On Quantum Mechanics : Stochastic is Not the Opposite of Deterministic

Most of the misconceptions about and misuses of quantum mechanics (QM) comes a misunderstanding of the stochastic nature of QM. In the 20th century, as we came to understand that QM is how the world works on a fundamental level, we had to abandon the idea that physics was in principle deterministic. Alas, too many have read too much into this abandonment. Recently, one of the typical creationist nutters was commenting on the Gonzales tenure denial, and had this to say:

Now, materialism is shot to pieces anyway, and has been ever since quantum mechanics began to be understood.

Er, no.

But, before that, we have had all sorts of mystics putting out stuff like the movie What the <Bleep> Do We Know, which took the non-deterministic nature of Quantum Mechanics to mean that we, somehow, get to decide the states that elementary particles will fall into.

Er, no.

Now, you might be excused for thinking that "stochastic" is the opposite of "deterministic" if you, for example, use the really cool Unix command-line utility "dict" (here is a web interface to the same program) to get a definition of "stochastic" off of one of the dictionary servers on the net. Among other things, you will see this:

2. random; chance; involving probability; opposite of deterministic

Hmm. However, stochastic does not mean simply "not deterministic." I shall attempt to elucidate.

The Clockwork Universe

Before QM was developed in the early 20th century, our understanding of physics was that the Universe was completely deterministic. The classical laws of physics tell us that if we know perfectly the position, velocity, rotation, mass, shape, and electric charge of every particle in a closed system (that is, there is nothing outside the system influencing it; the Universe would be an example), you could thereafter perfectly predict the motion of every particle for the rest of time. This is what is meant by the deterministic nature of Classical physics.

In practice, there are two caveats to this. First, we never perfectly know the position and velocity of everything simply because there is always limited precision to any measurement we make. However, this does not undermine the in principle deterministic nature of Classical Physics. Second, there is this thing called "chaos theory," which has shown us that in a "non-linear" (i.e., suitably complicated, and it doesn't take much) system, even infinitesimally small deviations in initial conditions can lead to macroscopic differences in the state of a system later on. So, even though we may have measured the positions and velocities of all particles to 10 decimal places, our uncertainty in the 11th decimal place will, in some systems, lead to uncertainties in our predictions about the first decimal place some time later. Again, though, this doesn't undermine the in principle deterministic nature of Classical Physics. The equations of Classical Physics say "if this particle is like this right now, it will be exactly like that at such and such later time."

The Stochastic Universe

Quantum mechanics throws a wrench in all that. The equations of QM tell us how particles and systems evolve through time, just like the equations of classical physics. However, the equations are very different. Instead of telling us where a particle will be, it tells us the probabilities of where a particle will be later, given the current state of the system. But, beyond that, there is this little thing called the Uncertainty Principle which tells us that we cannot, not even in principle, know the position and velocity of a particle perfectly at the same time. In practice, we can't do that even in classical physics, but in classical physics we can do that in principle. In QM, there is a fundamental limit from the laws of physics on how well we can know certain quantities in tandem with certain other quantities.

One of the neatest examples of this is the "Stern-Gerlach" experiment. In this experiment, you send particles— they used silver atoms, but I'll assume we're using electrons for the sake of simplicity— through a magnetic field. According to quantum mechanics, the spin (i.e. the intrinsic angular momentum) of an electron in a given direction (let's say the up/down direction for right now) always has the value 5.27×10-35 m2kg/s. It can have that value either pointing in the "up" direction (which would correspond to a counter-clockwise rotation as viewed from above) or the "down" (clockwise) direction. When you send an electron through the magnetic field of the Stern-Gerlach apparatus, it's path will be deflected either upwards or downwards based on the direction of its spin. As such, if you send a beam of electrons through a Stern-Gerlach apparatus, it will cleanly divide into two separate beams; the upper beam is all the electrons with their spin pointing one way, the lower beam is all the electrons with the opposite spin. No electrons are anywhere in between, because every electron must have exactly the same quantity of spin in either the up or down direction.

i-8fc9671b44b948b7e807298c3d5d84c8-sg1.png

In the picture above, electrons of unknown spin come into the Stern-Gerlach magnets from the left. Two beams come out. One beam has those electrons who have their spin along the vertical axis pointing up, the other has those electrons whose spin along the vertical axis points down.

