Hermitean Operators: Real Eigenvalues & Orthonormal Basis in QM

[hellfire]

In QM hermitean operators are used as observables, which provide real eigenvalues. According to a version of the spectral theorem, their eigenfunctions form an orthonormal basis. My question is about complex matrices and whether they might (may be under some restrictions) span also the entire space (i do not know exactly the spectral theorem). I try to understand the reasons for the usage of hermitean operators: in some references I found that they are used due to their real eigenvalues which correspond to measurements, but in others that they are used due to the fact that the eigenfunctions form an orthonormal basis. May be both?

Thanks.

 

[Ambitwistor]

An insightful question. In actuality, observables don't have to be Hermitian and have real eigenvalues (although some textbooks claim otherwise), because operators that have complex eigenvalues can also form an orthonormal basis. The most general class of operators which can be considered observables are the normal operators. A normal operator is one that commutes with its adjoint,

$$[A,A^\dagger] = 0$$.

See:

http://groups.google.com/groups?selm=6t3ve0$dtq$1@pravda.ucr.edu

 

[selfAdjoint]

So should I change my handle? But I did say, down on the discussion forum, that I was Normal!

 

[1100f]

Every self adjoint operator is normal, but there exist normal operators that are not self adjoint

 

[LeBrad]

observables don't have to be Hermitian and have real eigenvalues

Do you mean ovservables don't have to have real eigenvalues or hermitian operators don't have to have real eigenvalues? And depending on which you mean, could you give me an example of one?

 

[Ambitwistor]

Do you mean ovservables don't have to have real eigenvalues or hermitian operators don't have to have real eigenvalues? And depending on which you mean, could you give me an example of one?

The former. Trivial example: take any two commuting Hermitian observables A and B, and construct a new observable A+iB. It's a normal operator, and hence an observable, but it has complex eigenvalues.

 

[LeBrad]

I do not believe that an observable operator can have complex eigenvalues. Imaginary numbers do not exist in nature, they are simply a tool created by mathematicians to aid in manipulating formulas and such. It may be possible to construct a normal matrix with complex eigenvalues, but I don't think that necessarily makes it an observable. I was taught that real eigenvalues and an orthonormal basis are both necessary, but neither is sufficient on its own. An observable by definition has real expectation values, becuase they have to be able to be observed.

 

[lethe]

Originally posted by LeBrad
I do not believe that an observable operator can have complex eigenvalues.

its not a matter of belief. its a matter of definition. you are free to choose any definition you want, of course. if you want to define an observable to be a smaller class of operators than Ambitwistor, you can do that, but you are more restrictive.

Imaginary numbers do not exist in nature, they are simply a tool created by mathematicians to aid in manipulating formulas and such.

here are other examples of things that were invented by mathematicians: quaternionic numbers, matrices, real numbers, negative numbers, zero, irrational numbers, klein bottles, etc...

so what? what does this have to do with anything?

It may be possible to construct a normal matrix with complex eigenvalues, but I don't think that necessarily makes it an observable.

if you define "observable" to mean normal, than it does, regardless of what you might think.

I was taught that real eigenvalues and an orthonormal basis are both necessary, but neither is sufficient on its own. An observable by definition has real expectation values, becuase they have to be able to be observed.

why can t you observe complex numbers?

 

[Ambitwistor]

Originally posted by LeBrad
I do not believe that an observable operator can have complex eigenvalues. Imaginary numbers do not exist in nature, they are simply a tool created by mathematicians to aid in manipulating formulas and such.

You can say the same thing about real numbers, or negative numbers, etc. This is philosophy, not physics. You remind me of Asimov's essay on imaginary numbers:

http://groups.google.com/groups?selm=9kuknk$2rae$1@nntp1.u.washington.edu

(Later in the essay he describes how real numbers are useful for describing the magnitudes of physical quantities, but complex numbers are useful for describing other physical quantities, such as those involving directions.)

Physics itself does not say whether numbers "really exist in nature"; it provides a mathematical framework to describe what the outcome of experiments will be, and you can describe them equally well in terms of real numbers, complex numbers, or whatever.

e.g. in electromagnetism where you work with complex impedances and such, that is no less "reality" than working with the resistances and reactances (the real and imaginary components of impedence) individually. Either way is a valid mathematical representation of reality.

I was taught that real eigenvalues and an orthonormal basis are both necessary, but neither is sufficient on its own.

You can, if you want, simply define an observable to have real eigenvalues, but there is no logical necessity to do so.

An observable by definition has real expectation values, becuase they have to be able to be observed.

There's nothing stopping me from building an instrument to measure a complex observable. It's the same thing as measuring two real observables at the same time (which you can do if they commute).

See the link I referenced in my first post. The author of that article (in another article) also refers to this issue as the result of "textbook degenerative disease", although I can't find that article in its entirety:

http://groups.google.com/groups?selm=6tgosv$v67$1@manifold.math.ucdavis.edu

 

[LeBrad]

I understand that it is a matter of the definition of an observable, and an observable is something that can be observed and measured. Imaginary numbers just do not exist in nature.

