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We know in general a Hermitian operator is not guaranteed to have eigenvalues, but self-adjoint operator is(if I remember correctly). Then why we still claim all observables are hermitian instead of claiming them to be self-adjoint?
Indeed for bounded operators being Hermitian and being self-adjoint is equivalent.I think real observables corresponding at least remotely to what can be measured have to be bounded so that there is no difference as to whether we talk of them being hermitian and self-adjoint.
We don't. Only introductory books don't make the difference. Mathematically minded texts use terms as: unbounded, closed, symmetric, essentially self-adjoint and self-adjoint. For example Thirring's text, Teschl's text, Prugovecki's text, Reed & Simon 4 volume text and Galindo & Pascual QM text.We know in general a Hermitian operator is not guaranteed to have eigenvalues, but self-adjoint operator is(if I remember correctly). Then why we still claim all observables are hermitian instead of claiming them to be self-adjoint?
I am sorry if my question annoyed you. In fact I am familiar with the mathematical terminology; I was under the impression some people in this thread use different terminology in the context of physics. By the way:The definitions are discussed in at least 100 books and certainly appear on wikipedia.
Like I said, for me hermitian and self-adjoint are synonyms (although I personally prefer self-adjoint as it says what it is). This claim about what "mathematicians use" is not true. I don't want to call myself a mathematician, but you probably won't argue about Paul Halmos.Hermitean is a term used by mathematicians only when speaking about finite dimensional spaces endowed with a scalar product.
It didn't annoy me. And my remark with 100 books is justified, as you picked one of the very many which used a different terminology than the majority. It's a book published in 1951, which is exactly 60 years ago. However, the same decade saw the book of Riesz and Nagy which used the <symmetric> vs <self-adjoint> distinction. So it just might be that Halmos is only an exception...I am sorry if my question annoyed you. In fact I am familiar with the mathematical terminology; I was under the impression some people in this thread use different terminology in the context of physics. By the way:
Like I said, for me hermitian and self-adjoint are synonyms (although I personally prefer self-adjoint as it says what it is). This claim about what "mathematicians use" is not true. I don't want to call myself a mathematician, but you probably won't argue about Paul Halmos.
No. Hermitian operators that are not self-adjoint don't qualify as observables in the conventional sense. For example, their spectrum is not real.So the dilemma is, self adjoint operators have nice mathematical properties, but in physics we have some observables which can only be Hermitian but not self-adjoint, so it's still lack of a classification that covers all physical observables meanwhile maintains nice mathematical properties, is it?
No, not really. Actually, one achieves more. Observable would normally mean 'self-adjoint', however a set of different operators such as position and momentum cannot be self-adjoint on the same domain of L^2(R). They can be self-adjoint on D(x) and D(p), but D(x) is different than D(p). They don't contain exactly the same vectors. The fundamental commutation relation [x,p]=1 holds only on a dense subset of L^2(R) which is neither D(p), nor D(x). It turns out that we actually have 2 different sets of operators: x,p which are self-adjoint on D(x) and D(p), respectively, and x',p' which are not. The first set doesn't obey the commutation relation, the second does. What's the relation between the 2 sets ? Well, x,p are the closures in the norm topology of x',p'. As x,p are self-adjoint, x',p' are esentially self-adjoint, which is the nicest feature one can expect from 2 symmetric operators obeying commutation relations, as the operators are being forced to share a common domain which is imposed to be dense-everywhere in L^2(R).So the dilemma is, self adjoint operators have nice mathematical properties, but in physics we have some observables which can only be Hermitian but not self-adjoint
It isn't. E.s.a. operators are in a way equally nice to the s.a. ones, because they almost have the same domain (which is anyway dense everywhere in the Hilbert space) and on the common domain they have the same range.QUOTE=kof9595995 said:, so it's still lack of a classification that covers all physical observables meanwhile maintains nice mathematical properties, is it?
Well, I would disagree. It is not only matter of a mathematical rigor.Hi kof9595995,
I would say it is important as a matter of mathematical clarity to understand the formal difference between hermitian/symmetric and self-adjoint.
Yes, and I think that in introductory qm courses it is quite OK to skip this.The distinction between hermitean, maximal hermitean and hypermaximal hermitean was often dropped in introductory qm texts.
I think you're missing my main point. I agree that if you want to use the continuum language, then you may resort to this kind of formal technology at various points.Well, I would disagree. It is not only matter of a mathematical rigor.
In this discussion there was no mention of the problem of finding an actual self adjoint operator. That is, one usualy starts with some symmetric (some authors call it hermitian, even it is not everywhere defined) operator and tries to find its self adjoint extension. There might by many of those, or none. These are then typically given by some particular boundary conditions and might describe different physical situation.
See for example the case of energy of 1d particle confined to a half line (i.e. with an infinite barrier somewhere). There is a 1-parameter family of self-adjoint operators given by the same "differential expression", but with different boundary condition imposed in its domain. This parameter describes the quality of the barrier. In fact it gives the elasticity of the barrier (or reflection coefficient).
For those who are interested i would recommend: http://ajp.aapt.org/resource/1/ajpias/v69/i3/p322_s1?isAuthorized=no [Broken]