Re: Quantum mechanics and operators?

From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 11/17/04 

Date: Wed, 17 Nov 2004 16:54:46 +0000 (UTC)

Jeremy Price wrote:
> I'm trying to find out more about where the operators in quantum mechanics
> come from. From what I see, the justification for using them [...]

They come from Heisenberg and their justification is that they have
been immensely useful since he introduced them.

> Well, that's great, but that argument just goes in a big circle.

Justification of the basic postulates of any theory is necessarily
circular. If it were not, the postulates were not basic but derivable.

One must take all the basic postulates as a single foundation
on which everything else rests without circularity.
But the basic postulates themselves can only be motivated, but not
derived.

Most people simply trust that tradition selected good foundations.
If you want to probe that trust you can go into studying the sea of
publications on the foundations of quantum mechanics. But unless
you are very dedicated and spend a lot of effort on it,
it is likely that you'll drown there before having found satisfaction...

> I'm also interested in the use of complex numbers to write the equation.
> My quantum mechanics book mentions that they are used to simplify the
> equations, and that it doesn't physically mean anything.

Already Fourier analysis is most natural with complex numbers.
It looks quite 'meaningful' to me, though the term is somewhat
subjective.

And the time-independent Schroedinger equation defines the
Fourier components of real, measurable expectations. So it is
very natural that it is based on complex entities, too.

Arnold Neumaier

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