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234 7 QUANTUM THEORY
−1.0 −0.5 0.0 0.5 1.0
−0.5
0.0
0.5
1.0
x
N = 2
N = 5
Figure 7.2
−1.0 −0.5 0.0 0.5 1.0
0.0
0.5
1.0
x
N = 2
N = 5
Figure 7.3
(b) Figure 7.3 shows that, as before, the probability density narrows and sharp-
ens as the superposition is extended.
(c) In order to compute ⟨x2⟩, the expectation value of x2, it is necessary to
normalize the wavefunction. If attention is focused on the �nite range
−1 ≤ x ≤ +1 the normalizing factorN is found by evaluating
N 2 =
⎧⎪⎪⎨⎪⎪⎩
1
N2 ∫
+1
−1
[
N
∑
k=1
cos(kπx)]
2
dx
⎫⎪⎪⎬⎪⎪⎭
−1
Because the square of a sum involving (possibly) many terms is involved,
this appears to be a formidable task. However, some experimentation
with some modest values of N using mathematical so�ware implies that
the term in the braces is simply (1/N), so the normalizing factor isN =