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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 269
hence
∆x∆px = L(1/12 − 1/2n2π2)1/2 × nh/2L = (nh/2)(1/12 − 1/2n2π2)1/2
For n = 1
∆x∆px = L(1/12 − 1/2π2)1/2 × h/2L = (h/2)(1/12 − 1/2π2)1/2 ≈ 0.57ħ
and for n = 2 ∆x∆px ≈ 1.7ħ. �e Heisenberg uncertainty principle
is satis�ed in both cases, and it is evident that ∆x∆px is an increasing
function of n.�e principle is therefore satis�ed for all n > 1.
(b) In Problem P7E.17 it is shown that for a harmonic oscillator in a state with
quantum number υ
∆xυ∆pυ = (υ + 1
2 )ħ
�erefore, for the ground state with υ = 0, ∆x∆p = 1
2 ħ: the Heisenberg
uncertainty principle is satis�ed with the smallest possible uncertainty. It
follow that for υ > 0 the principle is also satis�ed because ∆xυ∆pυ is an
increasing function of υ.