6-6 ① The identities are:
(i) cos2x + sin2x = 1
(ii) cos(2x) = cos2x – sin2x
Rearranging (i) we get cos2x = 1 – sin2x. Hence, substituting back into (ii) we get:
cos(2x) = 1 – 2sin2x which, in turn, rearranges to give:
sin2x = ½(1 – cos(2x)).
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② We have:
We can take the a2 term outside the integral, since it is just a constant,
and replace sin2(nπx/L) with [1-cos(2nπx/L)]/2.
Hence our integral is now:
⑤ Since sin(2nπ) = 0 and sin(0) = 0, this simplifies to a2L/2 = 1. Hence:
Our new, improved, normalised, wavefunctions can now, therefore, be written as:
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