How to understand operators representing observables are Hermitian?

[George Jones]

When first learning quantum mechanics, I recommend against delving too deeply into these issues.

For the difference between "selfadjoint" and "hermitian", see

https://www.physicsforums.com/showthread.php?p=1887619#post1887619

 

[dextercioby]

The book written by the Nobel Prize winner is intended as a gentle presentation of the formalism/content of the theory and its main results. It's not about the mathematical foundations/formulation of quantum mechanics. That's why no distinction is made between hermitean and self adjoint. A distinction exists and is made at a higher level of rigor than the books by Ballentine, Grifftihs, Weinberg, Sakurai, Merzbacher, Messiah, Cohen-Tannoudji, etc.

(2) as you wrote is the second postulate (the 1st postulate regards the states, the second is about observables, the third is usually the Born rule and the 4th is the evolution equations for states/observables).

 

[bhobba]

When first learning quantum mechanics, I recommend against delving too deeply into these issues.

I fully concur.

When I first learned QM properly I became caught up with this damnable Dirac Delta function. It took me on a long detour into Rigged Hilbert spaces and such. I emerged with my questions answered but the cost was my understanding of QM took a hiatus.

Much better in my view is put such things aside at the moment, knowing they have an answer, but it requires a more advanced background. You can get that background by reading up on some functional analysis. I have a few texts, but one I like is Applied Analysis by John Hunter.

While doing that Ballentine is reasonably mathematically complete and would give you a good foundation in the physics.

Thanks
Bill

 

[shrodinger1226]

When first learning quantum mechanics, I recommend against delving too deeply into these issues.

For the difference between "selfadjoint" and "hermitian", see

https://www.physicsforums.com/showthread.php?p=1887619#post1887619

Excuse me, in your post #6 https://www.physicsforums.com/showthread.php?p=1887619#post1887619, you said "all Hermitian operators are self-adjoint, but not all self-adjoint operators are Hermitian. "

However my understanding is the opposite: "all self-adjoint operators are Hermitian, but not all Hermitian operators are self-adjoint". Did I get it wrong?

 

[George Jones]

Yes.

 

[George Jones]

For the momentum operator a particle in a 1-dimensional infinite well, see "Self-adjoint extensions of operators and the teaching of quantum mechanics",

http://arxiv.org/abs/quant-ph/0103153

 

[R136a1]

Yes.

No, I disagree. You have as definition that hermitian immediately implies that it's bounded. I don't think this is a very common definition. All definitions I've seen is that symmetric and hermitian are equivalent, and thus hermitian is not necessarily bounded. See for example Reed & Simon, page 255. So I would say that the OP has it exactly right.

 

[George Jones]

No, I disagree. You have as definition that hermitian immediately implies that it's bounded. I don't think this is a very common definition. All definitions I've seen is that symmetric and hermitian are equivalent, and thus hermitian is not necessarily bounded. See for example Reed & Simon, page 255. So I would say that the OP has it exactly right.

Well, I have (at least) a couple of books that only use "Hermitian" with respect to bounded operators:

1) Introductory Functional Analysis by Kreyszig (the text for a course I took as a student);

2)Hilbert Space Operators in Quantum Physics by Blank, Exner, and Havilcek.

Both of these references define symmetric unbounded operators, but don not associate the term "Hermitian" with such operators.

Some references (e.g., Riesz and Nagy) do not use "Hermitian" at all. In the recent book "Quantum Theory for Mathematicians" by Hall, "Hermitian" is used only in the sentence "Physicists refer to self-adjoint operators as Hermitian." Hall does not use "Hermitian" (but does use symmetric and sel-adjoint) when doing actual functional analysis.

 

[dextercioby]

That's my mind, too. <Hermitean> is not modern terminology in functional analysis. It belongs now only in linear algebra which is done in finite dimensional spaces, of course. It stands in front of the words matrix+matrices.