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Re: Imaginary reality



At 08:32 AM 4/22/00 -0700, Ben Crowell 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.

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.

Don't be fooled by the notion that it is "traditional" to classical E&M one
way and "traditional" to do QM the other way.

In general, the physics is what it is. We invent and/or borrow notation to
represent it (to a greater or lesser degree of precision).

To drive home this point, note that in the real world, a "quantum
mechanical electron's wavefunction" does _not_ obey the same rules as
complex arithmetic. Because it has spin, and because it has an
antiparticle, it obeys a more complicated arithmetic. Complex numbers are
not sufficient, but Hamilton and Pauli and Dirac cobbled up some machinery
to provide a reasonably elegant representation.

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

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.