Page 135 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

126 Harmonic oscillator

where k x , k y , k z are the respective force constants and ù x , ù y , ù z are the
respective classical angular frequencies of vibration.
È
The Schrodinger equation for this three-dimensional harmonic oscillator is
!
2
2
2
" 2 @ ø @ ø @ ø
2 2
2
1
2 2
2
ÿ ‡ ‡ ‡ m(ù x ‡ ù y ‡ ù z )ø ˆ Eø
y
x
z
2m @x 2 @ y 2 @z 2 2
where ø(x, y, z) is the wave function. To solve this partial differential equation
of three variables, we separate variables by making the substitution
ø(x, y, z) ˆ X(x)Y(y)Z(z) (4:52)
where X(x) is a function only of the variable x, Y(y) only of y, and Z(z) only
È
of z. After division by ÿø(x, y, z), the Schrodinger equation takes the form
!

2
2
" 2 d X " 2 d Y
2 2
2
1
1
ÿ mù x ‡ ÿ mù y 2
2mX dx 2 2 x 2mY dy 2 2 y

2
" 2 d Z
2 2
1
‡ ÿ mù z ˆ E
z
2mZ dz 2 2
The ®rst term on the left-hand side is a function only of the variable x and
remains constant when y and z change but x does not. Similarly, the second
term is a function only of y and does not change in value when x and z change
but y does not. The third term depends only on z and keeps a constant value
when only x and y change. However, the sum of these three terms is always
equal to the constant energy E for all choices of x, y, z. Thus, each of the three
independent terms must be equal to a constant
2
" 2 d X 1 2 2
ÿ mù x ˆ E x
x
2mX dx 2 2
" 2 d Y 2 2
2
1
ÿ mù y ˆ E y
y
2mY dy 2 2
2
" 2 d Z 1 2 2
ÿ mù z ˆ E z
z
2mZ dz 2 2
where the three separation constants E x , E y , E z satisfy the relation
E x ‡ E y ‡ E z ˆ E (4:53)
The differential equation for X(x) is exactly of the form given by (4.13) for a
one-dimensional harmonic oscillator. Thus, the eigenvalues E x are given by
equation (4.30)
1
ˆ (n x ‡ )"ù x , n x ˆ 0, 1, 2, ...
2
E n x
and the eigenfunctions are given by (4.41)