Heisenberg Principle and the Second Law of Thermodynamics

[Dennis Toy]

I was reading about the second law of thermodynamics and i read the rule that any isolated system becomes disordered. I was wondering does the Heisenberg Principle pertain to this or was it inspired by or based on this.

Now if you don't know, the heisenberg principle states that once a molecule is been measure, it has shifted position in the universe and cannot be measured. I was wondering is this because the molecule becomes disordered?

[Superboy]

I was wondering is this because the molecule becomes disordered?

No, it's because whatever device you use to measure the molecule also moves it.

[Surlethe]

Think about a beam of light. It has a wavelength, and an energy level, and the higher the energy, the smaller the wavelength. Now say you want to know where a proton is, and how fast it's going. The limit on how much you can know about its position is the wavelength of the beam of light, so you shine the light on the proton, and in order to get a good reading on its position, you make the wavelength smaller. However, this increases the energy of the beam, which, when it bounces off the proton, imparts some momentum to the proton, which means you don't know the momentum of the proton very accurately. Now, say you want to know the momentum of the proton pretty accurately; you have to give the proton a fairly low dose of energy, which means -- you guessed it -- a long wavelength. But this means you can't know the position as accurately.

That's the basic idea behind the uncertainty tradeoff.

[SpacedTeddyBear]

I think that the H.U.P pertains to the 2nd law, since the 2nd law is based on probability. If one takes a handful of coins, lay them out in a row, then find a way to toss them up and have them land in a row again, what would be the probability that the coins would land all heads? The same applies with an extremely large amount of molecules. In quantum mechanics, a free particle has an infinite number of allowed wave functions, since it may have an infinite "number" of allowed energies. If you were to measure either it's position or energy (or momentum), what would be the probability that you would find that free particle in that exact same spot again ( or with the same energy again) assuming that the next measurement is not made in an infintisimal (sp?) amount of time?

[kc8tbe]

The Heisenberg principle and Entropy are two different, independent concepts.

Suppose you have a chamber with three gas molecules in it. Suppose you conceptually divide the chamber into two imaginary halves. Suppose that all three gas molecules start out in one half of the chamber. Finally, suppose that this isolated system is not at absolute zero (the gas molecules are free to move).

At any given moment, what is the probability that all the gas molecules are in the half in the chamber they started in? The probability is .5*.5*.5*.5=one in sixteen. The probability that the gas molecules ended up in a more disordered state is therefore fifteen in sixteen. As you increase the number of gas molecules in the system, the probability of being in a more disordered state also increases. If you had one mole of gas then the probability of all of the gas molecules being in one half of the container is so small that you would not observe it over the current lifetime of the universe!

The Heisenberg uncertainty principle has to do with how precisely you can measure a quanton's position and momentum, as accurately described by Surlethe. The Heisenberg uncertainty principle is only related to the 2nd law in the sense that taking a measurement will increase the entropy of the universe (and probably your isolated system, too). But everything does that. Even picking your nose.

[Thinkmarble]

I always get a little stomach ache when talking abot probabilities in (classical) thermodynamics.
You have probability in quantum theory, which is "true" probability (an fundamental feature of or world), and you have probability in (classical)thermodynamics which expresses our ignorance of the system.

That is the gas molecules have indeed positions predetermined by their starting positions and equotations of motion, their is a probability of one for one specific configuration, and zero for all other, but we do not know which it is, so we use probabilities to describe our ignorance.
Entropy is just a measure about how many states our systems could be in, in agreement with our measurements.

The fun part is, that qt-thermodynamics looks much more natural, atleast to me (for a start the state counting and phase cell both arise naturally where as in c-thd they have to be inserted "by hand" )

And about Heisenberg, it is a general statement about non-commutating observables.
Simply put, the order of measurements matter, as when you measure observable A you are no longer in an eigenstate of observable B, and vice versa, so if you measure A first, then B, then A again you do not get the same result for A.
You can in fact measure momentum with an arbritary precision, and then you can measure position with an arbritary precision, it is just that after the position measurement your momentum measurement has become pretty useless (and Heisenberg tells you how useless).

*waiting for Kureneko to get corrected *

[Darth Wong]

You can illustrate entropy with an example: suppose you have a blue liquid and a red liquid in a tank, separated by a plastic divider sheet. When you pull out the divider, the two colours will tend to slowly run together over time, until you eventually get one uniform mixture. That's entropy in action; in a closed system there is no way it would ever go the opposite direction, and spontaneously separate from one big purple tank of liquid to the blue liquid being in one half of the tank and the red half liquid being in the other. It's about irreversible changes in a closed system, not just anything having to do with probabilities.

