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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 255
FromTable 7E.1 onpage 275 these polynomials have the property ∫
∞
−∞H
2
υe−y
2
dy =
π1/22υυ! so it follows that N2υ = 1/(π1/22υυ!). �e probability of �nding the
particle outside the range of the turning points is then
Pυ = 2∫
∞
√
2υ+1
∣ψυ ∣2 dy =
2
π1/22υυ! ∫
∞
√
2υ+1
H2υ e
−y2 dy
= 1
π1/22υ−1υ! ∫
∞
√
2υ+1
H2υ e
−y2 dy
�e integral is evaluated using mathematical so�ware to give the results in the
following which are plotted in Fig. 7.8. �e probability of a classical oscillator
being found in a non-classical region is, of course, zero and the correspondence
principle indicates that the quantum result must converge to this value as the
quantumnumber becomes large.�e results show that P is indeed a decreasing
function of υ, albeit rather slowly. �e probability reduces to about 0.02 for
υ = 200.
υ 0 1 2 3 4 5 6 7
Pυ 0.157 0.112 0.0951 0.0855 0.0789 0.0740 0.0702 0.0670
0 1 2 3 4 5 6 7
0.00
0.05
0.10
0.15
υ
P υ
Figure 7.8
P7E.16 Oscillations are expected about a minimum in the potential energy, because
this corresponds to the equilibrium arrangement. For a potential of the form
V = V0 cos 3ϕ, the (�rst) minimum is when 3ϕ = π, that is ϕ = π/3 or 60○.�e
form of the potential close to theminimum is found by expanding the function
in a Taylor series about the point ϕ = π/3
V/V0 = V(π/3) + (dV(ϕ)
dϕ
)
ϕ=π/3
(ϕ − π/3) + 1
2 (
d2V(ϕ)
dϕ2
)
ϕ=π/3
(ϕ − π/3)2
= cos(π) + (−3 sin 3ϕ)ϕ=π/3(ϕ − π/3) + 1
2 (−9 cos 3ϕ)ϕ=π/3(ϕ − π/3)2
= −1 + 1
2 × 9(ϕ − π/3)2