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QM and Classical System Coupled 
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#1
Mar612, 09:55 AM

P: 11

Hi
Consider a small system A which is described by quantum mechanics. A large system B is surrounding A and this large system is described by classical physics. What kind of interactions has the system B to the small qm system? Compared to B is A very small so I guess one can neglect the effects from A to B but the large system interacts in a dominant way to the small one. Greetings 


#2
Mar612, 10:05 AM

PF Gold
P: 3,085

That exact scenario is more or less the empirical basis of quantum mechanics, where A is the measuring apparatus and B is the system being understood. We use our understanding of how A works, empirically, to obtain predictive power over B. All the ways that B affects A (or more correctly, all the zillions of little Bs in A) is already factored into our understanding of how A works (like how to make a measurement and what we are measuring if we do it in different ways). That makes it hard to use A to study how B affects A, but we get away with it because the effects of B on A (which are not small, B influences the outcome that A is registering) seem to follow patterns that we can understand under the general term "measurement." If that seems a bit vague, welcome to empiricism! Can't live with it, can't live without it.



#3
Mar612, 10:46 AM

P: 11

Hi
Thanks for the quick answer. I think about a little molecule surrounded by water and the molecule is described by quantum mechanics and the water with classical physics. What interactions do the water have on the molecule? Greetings 


#4
Mar612, 01:48 PM

PF Gold
P: 3,085

QM and Classical System Coupled
The water "measures" the molecule, in the sense that it monkeys with the phase of the molecular quantum wave in ways that are too diffucult for us to track (since we are treating the water classically). This "monkeying" has the effect of "decohering" the molecule's phase, which in turn takes the molecule from its "pure" originally prepared state and puts it into a "mixed" state where the different possible states of the molecule no longer interfere with each other in any experiment on the molecule (to recover the interference we'd have to include the entire state of the water in ways that we just can't do). When this holds, we take the philosophical stance that the "molecule is in one state or another, but we just don't know which, prior to measuring it ourselves." Hence interactions with the water, when the water is treated classically, act like a kind of intermediate measurement on the molecule, even when we don't know its outcome until we do a real measurement.
Of course, all of this is predicated on treating the water classically, which is indeed the practical thing to do. But where you get into all the mindbending different interpretations of what is "really going on" there is when you require that even the water be a quantum mechanical system, we are just not choosing to treat it that way for practical reasons. In Bohr's view, we have no choice the water must be treated classically, so it is classical, there's just no difference there. Less empiricist approaches hold that the water is "really" a quantum mechanical system, although I would argue (in agreement with Bohr) that this is a classic example of mistaking the map for the territory. 


#5
Mar612, 01:54 PM

P: 1,583

As Ken G mentioned, look up "deocherence".



#6
Mar812, 07:11 AM

Sci Advisor
P: 1,943

http://www.chem.utoronto.ca/~rkapral/Papers/mqc.pdf answers your question. From the abstract: ''Mixed quantumclassical equations of motion are derived for a quantum subsystem of light mass m particles coupled to a classical bath of massive ͑mass M particles.'' 


#7
Mar812, 08:28 AM

PF Gold
P: 3,085

Not quite that paper requires that the bath molecules have much larger mass than the quantum particles, whereas I get the sense the OP is interested in just more water inside of water. In other words, the paper seems to focus on situations where the "cut" is imposed by a greater "classicalness" of the individual particles in the bath, rather than by the sheer weight of numbers of the particles in the bath. It's still clearly an interesting and relevant reference though! It would seem the next step is to try to generalize that type of approach for systems of large numbers of molecules rather than high mass molecules, but it will require an approach that doesn't expand in a small parameter (or finds some other clever way to do that).



#8
Mar812, 08:51 AM

Sci Advisor
P: 1,943

But a water molecule surrounded by water is different as the different water molecules are indistinguishable. (A practical question is: How does one remember which molecule was singled out, as all are indistinguishable?) In any case, the answer is here given by the 1particle reduced density operator of statistical mechanics (if the water molecule is considered as asingle particle). See. e.g., Chapter 7F of Reichl's book on statistical Physics. 


#9
Mar812, 09:02 AM

PF Gold
P: 3,085




#10
Mar812, 09:55 AM

Sci Advisor
P: 1,943

For a treatment of a single electron in water, see http://jcp.aip.org/resource/1/jcpsa6/v116/i19/p8418_s1 


#11
Mar912, 09:02 PM

PF Gold
P: 3,085




#12
Mar912, 09:17 PM

P: 198




#13
Mar1112, 03:25 AM

Sci Advisor
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#14
Mar1112, 10:31 AM

PF Gold
P: 3,085




#15
Mar1212, 06:46 AM

Sci Advisor
P: 1,943

In particular, although all electrons in the universe are indistinguishable, people routinely talk about ''this'' electron if it figures in their experiment, without running ever into inconsistencies. The label 'electron' applies on many levels, not only at the most fundamental ones, where indistinguishability reigns. 


#16
Mar1212, 10:15 AM

PF Gold
P: 3,085




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