I didn't always feel this way, in fact, earlier this year I was arguing with some people about this very subject, and I was on the other side. I was insisting that imaginary numbers are real and exist in things. The argument essentially ended when my teacher told me I was an idiot for thinking that, and that imaginary numbers don't exist in nature, they are just a tool. Imaginary numbers are extremely important and useful, but they do not represent anything real.

I am far from an expert in quantum mechanics, but until I see somebody eat 3i hotdogs, or give me sqrt(-1) pieces of chalk, I can't believe that an observable can have complex eigenvalues.

 

[jcsd]

Originally posted by LeBrad
I understand that it is a matter of the definition of an observable, and an observable is something that can be observed and measured. Imaginary numbers just do not exist in nature.

I didn't always feel this way, in fact, earlier this year I was arguing with some people about this very subject, and I was on the other side. I was insisting that imaginary numbers are real and exist in things. The argument essentially ended when my teacher told me I was an idiot for thinking that, and that imaginary numbers don't exist in nature, they are just a tool. Imaginary numbers are extremely important and useful, but they do not represent anything real.

I am far from an expert in quantum mechanics, but until I see somebody eat 3i hotdogs, or give me sqrt(-1) pieces of chalk, I can't believe that an observable can have complex eigenvalues.

LeBrad it has been known for a couple of hundred years that certain physical problems (or at least problems that can be put in the form of a physical argument) NEED imaginary numbers in order to be solved. Have you ever seen someone eating pi hot dogs?

 

[Hurkyl]

And just how often have you been given -1 hot dogs, or dug half a hole?

 

[LeBrad]

I know that you may need imaginary numbers to solve certain problems, but that doesn't mean that you can measure them.

And maybe eating hotdogs was a bad example, but i could find a stick Pi meters long, or half a meter long, but never 1+2i meters long.

 

[Hurkyl]

Riddle me this: why do you think the hot dogs were a bad example?

 

[LeBrad]

Point well taken, although hotdogs aren't really a quality of a system, like position or momentum. Maybe I should only think of things as vectors with a magnitude and a direction, and not as positive or negative. Now if you want to measure and imaginary number you have to transform yourself into the complex plane to find its vector.

You make good points, and have forced me to look at this in a new light, but I still think that an observable quantity has to be real.

 

[slyboy]

Related to complex observables and non-Hermitian Hamiltonians, you might be interested in the following two pieces of theory:

Firstly, there is a sense in which an ordinary Hermitian observable can give you a complex number for a measurement result. It was developed by Aharonov and co-workers at Tel-Aviv university. The experiment works as follows:

1) Prepare a particular quantum state [ tex ]|\psi \rangle[ /tex ].
2) Perform a 'weak' measurement of the observable of interest.
3) Post-select by performing another measurement and only counting those that come out to be another particular state [ tex ]|\phi \rangle [ /tex ].

The 'weak' measurement is performed by using a quantum system as the measuring device, which has an initial uncertainty in position that is far greater than the change in position caused by the measuring interaction. Therefore, the procedure has to be performed a large number of times to get any sensible results.

Anyway, the resulting 'weak-value' of the measurement will br [ tex ]\langle \psi | O | \phi \rangle [ /tex ] where [ tex ] O [ /tex ] is the observable of interest. This value is generally complex, but the real part is the average position of the measuring device and the imaginary part shows up in the momentum of the measuring device.

Secondly, about non-Hermitian Hamiltonians. There are certain non-Hermitian Hamiltonians which actually have entirely real eigenvalues. Their eigenstates may not be orthogonal according to the usual inner product, but they are orthogonal with respect to a different inner product related to certain symmetry properties of the Hamiltonian. It has been conjectured that these Hamiltonians might have something to do with the part of the Standard Model that deals with the Higgs boson.

Studying these things has become a mini-industry at the moment, so just do a search for "pseudo Hermitian Hamiltonians" if you are interested. Otherwise:

"Must a Hamiltonian be Hermitian?"
C. M. Bender, D. C. Brody, and H. F. Jones
American Journal of Physics 71, 1095-1102 (2003)

is a good entry to the literature.

 

[jcsd]

slyboy, your latex coding didn't work because the brackets should be written without spaces (e.g. [\tex], [tex]).

 

[Poska]

There is a big differnce between operators and observebles. The set of operators fo standard QM is B(H) (bounded linear op.s on Hilbert space) but this set is not equal to the set of observables. We choose observables whwn we have our system (for example. we have particele and magnetic field (Stern - Gerlach experiment))and we measure spin of particle, so we choose an obserwable (pauli matrices) in 2 dim H space. And our result is some number. And this have nothing to do with hermitian or not hermitian staff. We chose hermitian obserwables because they are easy. And we have good math. tools to get their spectrum, but there is no specyfic rule it's only rutine.