Re: Practical applications of complex numbers

Johann 'Myrkraverk' Oskarsson <[email protected]> wrote or quoted:
>This is some beautiful ASCII art rendering of waves.  Do you have a tool
>that does this, or do you keep this pre-rendered in a text file for just
>such occasions?

  I've written a Python script that renders pixels to an array
  and then tries to match rectangles with such pixels to ASCII
  characters; it is using a specific raster font. The raw results
  of this approach did not look very good, and I found out that
  I can improve the result by restricting the set of characters to
  just a few selected characters like ".". Also, I take the slope
  of the curve into account. For example, the downward moving
  accent "`" is only used where the curve does move downward with
  approximately this angle (as can be seen in the sine plots).
  But this Python script is not yet ready for publication. I also
  edited two characters of the plots manually in my previous post.

>>If we combine /two/ plates of thickness d/2 we still get a multipli-
>>cation by -1. So what does /one/ plate of thickness d/2 multiply
>>the amplitude with?
>I have a feeling the answer should be /i/, but I'm not sure.  Feel free
>to recommend books, websites, or PDF files where I can brush up on light
>physics.

  This example was taken from a book about quantum physics that is
  as easy and readable as a book about this topic can possibly be:

"Quantum Processes, Systems, and Information" (2010) -
Benjamin Schumacher and Michael D. Westmoreland.

  (Schumacher is known for his coinage of the word "qubit".)

  The authors write in section 2.1:

|Glass plates can be made in a continuous range of thicknesses,
|producing a continuous range of phase shifts. For this to be
|possible, the beam phases a must be complex quantities, with
|both real and imaginary parts. A plate with thickness d/2 may
|multiply the amplitude by a factor of i = sqrt −1. This does not
|change the magnitude of the complex phase a, since |a| = |ia|.
|Two such plates (or a single plate of thickness d) multiply
|the phase by i^2 = −1, as required.

  .