🚀 go-pugleaf

In article <3794@oberon.USC.EDU> mathur@pollux.usc.edu (Samir Kumar Mathur) writes:
>Here is a thought experiment which seems to violate Uncertainity Principle (UP)
>that came up at a casual discussion with a friend.
>
>Scenario : 
>	There are two particles A and B and I am trying to measure their
>	momentum and positions accurately.
>
>Experiment :
>	(1) At time t:
>		(a) I measure the momentums p(A,t) & p(B,t) and hence
>			p(A&B,t) = p(A,t) + p(B,t) as accurately as I wish.
>
>	(2) At time t+dt:
>		(a) I measure the momentum of A, p(A,t+dt) as accurately
>			as I wish.
>		(b) I measure the position of B, q(B,t+dt) as accurately
>			as I wish.
>		(c) I calculate p(B,t+dt) = p(A&B,t) - p(A,t+dt) 
>			{conservation of momentum} as accurately as I wish.
>
>Conclusion:
>	From steps 2(b),(c) I conclude that I can measure p(B,t+dt),q(B,t+dt)  
>	as accurately as I wish. This obviously violates Heisenberg's UP.
>
>-----------------------------------------------------------------------------

A simple answer to this paradox could center around several points:
	1)	How repeatable are your results? 
	2)	Exactly how is your system configured?

	In answer to 1), you may be able to measure q(B,t+dt), but if
	you run several experiments, you'll find that the value of q
	is not predictable to within a factor of hbar/2/deltap where
	deltap is the uncertainty in p(A) and p(B).  If you adjust the
	experiment to improve the measurement of q, then the values of
	the p's will deteriorate (assuming your experiment is capable of
	measureing to the accuracies required to observe these).

	For point 2) it looks like you have two particles A and B which
	are interacting only with each other, and that they may exchange
	momentum between t and t+dt.  If this is so, how did you measure
	the momentum at t so that you knew what the momentum was after the
	measurement?  (I won't suggest that the act of measurement produces
	effects which can't be determined; limitations in measurements are
	inherent in the system, and should be determinable, but it is
	important to consider that your measurement techniques should be
	reflected in the conservation of momentum equation (have you
	added or changed momentum?)).

	Lastly, I'd like to point out that the Heisenberg Uncertainty
	Principle is mathematically equivalent to the restriction of the
	rate of transmission of information via radio.  If you send
	a signal with amplitude modulation (AM) or frequency modulation 
	(FM -- though this is a little more complicated) a pure tone
	(the carrier) contains no information.  When you modulate the
	carrier, you induce new frequencies.  If you want to localize a
	signal (information) in a smaller and smaller dt, then you
	need a wider and wider dfrequency (bandwidth of induced
	frequencies) to handle the information.  This is equivalent to
	trying to localize the location of a particle to dq; it requires
	a larger and larger bandwidth of dp's to handle the information
	required to define dq.  The reason for this is that in quantum
	mechanics, the p,q pair are 'conjugate' to each other in the
	same way that frequency,t are 'conjugate' to each other in
	radio transmission.  If you accept the DeBrogie hypothesis (which
	has been experimentally confirmed in any experiment dependant on
	quantum effects), then the Heisenberg principle follows.  It's
	built into the structure of QM.

	Hope this is a help.


	Dan