QM and Classical System Coupled

Joschua_S

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

Ken G

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.

Joschua_S

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

Ken G

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 mind-bending 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.

lugita15

As Ken G mentioned, look up "deocherence".

A. Neumaier

The paper
http://www.chem.utoronto.ca/~rkapral/Papers/mqc.pdf
answers your question. From the abstract:

''Mixed quantum-classical equations of motion are derived for a quantum subsystem of light mass m particles coupled to a classical bath of massive ͑mass M particles.''

Ken G

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).

A. Neumaier

If the environment consists of different particles than the single molecule, the situation is similar, even if the molecule is as heavy or heavier than the particles in the environment.

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 1-particle 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.

Ken G

That cannot be true about that paper, because it uses an expansion in a "small parameter" relating to the mass ratio. If the mass ratio is unity, their approach does not achieve its objective.
Yes, indistinguishability would be an interesting wrinkle to an exercise like that, it would interesting if it mattered in some way as to how that "cut" comes about!
If they are looking for a mathematical answer, then yes, that would be a good way to go. If they are looking for a physical description of what is going on, or if they don't have that text handy, then other kinds of answers could be useful, perhaps along the lines of those given above.

A. Neumaier

I was speaking qualitatively, assuming the OP was interested in general in how to set up a quantum-classical framework. The details of how they proceed is of course different depending on the mass ratio, and if the mass ratio is one then one needs a different small parameter. (1/N woud do.)

For a treatment of a single electron in water, see
http://jcp.aip.org/resource/1/jcpsa6/v116/i19/p8418_s1
As one considers the average over all other particles with one particle fixed, it doesn't matter, as every particle gives the same result due to indistinguishability.
Any good textbook on statistical mechanics should have a discussion of reduced density operators. If you think my answer should be expanded, why don't you look it up and then write something?

Ken G

Yet one could argue that such a mindset is fundamentally inconsistent with the concept of indistinguishability. One cannot average over the "other particles" because there is no such "otherness" in the first place-- one can only pretend it and hope for the best. We often do that in quantum mechanics, out of necessity, but it's not formally correct, it violates the postulates without additional assumptions that may or may not be true in general.
My goal was not to expand your answer. I have no intentions on your answer at all. I was pointing out that there are answers from a mathematical perspective, and there are answers that have a more descriptive flavor. I'm sure your answer involving reduced density operators is not intended to be dismissive of answers that might (or might not) more directly relate to the issues the OPer has in mind. Thus multiple answers are often of value.

questionpost

With classical systems, wave functions come to a near zero, so you will not see as much statistical randomness at all, but if you have something small like a molecule, you can't really keep track of where that individual molecule will be especially in a substance with many other identical molecules, but you can assume that no matter where it is that it's physics will be the same, and that it's not likely to disappear unless your in some kind of vacuum, so you can kind of predict what will happen with it. Although just a single molecule in reality is something you can't keep track of at all unless you have a very isolated system and even then you just see random points popping up.

A. Neumaier

It can be done and is formally correct. Indeed, it is done in all textbooks on statistical mechanics, andis the basis of deriving nonequilibrium thermodynamics from molecular qnautm mechnaics.

Ken G

Well, if it is "formally correct," then it is not averaging over "other electrons" but rather over other degrees of freedom than the ones we are associating with the observations we are (formally incorrectly) describing as relating to "this electron." But even if we do frame it as averaging over "other electrons", we can probably get away with it, as a kind of approximation. Physics is about results, not formalities, but we must walk that line in a consistent way. If a question is framed from the perspective of "what quantum mechanics says" rather than "what works in practice", the answer can be quite different.

A. Neumaier

In explaining formal things on an informal level (as often here on PF), there is a little more freedom than you'd like to accept.

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.

Ken G

Actually, I can accept it fine. I felt you were the one who was standing on formality. But here the question is about separating the difference between "this water" and "that water", so we must be careful about that-- an issue that you brought up yourself. I am not objecting to the answer you gave, it's a very good answer. I'm saying it's not "the" answer. Answers stem from the assumptions we make.