MadSci Network: Physics

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

Date: Mon Sep 20 09:49:18 2004
Posted By: Guy Beadie,
Area of science: Physics
ID: 1095094110.Ph

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

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 

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
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 
the links outlined at the end of this answer.

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-

Technicality concerning the ‘uncertainty’ relationship between E and t:

Also D. Shalitin, “On the time-energy uncertainty relation,” American 
Journal of Physics, vol 52, iss 12, p 1111, 1984, link (for those w/ 
access to library subscriptions) at:

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