Message from discussion
eigenstates of position
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From: George Jones <george_llew_jo...@yahoo.com>
Newsgroups: sci.physics
Subject: Re: eigenstates of position
Date: Tue, 16 Apr 2002 10:22:34 -0400
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"Gregory L. Hansen" wrote:
> In article <3CB82464.736AA...@yahoo.com>,
> George Jones <george_llew_jo...@yahoo.com> wrote:
> >franz heymann wrote:
> >
> >> <me...@cars3.uchicago.edu> wrote in message
> >> news:lzIt8.149$S4.9495@news.uchicago.edu...
>
> >> Oh?
> >> Use the momentum eigenstates to define a Hilbert space. Then the delta
> >> function eigenstate of position is, putting it crudely, the state with
> >> equal components from all possible momenta.
> >
> >It is true that delta functions and plane waves are (in a distributional
> >sense) Fourier transforms of each other, but neither delta functions nor
> >plane waves live in Hilbert space. Also, delta functions and plane waves
> >are "idealized" states of a system, and are not physical states of a
> >system.
>
> But have uncounted students been working along in the mistaken belief that
> there are position and momentum eigenstates, or have they been working
> along in the mistaken belief that they're using a Hilbert space?
Both of these positions have rigourous math versions that give the same
results, but the rigorous version of second position captures more of
the spirit of what physicists actually do.
Why, then, have almost all physics students been told that the rigourous
math underlying the formalism of quantum theory is based solely on
Hilbert spaces? I think the reasons are historical.
Dirac's 1926 Ph.D. thesis began the synthesis and formalization of
quantum theory that became the intellectual tour de force "The
Principles of Quantum Mechanics." This work was very beautiful, formal,
and general, but it was not very rigorous.
In the early 1930s, von Neumann came up a rigorous Hilbert space
formalism for quantum theory, but this formalism did not include states
like delta functions or plane waves. At this time there was no
mathematical theory of delta functions. Physicists then started telling
their students that the rigourous verion of quantum theory was based
on Hilbert spaces, and went on calculating things using Dirac's
formalism, which looked and felt a lot different than von Neumann's
formalism. Probably many physicists didn't realize that there was a
difference, apart from rigour.
In the 1930s, 40s, and 50s a rigourous general theory of generalized
functions (or distributions) that included delta functions was deveoped,
mainly L. Schwartz in the West and I.M. Gelfand in the USSR. This theory
of generalized functions was applied to quantum theory in 1960s and
1970s, resulting in a rigourous rigged Hilbert space (also called a
Gelfand triple) version of Dirac's formalism.
This rigourous version of Dirac's formalism has gone largely (though not
completely) unnoticed. There are at least a couple of reasons for this.
After von Neumann, profs told their students about Hilbert spaces; some
of these students became profs and told their students about Hilbert
spaces; etc. This went on for about 40 years before the other version
was introduced. After this, few people cared because they thought things
had been worked out long ago, and because this stuff doesn't often affect
day to day calculations.
Another reason is that the mathematics of rigged Hilbert spaces is
taught by math departments far less often than the mathematics of
Hilbert spaces. For example, Hilbert space theory was included in a
functional analysis course that I took, but no course that covered
rigged Hilbert spaces was available for me to take.
I think if the rigged Hilbert space version of quantum theory had come
along before the Hilbert space version of quantum theory, then the
Hilbert space version would might today be even less well-known than the
rigged Hilbert space version actually is. Students would now be hearing
vague mutterings about "making things rigourous with Gelfand triples"
instead of hearing vague mutterings about "making things rigourous with
Hilbert spaces."
To me, all this illustrates the brilliance of Dirac. It took
mathematicians 30 years (or more) to catch up with him!
Einstein had a couple of interesting things to say about Dirac. About
Dirac's quantum mechanics: "Dirac, to whom we owe the most logically
perfect presentation of mechanics ..." About the genius of Dirac: "I
have trouble with Dirac. This balancing on the dizzying path between
genius and madness is awful."
Regards,
George
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