Endnote on Imaginary Numbers

In Appendix 6 we made a quick note that our wavefunctions may turn out to contain imaginary numbers, i.e. they may contain i = V. For the electron wavefunction problems studied in this appendix, imaginary numbers become more important, and so this section will give a brief overview to show how they should be handled. 

In this appendix we go a little futher and use the complex exponential in some of the wavefunctions. We are used to the idea that an exponential function describes rapidly increasing or rapidly decaying behaviour. Now we have to accept that a complex exponential is actually a periodic function, since it obeys the identity 

This is actually the way we identify the real and imaginary parts of the complex exponential. 

The usual method to explain Equation (A9.85) is to use the standard Taylor expansions of the three functions and show that they are equivalent to one another (see Further Reading section in this appendix). In this section we will just look at a few examples that confirm Equation (A9.85) is correct and that exp( i(p) is periodic. 

using the complex conjugate approach to obtain the square of the magnitude of exp( i ) gives 

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