I am a novice on QM and until now i have allways been using sinusoidal form of wave equation: A=A0sin(kx−ωt)
Well in QM everyone uses complex exponential form of wave equation: A=A0ei(kx−ωt)
QUESTION: How do i mathematically derive exponential equation out of sinusoidal one? Are there any caches? I did read Wikipedia article where there is no derivation.
Answer
You asked about the second equation. See below:
eix=1+ix+(ix)22!+(ix)33!+(ix)44!+(ix)55!+(ix)66!+(ix)77!+(ix)88!+⋯=1+ix−x22!−ix33!+x44!+ix55!−x66!−ix77!+x88!+⋯=(1−x22!+x44!−x66!+x88!−⋯)+i(x−x33!+x55!−x77!+⋯)=cosx+isinx .
To calculate the expansions I have used in the above equation, you need to understand the procedure for finding Taylor expansions of functions. This youtube video teaches the procedure: http://www.youtube.com/watch?v=GUtLtRDox3c
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