Of course, if it's not a closed system, then it's possible that such re-ordering could take place. If there's a slight difference in specific gravity then a centrifuge could re-segregate the liquids, or if there's some magnetic sensitivity you might be able to use powerful electromagnets, or if there's some chemical reaction you can exploit which only affects one of the two dyes, etc.

[Eframepilot]

Think about a beam of light. It has a wavelength, and an energy level, and the higher the energy, the smaller the wavelength. Now say you want to know where a proton is, and how fast it's going. The limit on how much you can know about its position is the wavelength of the beam of light, so you shine the light on the proton, and in order to get a good reading on its position, you make the wavelength smaller. However, this increases the energy of the beam, which, when it bounces off the proton, imparts some momentum to the proton, which means you don't know the momentum of the proton very accurately. Now, say you want to know the momentum of the proton pretty accurately; you have to give the proton a fairly low dose of energy, which means -- you guessed it -- a long wavelength. But this means you can't know the position as accurately.

That's the basic idea behind the uncertainty tradeoff.

Yeah. The more complicated idea of the uncertainty principle is that the momentum and position of a particle do not have a separate existence, at least in the Copenhagen interpretation. Because the position and momentum operators do not commute, it is impossible for the wavefunction of a particle to simultaneously be in an eigenstate of position and momentum, i.e. the particle simply cannot have a definite position and momentum at the same time.

This raises the question, "Does the particle actually have no separate position and momentum, or is this just some weirdness arising from the mathematics of quantum mechanics?" Some, including Einstein, believed the latter and said that quantum mechanics was not a complete theory of nature because of it. They said that there existed "hidden variables" of definite position and momentum that could not be observed but still existed, hidden by the limit of our measurement systems. However, it has been demonstrated by mathematics and experimentation (Bell's Inequality) that any theory incorporating hidden variables must also be nonlocal, which means it must contain unpleasant action at a distance, something that is undesirable in a theory of nature.

This is why Feyman said that nobody really understands quantum mechanics.

Fortunately, none of this is important to the operation of the Second Law of Thermodynamics, which would operate the same in a completely deterministic, Newtonian universe.

[drachefly]

Yeah. The more complicated idea of the uncertainty principle is that the momentum and position of a particle do not have a separate existence, at least in the Copenhagen interpretation.

or in Many Worlds, for that matter. It's just Bohm's Global Hidden Variables theory (the one you mention) that disagrees. Also note that the Copenhagen Interpretation is nonlocal -- when the wavefunction collapses on a separated entangled system (quantum teleportation), this collapse is superluminal. MW doesn't have this issue.

Also, it's not an accident that the Observer Effect (what Surlethe described) and the Heisenberg Uncertainty Principle provide the same constraint, but saying why would involve some mid-advanced calculus (hint for those who have such: it's about the Fourier Transform).

[Drooling Iguana]

You can illustrate entropy with an example: suppose you have a blue liquid and a red liquid in a tank, separated by a plastic divider sheet. When you pull out the divider, the two colours will tend to slowly run together over time, until you eventually get one uniform mixture. That's entropy in action; in a closed system there is no way it would ever go the opposite direction, and spontaneously separate from one big purple tank of liquid to the blue liquid being in one half of the tank and the red half liquid being in the other. It's about irreversible changes in a closed system, not just anything having to do with probabilities.

Of course, if it's not a closed system, then it's possible that such re-ordering could take place. If there's a slight difference in specific gravity then a centrifuge could re-segregate the liquids, or if there's some magnetic sensitivity you might be able to use powerful electromagnets, or if there's some chemical reaction you can exploit which only affects one of the two dyes, etc.

Doesn't that depend on a pretty arbitrary definition of "orderly"? It seems to me that in this situation the water is simply finding its own "order", based on things like the temperature of various portions of it and the chemical properties of the dye, rather than the order that was imposed on it earlier.

And you could take an opposite example, where you have oil dyed one colour and water dyed another, which have been mixed heavily so that they appear to be homogenous. If you leave that alone in a closed system the two liquids will seperate and become more "orderly."

Is there some accepted definition of "order" in physics that I'm missing, or something?

[drachefly]

There is an requirement for normal statistical mechanics, that the system is free to change over a range of states, and that each individual state within that range is equally likely. This applies to a great many systems.

So, take your system. Consider all of its possible configurations, under any set of constraints you feel like. How many of them are there?
Take the natural logarithm of this number -- that's the entropy.

