### Re: Re: What are all of the known conjugate attributes of a quantum particle?

Date: Mon Sep 20 09:49:18 2004
Area of science: Physics
ID: 1095094110.Ph
Message:

Hi, Jeremy.

You’re asking a very interesting question – one that goes to the heart of
quantum mechanics.  To begin with, let’s define what ‘conjugate
attributes’ means.  First, I’m going to refer to ‘conjugate variables’
rather than attributes.  (A Google search under conjugate attributes turns
up philosophy sites, rather than physics sites.)  The generally-accepted
definition that I’m going to choose can be seen, for example, at
http://www.campusprogram.com/reference/en/wikipedia/uncertainty_
principle.html

and states that two variables are conjugates if their quantum operators
have a non-zero commutation relation.  That’s it – all variables with non-
commuting operators are conjugate variables.  Because there are so many
(an infinite number, actually) I won’t go into even a partial list here.
For any operator that you can think of, angular momentum, spin, intensity,
etc., any other operator that doesn’t commute with it forms a conjugate
pair.

I imagine that you associate conjugate variables with the uncertainty
principle.  There’s a good reason for that: all conjugate variables have
an associated uncertainty relation, with the minimum uncertainty related
to their commutator.  See the link at
ht
tp://scienceworld.wolfram.com/physics/UncertaintyPrinciple.html
for the proof.

There’s a technicality concerning the uncertainty relation between E and
t, since t is not a variable with an associated operator.  I leave that to

Lastly, if I understand your question, “Oh and also if you could tell me
what wave form represents the attribute” correctly, then you’re referring
to an eigenstate of the corresponding operator.  For example, the
eigenstate of the position operator is a delta function, while the
eigenstate of the momentum operator is a plane wave.

However, I feel compelled to point out that both of these are just the
spatial representations of these wavefunctions, and the power of the
wavefunction is that it isn’t confined to just one set of coordinates.
You can equally-well describe a wavefunction in momentum space, in which
case the eigenstate of the momentum operator is a delta function and that
of the position operator is a plane wave.  It’s the same thing, the same
eigenstate representing the same information, but viewed from a different
set of coordinates.

Good luck, and keep asking questions!

Good intro QM book:
================
My favorite: “Principles of Quantum Mechanics,” R. Shankar ISBN 0-306-
40397-8

Technicality concerning the ‘uncertainty’ relationship between E and t:
===================================================
http://math.ucr.edu/h
ome/baez/uncertainty.html

Also D. Shalitin, “On the time-energy uncertainty relation,” American
Journal of Physics, vol 52, iss 12, p 1111, 1984, link (for those w/
http://scitat
ion.aip.org/getpdf/servlet/GetPDFServlet?
filetype=pdf&id=AJPIAS000052000012001111000001&idtype=cvips

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