Talk:Position operator

Untitled

There needs to be a discussion of the "classical equivalent" to the Q.M. position operator. Also would be nice if someone would explain what it means for the position operator to act on a wave-function (i.e. R|ψ> or <ψ|R|ψ> where R is the position operator). — Preceding unsigned comment added by 144.32.53.52 (talk) 12:40, 30 June 2011 (UTC)

First equation

Shouldn't it be Q(ψ(x)) instead of Q(ψ)(x)?

it's a matter of which notation one prefers. in the current version, the image of ψ under Q is denoted by Q(ψ). so Q(ψ)(x) specifies Q(ψ) as a function, the value Q(ψ) takes at x. Mct mht 00:26, 5 January 2007 (UTC)

I changed it back to Q(ψ)(x). Q operates on ψ, not x; Q(ψ(x)) implies that Q is a function of x and that Q(ψ(x)) is the composition of Q and ψ acting on x; ψ(x) is a number, not a function, ψ is the function. Phoenix1177 (talk) 03:44, 18 July 2010 (UTC)

Notation

Isn't the position operator more widley known as

$\mathbf {\hat {r}} \psi =\mathbf {r} \psi$

(simply and elagantly) instead of all the weirdy notation with - Q(ψ(x)), Q(ψ)(x), or even Q or WTF jibber is currently in the article? the notation I just provided is far more common in more elementary books and more recognizable to a wider audeince when they see other QM operators. I can't find any referances for the current notation. I'm not, in any way, shape or form saying the above is wrong, only strange to use Q when in QM usually r is used (I guess this is an implication of generalized coordinates...).

If no one objects within (say) a couple of weeks - i'll tweak it to the hatted notation. -- F = q(E + v × B) 13:38, 27 December 2011 (UTC)

I agree, it seems like most texts use either $\mathbf {\hat {r}}$ or $\mathbf {\hat {x}}$, although we should still mention the other (older?) notation. a13ean (talk) 03:24, 28 December 2011 (UTC)

Though it's all good to use more common notation, the problem with the above notation is that the operater $\mathbf {\hat {r}}$ goes from a dense subspace of $L^{2}$ into $L^{2}$. As such, $\mathbf {\hat {r}} \psi$ should be a function in $L^{2}$. The correct way to write the action of $\mathbf {\hat {r}}$ is thus

$(\mathbf {\hat {r}} \psi )(\mathbf {r} )=\mathbf {r} \psi (\mathbf {r} ).$

I know that physics textbooks often supress the $(\mathbf {r} )$ part, but it really doesn't make any sense to multiply $\psi$ with $\mathbf {r}$ without specifying what $\mathbf {r}$ is. If you supress the $(\mathbf {r} )$ part, the reader actually has to know beforehand what the position operator is if he wants to make sense of the definition. Notice how the notation is now the same as the one used in the article, other than a renaming of the operator. Isdatmaths (talk) 12:37, 10 March 2012 (UTC)

Confused article

Article needs to keep what's rigorous, meant for the mathematical audience, and what's not, as in a typical introductory physics text, straight. Mixing the two is bad. Mathematically, the position operator as an operator on L^2(R) has no eigenvalues and measurement is treated using a projection valued measure. This is standard for observables/self-adjoint operators (via the spectral theorem, although the position operator is a very simple case and the spectral theorem is not needed). Physics literature, e.g. Sakurai, plays loose with the notion of spectrum. This is incorrect from a mathematical point of view but generations of physicist have been trained this way and it seems to suffice for what they need. In any case, it is bad to speak of "eigenvalues" in the intro when it's pointed out in the next section that the operator has no eigenvalues.

One way to fix this is present the physics formulation first (as most readers are probably from physics) then give the mathematical formulation and the necessary corrections. Mct mht (talk) 09:45, 24 August 2012 (UTC)

Reorganization

I have moved the old "Statement" section lower in the article and renamed it "formalism" and have also written a more accessible introduction in line with the suggestions in the above two sections, namely "Notation" by User:F=q(E+v^B) and "Confused article" by User:Mct mht. Please discuss the change here if you have any comments or objections to it. In my opinion, the last two sections still need to be slightly edited to fit in better as well as integrate what User:Mct mht said about rigour.

Xuanji (talk) 15:25, 25 August 2012 (UTC)

Problem with dimension greater than 1

There is a problem in the definitions given here when dimension is greater than 1: the d-dimensional position operator is not an endomorphism of the Hilbert space of states, so it cannot be self-adjoint, and its spectrum is not defined with the standard definition. Of course we can modify the definitions, and define some kind of eigenvalue for operators between different Hilbert spaces by fixing some additional natural morphisms but it is not entirely obvious how to do it in general -especially when one tries with other "quantities" than the d-position that should be observables.

I am not going to start a general discussion here of the formalism of quantum mechanics, but it seems to me that requiring that observables be self-adjoint operators is too restrictive. I imagine that some people have already generalized that notion, but that has not made it into the mainstream formalism. If one thinks of the algebraic side of it, i don't think that observables should form an algebra, but a more generalized notion which allows for a composition/multiplication that is only partially defined -depending on compatibility of codomain on the right with domain on the left. Plm203 (talk) 03:18, 16 June 2023 (UTC)