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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 257
−2 −1 0 1 2
−2
0
2
y
H
υ(
y)
υ = 3
υ = 4
υ = 5
Figure 7.10
7F Rotational motion
Answers to discussion questions
D7F.2 Rotational motion on a ring and on a sphere share the following features: (a)
quantization arising as a result of the need to satisfy a cyclic boundary con-
dition; (b) energy levels which go inversely with the moment of inertia; (c)
the lack of zero-point energy; (d) degeneracy; (e) quantization of the angular
momentum about one axis.
Solutions to exercises
E7F.1(b) �e magnitude of the angular momentum associated with a wavefunction with
angular momentum quantum number l is given by [7F.11–288], magnitude =
ħ[l(l + 1)]1/2. Hence for l = 2 the magnitude is ħ[2(2 + 1)]1/2 = 61/2ħ .
�e projection of the angularmomentumonto the z-axis is given by [7F.6–284],
ħm l , where m l is a quantum number that takes values between −l and +l in
integer steps, m l = −l , −l + 1, . . . + l . Hence the possible projections onto the
z-axis are −2ħ,−ħ, 0, ħ, 2ħ .
E7F.2(b) �e wavefunction of a particle on a ring, with quantum number m l is ψm l =
eim l ϕ = cos(m lϕ)+ i sin(m lϕ) in the range 0 ≤ ϕ ≤ 2π.�e real and imaginary
parts of the wavefunction are therefore cos(m lϕ) and sin(m lϕ) respectively.
Nodes occur when the function passes through zero, which for trigonometric
functions are the same points at which the function is zero. Hence in the real
part, nodes occur when cos(m lϕ) = 0, and so when m lϕ = (2n + 1)π/2 for
integer n, which gives ϕ = (2n + 1)π/2m l . In the imaginary part, nodes occur
when sin(m lϕ) = 0 and so when m lϕ = nπ for an integer n, which gives ϕ =
nπ/m l .