Now, consider a system in which certain of these constraints can change -- the length scale of mixing of two fluids, for example (oil/water, or the dyes listed).
Now, list ALL the states, for all possible values of the constraint.

For the dyes, it works out that the states with a tiny tiny mixing length scale have almost all the states (the fraction of states that do not have this tiny mixing length is something like 10^-(10^26) ). For oil and water, it turns out to be the reverse. This difference is due to the microscopic behavior of the constituents.

So, as time goes on, the state will change. Simply under the influence of random changes and nothing else, it will wander into the constraint regime which has almost all the states. It could wander out, but it has an overwhelming probability not to.

[Kuroneko]

The fun part is, that qt-thermodynamics looks much more natural, atleast to me (for a start the state counting and phase cell both arise naturally where as in c-thd they have to be inserted "by hand" )

Just to elaborate on what Thinkmarble is getting at, the classical entropy approach is roughly as follows. The configuration of the system is described as a single point in its phase space. The relevant information could be position and velocity/momentum for each particle (this is in fact the most convenient form of representation, esp. from the point of view of Lagrangian or Hamiltonian formalism). Because the state of the entire system is represented by a single point in the phase space, it makes sense to ask which region in the phase space are compatible with the observed macroscopic properties (e.g., energy/temperature). Entropy is a measure of this.

When one counts microstates compatible with the observed macrostate, what is taken to be as the "same" microstate is somewhat arbitrary. One can partition the phase space into equal volumes, with each individual volume counting as the "same" microstate. The entropy is a logarithm of the number these volumes compatible with the macrostate. The size of these volumes is arbitrary, but that's not a problem since it makes the logarithm differ only by a constant, and we're really interested in entropy differences anyway.

In quantum mechanics, things are a bit better. There's a lowest possible meaningful phase space volume enforced by the uncertainty relation, so that one's choice doesn't have to be arbitrary. (The actual computations are different, but with the Planck constant, the classical treatment often gives the same result.)

Doesn't that depend on a pretty arbitrary definition of "orderly"?

Classically, sort of, since the choice of the phase space volume is arbitrary. But that's not what you have in mind.

It seems to me that in this situation the water is simply finding its own "order", based on things like the temperature of various portions of it and the chemical properties of the dye, rather than the order that was imposed on it earlier.

The interpetation of entropy with disorder doesn't really do it justice. Instead of molecules in vessel, let's have lights that may either be on or off independently at random. Say, one configuration is this: 1101001001. Compare with this: 1111100000. Intuitively, the former is more "disordered" than the latter. Physically, the phase space is the space is the space of all possible configurations, of which there are 2^10 = 1024. Both arrangements are equally likely, so that both have the exact volume in the phase space and thus the same entropy.

There is an requirement for normal statistical mechanics, that the system is free to change over a range of states, and that each individual state within that range is equally likely. This applies to a great many systems.

I suppose one could treat entropy with a general measure on the phase space and a probability density function, which would also circumvent the abitrary-volume problem, but I suspect that sort of thing would give Physics 101 students heart attacks.

[drachefly]

Physics 101 students don't learn Statistical Mechanics anyway.

I agree that the density approach is generally cleaner, because instead of having an irrelevant arbitrariness, it has none.

[Darth Wong]

Doesn't that depend on a pretty arbitrary definition of "orderly"? It seems to me that in this situation the water is simply finding its own "order", based on things like the temperature of various portions of it and the chemical properties of the dye, rather than the order that was imposed on it earlier.

And you could take an opposite example, where you have oil dyed one colour and water dyed another, which have been mixed heavily so that they appear to be homogenous. If you leave that alone in a closed system the two liquids will seperate and become more "orderly."

Is there some accepted definition of "order" in physics that I'm missing, or something?

Well you see, that's the problem with analogies, in that they are usually not precise.

If you so violently mix the oil and water together so that they appear homogeneous (although I'm not sure this is even possible under normal conditions), then you will create a lot of entropy. Over time the entropy level of the solution will drop, but of course, it's cooling off and it's not a closed system.

[drachefly]

If you violently mix the oil and water, as above, but close the system by putting it in an ideal thermos, then it will separate, but it will have a higher temperature. This higher temperature will increase the entropy more than the mixing would have.

[drachefly]

If you violently mix the oil and water, as above, but close the system by putting it in an ideal thermos, then it will separate, but it will have a higher temperature.
Though the energy was provided when you shook it, the temperature will rise as the solution separates as more of the energy becomes thermal instead of potential.
This higher temperature increases the entropy more than the loss of mixing decreases it.