120
2 Quantum Mechanics – I
2.56
n(E 2 )
n(E 1 )
= e
−(E2−E1)/kT
=
1
10
T =
E 2 − E 1
kln 10
=
2.26
8.625 × 10 −5 × 2.3
= 1.14 × 10
4 K
2.3.6 Molecules
2.57 The two modes of motion of a diatomic molecule are (i) rotation and (ii) vibration.
The first order rotational energy is
2 J (J + 1)/2I 0 , where I 0 = M R
2
0 is
the moment of inertia of the molecule about an axis perpendicular to the line
joining the nuclei; the energy being the same as for the rigid rotator. Clearly
the spacing between successive levels is unequal; it progressively increases
with the increasing value of J , where J = 0, 1, 2 . . . The spectrum called
band spectrum arises due to optical transitions between rotational levels. The
band spectrum is actually a line spectrum, but is thus called because the lines
are so closely spaced and unresolved with an ordinary spectrograph, and give
the appearance of a band.
The second mode consists of to and fro vibrations of the atoms about the
equilibrium position. The motion is described as simple harmonic motion.
The energy levels are given by E n = ω (n + 1/2), where n = 0, 1, 2 . . .
and are equally spaced. However as J or n increases, the spacing between
levels becomes smaller than that predicted from the simple rigid rotator and
harmonic oscillator.
2.58 The rotational energy levels are given by
E J =
2 J (J + 1)/2I o
where I o is the rotational inertia
ΔE = E 1 − E 0 =
2
/I o
If μ is the reduced mass,
I = μr
2
=
m H m D r
2
m H + m D
=
m H . 2m H r
2
m H + 2m H
=
2
3
m H r
2
(because m D ≈ 2m H )
ΔE =
3
2
2m H r 2 =
3
2
.
(c MeV − fm)
2
m H c 2 (0.075 × 10 −9 m) 2
=
3
2
(197.3 × 10
−15 MeV − m)
2
938(0.075 × 10 −9 m) 2
= 0.011 × 10
−6 MeV
= 0.011 eV.
λ = 1,241/0.011 = 1.128 × 10
5 nm
= 0.113 mm
2.59 All nuclei of even A, with zero or non-zero spin obey Bose statistics and
all those of odd A obey Fermi statistics. The result has been crucial in
2 Quantum Mechanics – I
2.56
n(E 2 )
n(E 1 )
= e
−(E2−E1)/kT
=
1
10
T =
E 2 − E 1
kln 10
=
2.26
8.625 × 10 −5 × 2.3
= 1.14 × 10
4 K
2.3.6 Molecules
2.57 The two modes of motion of a diatomic molecule are (i) rotation and (ii) vibration.
The first order rotational energy is
2 J (J + 1)/2I 0 , where I 0 = M R
2
0 is
the moment of inertia of the molecule about an axis perpendicular to the line
joining the nuclei; the energy being the same as for the rigid rotator. Clearly
the spacing between successive levels is unequal; it progressively increases
with the increasing value of J , where J = 0, 1, 2 . . . The spectrum called
band spectrum arises due to optical transitions between rotational levels. The
band spectrum is actually a line spectrum, but is thus called because the lines
are so closely spaced and unresolved with an ordinary spectrograph, and give
the appearance of a band.
The second mode consists of to and fro vibrations of the atoms about the
equilibrium position. The motion is described as simple harmonic motion.
The energy levels are given by E n = ω (n + 1/2), where n = 0, 1, 2 . . .
and are equally spaced. However as J or n increases, the spacing between
levels becomes smaller than that predicted from the simple rigid rotator and
harmonic oscillator.
2.58 The rotational energy levels are given by
E J =
2 J (J + 1)/2I o
where I o is the rotational inertia
ΔE = E 1 − E 0 =
2
/I o
If μ is the reduced mass,
I = μr
2
=
m H m D r
2
m H + m D
=
m H . 2m H r
2
m H + 2m H
=
2
3
m H r
2
(because m D ≈ 2m H )
ΔE =
3
2
2m H r 2 =
3
2
.
(c MeV − fm)
2
m H c 2 (0.075 × 10 −9 m) 2
=
3
2
(197.3 × 10
−15 MeV − m)
2
938(0.075 × 10 −9 m) 2
= 0.011 × 10
−6 MeV
= 0.011 eV.
λ = 1,241/0.011 = 1.128 × 10
5 nm
= 0.113 mm
2.59 All nuclei of even A, with zero or non-zero spin obey Bose statistics and
all those of odd A obey Fermi statistics. The result has been crucial in