Imaginary reality

[Ben Crowell <bcrowell@SPRINTMAIL.COM>]

I wrote
> In classical E&M, it seems to me that the only point of using
> complex numbers is to simplify notation, e.g. to be able to write
> a Fourier analysis with only one function, the exponential, rather
> than 2, sines and cosines.
>
> On the other hand, a quantum mechanical electron's
> wavefunction happens to obey the same rules as complex
> arithmetic. The theory is actually incomplete without
> complex numbers, in the same way that you can't make
> a complete theory of the quadratic equation without them.

John Denker wrote:
> Technically and formally, the foregoing distinction is not valid.

> As I discussed at length in a previous posting, complex numbers are always
> optional.  In QM just as in classical E&M, you can write the complex
> number   (p+iq) as a vector in the (p, q) plane.  And vice versa.

Of course the _names_ of the math concepts aren't attached to the
physical phenomena with little post-it notes. For that matter, real
numbers are optional: replace the coordinate x with an arrow pointing from
the origin to a point in space.

I wrote:
> Since the original question was about pedagogy, this seems
>  to me a good argument for waiting until QM to introduce
> complex waves.

John Denker wrote:
> That's a separate question.  And I'm not sure I agree with the conclusion.

> As an employer, if I interviewed a person who had taken a college-level
> physics course that covered classical E&M waves (or even an engineering
> course that covered circuit analysis), yet who couldn't handle the
> complex-number representation, I'd be pretty disappointed.

Sounds like we're talking about two different things here. I was referring
to a freshman-sophomore introductory survey course, not an upper-division
physics or engineering course. I assume nobody is going to get a bachelor's
degree in physics or engineering without working with the complex number
formalism.