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Topic: Little Quantum Mechanics Question
Replies: 22   Last Post: Sep 22, 2003 12:50 AM

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Posts: 914
Registered: 12/6/04
Re: Little Quantum Mechanics Question
Posted: Sep 16, 2003 2:53 PM
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In article <bk75eq$eoi$1@bunyip.cc.uq.edu.au>, D.McAnally@i'm_a_gnu.uq.net.au (David McAnally) writes:
>meron@cars3.uchicago.edu writes:

>>>>>But, also, the postion operator, X , is an observable:
>>>>>X|x> = x|x>
>>>>>So one can expand |Psi> as:
>>>>>|Psi> = Integrate[ f(x)|x> , {x,0,L} ]
>>>In physics, maybe, but in math we tend to be more careful.

>>Yes, true. I admit that physicists tend to be kinda sloppy about
>>these things. Usually they can get away with this, but not always.
>>So, I plead guilty as charged.

>>>The position operator has a continuous spectrum. As George Jones
>>>observed, |x> is not a member of the Hilbert space, and X has
>>>_no_ eigenvectors, let alone an orthonormal basis of them.
>>>Mathematicians (or mathematical physicists) use various devices
>>>to deal with this, e.g. the "projection-valued measure" form of the
>>>Spectral Theorem.

>As I noted elsewhere, |x> actually DOES exist in the largest space of
>a Gelfand triple, the middle space of which is the Hilbert space.

Yes, only it doesn't exist within Hilbert space proper. In many
situations this distinction can be ignored. Sometimes, it cannot.

Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"

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