ p ˆÿi" d=dx
2
^ p 2 " d 2
^ 1 2 2 1 2 2
H ˆ ‡ mù x ˆÿ ‡ mù x (4:12)
2m 2 2m dx 2 2
The Schrodinger equation is, then
È
2
2
" d ø(x) 1 2 2
ÿ ‡ mù x ø(x) ˆ Eø(x) (4:13)
2
2m dx 2
It is convenient to introduce the dimensionless variable î by the de®nition
1=2
mù
î ˆ x (4:14)
"
so that the Hamiltonian operator becomes

"ù d 2
^ 2
H ˆ î ÿ (4:15)
2 dî 2
Since the Hamiltonian operator is written in terms of the variable î rather than
x, we should express the eigenstates in terms of î as well. Accordingly, we
de®ne the functions ö(î) by the relation
1=4
"
ö(î) ˆ ø(x) (4:16)
mù
If the functions ø(x) are normalized with respect to integration over x