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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 231
�e Heisenberg uncertainty principle, [7C.13a–258] is rearranged to give the
uncertainty in the position as ∆q ≥ ħ/(2∆p), which gives a minimum uncer-
tainty of ∆qmin = ħ/(2∆p).�is is evaluated as
1.0546 × 10−34 J s
2 × (9.06... × 10−30 kgms−1)
= 5.82 × 10−6 m
Solutions to problems
P7C.2 (a) Consider the integral I = ∫
L
0 sin(nπx/L) sin(mπx/L)dx. Using the iden-
tity, sinA sinB = 1
2 cos(A − B) −
1
2 cos(A + B) with A = nπx/L and
B = mπx/L, this can be rewritten as
I = 1
2 ∫
L
0
cos[(n −m)πx/L]dx − 1
2 ∫
L
0
cos[(n +m)πx/L]dx (7.1)
(b) In the case of n = 2,m = 1 the two integrands are cos(πx/L) and cos(3πx/L)
which are plotted in Fig. 7.1
0.2 0.4 0.6 0.8 1.0
−1.0
−0.5
0.5
1.0
x/L
cos(πx/L)
cos(3πx/L)
Figure 7.1
(c) It is seen that each of these functions are antisymmetric about x = L/2,
such that the value of the function as L/2+ δ is minus that at L/2− δ. As
a result, the integral of these functions over a symmetrical region about
x = L/2 is zero.
An alternative way of coming to the same conclusion is to note that the
integral of the �rst half of a cosine wave is zero on account of the enclosed
area above and below the x-axis being the same (the solid curve). Sim-
ilarly, the integral of three complete half cosine waves is also zero (the
dashed curve).
(d) For the general case n and m and both integers, and so n ± m are also
integers. �e two integrands in eqn 7.1, when considered over the range
x = 0 → L, will each comprise a complete number of half cosine waves.
By the same argument as in (c), these functions will integrate to zero and
hence the wavefunctions are orthogonal for n ≠ m.