Why most observables have real eigenvalues

[aaaa202]

I have always been quite confused about the fact that any measurement MUST yield a real number. What says it must so? Don't we modify our measurement apparatus to yield something which is consistent with the theory. So coulnd't we just imagine having complex values for momentum and position. All in all: I never found the reason satisfying that "observables are represented by hermitian operators because their eigenvalues are real".

 

[jedishrfu]

This is an interesting and deep question. On the one hand we have devices to measure various quantities and on the other we have theories to model these processes.

The use of complex numbers within a theory are used as a means to an end. The end being a real number, a measurable quantity. Just because we use complex numbers within a theory does not mean the physical system we are measuring uses complex numbers to operate. The physical system is still ultimately a mystery that we can model quite well.

 

[aaaa202]

But who says that a measurement should yield a real number? I mean the only thing that I can see supports this is that we want to build up QM on the knowledge of classical physics we already know. And there by definition, everything is real.

 

[cattlecattle]

But who says that a measurement should yield a real number? I mean the only thing that I can see supports this is that we want to build up QM on the knowledge of classical physics we already know. And there by definition, everything is real.

It's just a reasonable postulate. I couldn't imagine anything physical to be complex number, if you can, please give an example (even hypothetical ones). I would personally visualize physical attributes to be something you can read from an equipment. What does it even mean if you say a physical attribute is a complex number?

 

[DrDu]

A measurement might well give a complex number. The most general operators in QM are represented by normal operators (i.e. operators which can be represented as the sum of a hermitian and an anti-hermitian operator which commute with each other and can thus be simultaneously diagonalized).

 

[mfb]

A measurement might well give a complex number. The most general operators in QM are represented by normal operators (i.e. operators which can be represented as the sum of a hermitian and an anti-hermitian operator which commute with each other and can thus be simultaneously diagonalized).

How can you measure a complex number?

I think it is just a design issue - our physics is designed to interpret all measurements as real values. You could combine position and momentum into a single complex number, and measure this (easy in classical mechanics). It would not change physics at all.
It is interesting that quantum mechanics has to introduce things which we cannot measure directly (like phases of wave functions). But that is how physics works, and "why" is a question for philosophy.

 

[Jorriss]

Don't we modify our measurement apparatus to yield something which is consistent with the theory.

If you mean what I think this means, we do the opposite. Theory has to conform to experiment.

 

[bhobba]

But who says that a measurement should yield a real number

No one. Its just that fundamental things like distance and time are defined in terms of real numbers and stuff like force and energy are derived from it - probably from the fact in the past it boiled down to reading of some kind of distance which is in 1-1 correspondence with the reals. Of course in modern times we have digital readouts etc which are rational numbers and they are dense in the reals.

Thanks
Bill

 

[Hurkyl]

All in all: I never found the reason satisfying that "observables are represented by hermitian operators because their eigenvalues are real".

There isn't one.

However, keep in mind that to best understand the complex numbers, one must study its subfield of real numbers -- especially the set of positive real numbers -- which has many nice properties.

Hermitian operators play a very similar role in this setting that the real numbers play in the setting of complex numbers, and there is a lot of analogy.

The setting is sufficiently more abstract than complex numbers, however, that the centuries of realization that complex numbers are a good idea has been slower to penetrate -- especially because QM introduces yet another word (observable) into the collection of technical terms that are homonyms for English words, thus causing people to confuse their connotations. (examples of other such words are "real", "imaginary", "natural")

And to be honest, the idea of a complex-valued observable are rather mild. There is no problem -- other than mathematical sophistication -- with going much further and having things like kitten-valued observable, for example.

 

[f95toli]

How can you measure a complex number?

Using a vector network analyzer? Many microwave measurements measure complex numbers (e.g. S parameters) directly since we are interested in both amplitidude and phase.
There are also passive components called IQ demodulators which will give you two voltages out, proportional to the real and imaginary part of the signal, respectivly (assuming you adjust the phase of the LO properly).

Also, it is possible to calculate the S parameters of QM systems (using e.g. linear response theory), and the result of these calculations are (obviously) complex.