Re: 10 questions on QM postulates

On 2005-05-15, Hendrik van Hees <hees@xxxxxxxxxxxxx> wrote:

>> 1) How can the mathematical concept of hermitian operators be
>> translated into physical terms? What are all the conditions which
>> ultimately restrict operators such that they can be only hermitian as
>> opposed to any other type?
>
> That's a very deep question, and I also asked it to my professor when he
> introduced the concept. His answer was: "The operators, describing
> observables, must have only real eigenvalues, and Hermitean operators
> have only real eigenvalues."
>
> In my opinion, that's a short but not really satisfactory answer. After
> thinking about the foundations of quantum theory from time to time (I
> apply quantum theory every day, but one does not think about its
> foundations too often in this kind of work, so it's more a hobby of
> mine to think about it from time to time, especially when I hear
> something about modern quantum-optics experiments, which help a lot to
> understand the foundations of quantum theory), I think the real
> mathematical reason that the observables can be described without any
> inconsistencies by Hermitean (or to be more precise by essentially
> self-adjoint) operators is the spectral theorem, i.e., each quantum
> state can be described by a superposition of a complete set of
> generalized eigenvectors of an self-adjoint operator.
>
> I do not claim that an operator must be necessarily self-adjoint to meet
> this property. I'm not sure wheter it is or not, it might be sufficient
> that an operator commutes with its adjoint, which is usually called a
> "normal operator". Perhaps somebody more familiar with the mathematical
> details may answer this question.

One way to look at self-adjointness is that self adjoint operators are
the only ones that can be full recovered just from their expectation
values.

Let A be a self-adjoint operator, then we can define a quadratic form

a(psi) = <psi|A|psi>.

In turn, if we know a(psi) for all |psi>, not just the eigenstates of A,
then if a is real valued and satisfies the parallelogram law

a(psi+phi) + a(psi-phi) = 2(a(psi) + a(phi)),

Then the matrix elements of A can be recovered via the so-called
polarization formula

<psi|A|phi> = (1/4) sum_{n=0..3} i^n a(psi + i^n phi).

Hmm, I'm not 100% about the constant 1/4. I know that's what it is in
the real vector space case, but it might be somewhat different in the
complex case.

If A is self-adjoint, then the reality and parallelogram law conditions
are always satisfied. In the infinite dimensional case, the quadratic
form a(psi) should also satisfy some sort of continuity condition.

Once we have a self-adjoint operator, the spectral theorem gives us a
decomposition of the Hilbert space into orthogonal eigenspaces. Thus
this property can be derived just from some properties of observables,
i.e. expectation values.

Igor

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