226
3 Quantum Mechanics – II
Exchange of x → −x, y → −y, z → −z implies θ → π − θ and
ϕ → π + ϕ, so that
P 1
m (cos θ ) → (−1)
l+m
and e
imϕ
→ (−1)
m e
imϕ
, Thus Pρ = (−1)
2m (−1)
l
= (−1)
l
ρ
where m is an integer. Thus ρ is symmetrical for even l and antisymmetrical
for odd l.
First consider zero nuclear spin. The total wave function ψ is antisymmetrical for odd l and symmetrical for even l. As the nuclei must obey either
Fermi or Bose statistics, either only the l = odd states must exist or only the
l = even states must exist. It turns out that for nuclei with zero spin only the
even rotational states exist and odd rotational states are missing.
Next consider the case of non-zero spin. A nucleus of total angular momentum I can have a component M in any prescribed direction taking 2I +1 values
in all (I, I − 1, . . . − I ), that is 2I + 1 states exist. For the two identical nuclei
(2I + 1)
2 wave functions of the form ψ M1 (A)ψ M2 (B) can be constructed. If
the two nuclei are identical, these simple products must be replaced by linear combination of those products which are symmetric or antisymmetric for
interchange of nuclei. If M 1 = M 2 , the products themselves are (2I + 1)
symmetric wave functions, the remaining (2I + 1)
2
− (2I + 1) = 2I (2I + 1)
functions with M 1 = M 2 have the form ψ M1 ( A)ψ M2 (B) and ψ M2 ( A)ψ M1 (B).
Each such pair can be replaced by one symmetric and one antisymmetric
wave function of the form ψ M1 (A)ψ M2 (B) ± ψ M2 ( A)ψ M1 (B). Thus, half of
2I (2I + 1) functions, that is I (2I + 1) are symmetric and an equal number antisymmetric. Therefore, total number of symmetric wave functions
= (2I + 1) + I(2I + 1) = (2I + 1)(I + 1). Total number of antisymmetric
wave functions = I (2I + 1). Therefore, the ratio of the number of symmetric
and antisymmetric functions is (I + 1)/I .
From the previous discussion it was shown that for the symmetric electronic wave function of the molecule the interchange of nuclei produces a
factor (−1)
l in the molecular wave function. Thus, for nuclei obeying Bose
statistics symmetric nuclear spin functions must combine with even l. Because
of the statistical weight attached to spin states, the intensity of even rotational lines will be (I + 1)/I as great as that of neighboring odd rotational
lines.
For nuclei obeying Fermi statistics, the spin and rotational states combine
in a manner opposite to the previously described and the odd rotational lines
are more intense in the ratio (I + 1)/I.
Thus, by determining which lines are more intense, even or odd, the nuclear
statistics is determined and by measuring the ratio of intensities of adjacent
lines the nuclear spin is obtained. The reason for comparing the intensity of
neighboring lines is that the intensity of rotational lines varies according to
the occupation number of rotational states governed by the Boltzmann distribution.
3 Quantum Mechanics – II
Exchange of x → −x, y → −y, z → −z implies θ → π − θ and
ϕ → π + ϕ, so that
P 1
m (cos θ ) → (−1)
l+m
and e
imϕ
→ (−1)
m e
imϕ
, Thus Pρ = (−1)
2m (−1)
l
= (−1)
l
ρ
where m is an integer. Thus ρ is symmetrical for even l and antisymmetrical
for odd l.
First consider zero nuclear spin. The total wave function ψ is antisymmetrical for odd l and symmetrical for even l. As the nuclei must obey either
Fermi or Bose statistics, either only the l = odd states must exist or only the
l = even states must exist. It turns out that for nuclei with zero spin only the
even rotational states exist and odd rotational states are missing.
Next consider the case of non-zero spin. A nucleus of total angular momentum I can have a component M in any prescribed direction taking 2I +1 values
in all (I, I − 1, . . . − I ), that is 2I + 1 states exist. For the two identical nuclei
(2I + 1)
2 wave functions of the form ψ M1 (A)ψ M2 (B) can be constructed. If
the two nuclei are identical, these simple products must be replaced by linear combination of those products which are symmetric or antisymmetric for
interchange of nuclei. If M 1 = M 2 , the products themselves are (2I + 1)
symmetric wave functions, the remaining (2I + 1)
2
− (2I + 1) = 2I (2I + 1)
functions with M 1 = M 2 have the form ψ M1 ( A)ψ M2 (B) and ψ M2 ( A)ψ M1 (B).
Each such pair can be replaced by one symmetric and one antisymmetric
wave function of the form ψ M1 (A)ψ M2 (B) ± ψ M2 ( A)ψ M1 (B). Thus, half of
2I (2I + 1) functions, that is I (2I + 1) are symmetric and an equal number antisymmetric. Therefore, total number of symmetric wave functions
= (2I + 1) + I(2I + 1) = (2I + 1)(I + 1). Total number of antisymmetric
wave functions = I (2I + 1). Therefore, the ratio of the number of symmetric
and antisymmetric functions is (I + 1)/I .
From the previous discussion it was shown that for the symmetric electronic wave function of the molecule the interchange of nuclei produces a
factor (−1)
l in the molecular wave function. Thus, for nuclei obeying Bose
statistics symmetric nuclear spin functions must combine with even l. Because
of the statistical weight attached to spin states, the intensity of even rotational lines will be (I + 1)/I as great as that of neighboring odd rotational
lines.
For nuclei obeying Fermi statistics, the spin and rotational states combine
in a manner opposite to the previously described and the odd rotational lines
are more intense in the ratio (I + 1)/I.
Thus, by determining which lines are more intense, even or odd, the nuclear
statistics is determined and by measuring the ratio of intensities of adjacent
lines the nuclear spin is obtained. The reason for comparing the intensity of
neighboring lines is that the intensity of rotational lines varies according to
the occupation number of rotational states governed by the Boltzmann distribution.