Page 136 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4.6 Three-dimensional harmonic oscillator 127
1=4

2
(x) ˆ (2 n x !) ÿ1=2 mù x (î)e ÿî =2
n x
ð"
X n x H n x
1=2

mù x
î ˆ x
"
Similarly, the eigenvalues for the differential equations for Y(y) and Z(z) are,
respectively
1
ˆ (n y ‡ )"ù y , n y ˆ 0, 1, 2, ...
2
E n y
1
ˆ (n z ‡ )"ù z , n z ˆ 0, 1, 2, ...
E n z
2
and the corresponding eigenfunctions are
1=4

2
(y) ˆ (2 n y !) ÿ1=2 mù y (ç)e ÿç =2
n y
ð"
Y n y H n y
1=2

mù y
ç ˆ y
"
1=4
2
(z) ˆ (2 n z !) ÿ1=2 mù z (æ)e ÿæ =2
n z
Z n z H n z
ð"
1=2

mù z
æ ˆ z
"
The energy levels for the three-dimensional harmonic oscillator are, then,
given by the sum (equation (4.53))
1 1 1
ˆ (n x ‡ )"ù x ‡ (n y ‡ )"ù y ‡ (n z ‡ )"ù z (4:54)
2 2 2
E n x ,n y ,n z
The total wave functions are given by equation (4.52)
m 3=4
(x, y, z) ˆ (2 n x ‡n y ‡n z n x !n y !n z !) ÿ1=2 (ù x ù y ù z ) 1=4
ð"
ø n x ,n y ,n z
2
2
2
(æ)e ÿ(î ‡ç ‡æ )=2 (4:55)
3 H n x (î)H n y (ç)H n z
An isotropic oscillator is one for which the restoring force is independent of
the direction of the displacement and depends only on its magnitude. For such
an oscillator, the directional force constants are equal to one another
k x ˆ k y ˆ k z k
and, as a result, the angular frequencies are all the same
ù x ˆ ù y ˆ ù z ù
In this case, the total energies are