A body can rotate in three independent directions. You can think of this as rotations around the X, Y, and Z axis. Something can be rotating counter-clockwise as viewed from above (that would be spin "up"), and also counter-clockwise as viewed from the left (that would be spin "left"). Let us now suppose that we block out the beam of spin down electrons, and measure the left/right spin of the beam of spin up electrons. Since we haven't yet measured the left/right spin, unsurprisingly the spin up electrons are a mix of spin left and spin right electrons:

i-8eaf9ac180dafc6ac83dbc4a7f4d1bd1-sg2.png

No surprise there. Deterministic physics and QM predict the same thing. Here's where things get interesting, though. Take the beam that was all spin up electrons, and which has been sent through the left/right divider. Block out the spin right electrons, and just keep the spin left electrons. Run them through another up/down measurer. What do you expect?

If physics is purely deterministic, given that the left/right and up/down spin axes are independent directions of spin, you'd expect the beam to come out all spin up. After all, we've measured all of these electrons to be spin up before, so they should still be that way. In quantum mechanics, however. it turns out that spin along different axes are orthogonal quantities, which means you cannot know both of them at the same time. Just as you can't know the position and velocity of a particle perfectly at the same time, you can't know the up/down and left/right spin of a particle at the same time. We know that all of the particles in the "spin left" beam have left/right spin to the left... so we can't know their up/down spin. The result we get is this:

i-00985dcc5241bc0e22d30d84074c2d39-sg3.png

Determinism is out the window. If you know the particle has spin in the left direction, you cannot know the spin in the up/down direction even if you have measured the up/down spin before. All QM lets you calculate is the probability of measuring an up spin or a down spin; it turns out that there is a 50% chance for each.

The same holds for all quantum mechanical calculations. It deals with the propagation of probability densities, not with the propagation of the positions, velocities, rotations, and so forth of particles.

You fire an electron of known speed at a wall. After you've given enough time for it to bounce back, where do you find it? QM doesn't give you a single answer. It gives you the probabilities of finding it a number of different places. Most likely it's bounced straight back and is at a detector you have right next to where you fired it from. However, there is some small probability that it's somewhere else; indeed, there is some small probability that it went right through the wall!

Stochastic Is Not the Opposite of Deterministic

Clearly the two pictures— the deterministic picture of classical physics and the stochastic picture of QM— are completely different. However, it is a vast misinterpretation of QM to take its stochastic nature as evidence that reality works in any other sort of non-deterministic way you can think of.

To address the two points above: QM does not, at all, take away the materialistic nature of science. Just because the direction an electron goes isn't uniquely specified by the laws of physics, that does not mean that we need to invoke some sort of "higher power" that decides which way each electron goes! Quantum mechanics is deterministic about the probabilities. Each electron that goes all the way through the experiment above has a 1/2 chance of being spin up at the end, a 1/2 chance of being spin down at the end. QM is unambiguous about that, and these are just the cold, unfeeling, materialistic laws of physics at work.

Second, the fact that the laws of physics do not uniquely determine the final measured state of a system does not mean that we get to decide that state. This is the mistake that What the <Bleep> Do We Know?, and much other QM-inspired mysticism, makes. Yeah, physics is non-deterministic. But, once again, it is deterministic in the probabilities. The probabilities are what they are, no matter what we want, no matter what we visualize, no matter what we meditate upon while imbibing inspirational substances. It doesn't matter how much you want that electron to be spin up, it doesn't matter how many good "vibes" you give the electron about being spin up, it always has exactly a 1/2 chance of being spin up, a 1/2 chance of being spin down.

QM is stochastic. Determinism is out the window, but that doesn't mean that every damn thing you can think of is in the window.

Tags

More like this

I was looking at some polling about science over the weekend, and discovered that they helpfully provide an online quiz consisting of the factual questions asked of the general public as part of the survey. Amusingly, one of them is actually more difficult to answer correctly if you know a lot…
I got way behind on my reports from my Modern Physics class-- the last one was over month ago, and the class has since ended. There's enough material left to be really awkward as a single post, though, so I'm going to take my cue from Brandon Sanderson and split it into three parts. The remaining…
There have been some good comments on last week's post about the Many-Worlds Interpretation, which I find a little surprising, as it was thrown together very quickly and kind of rant-y on my part, because I was annoyed by the tone of the original Phillip Ball article. (His follow-up hasn't helped…
Let's say you have a thousand of your friends in a large field, and every one of them has exactly two coins - one penny and one dime - in his pocket. Or her pocket, as the case may be. No exceptions, they all have those and only those two coins. Now imagine nearly the same situation with a…

"QM is stochastic. Determinism is out the window, but that doesn't mean that every damn thing you can think of is in the window."

Stochastic may not be the "opposite" of deterministic, but it certainly means "not deterministic" as your final sentence suggests. Creationist claptrap aside (and the question of materialism has no bearing on deterministic or stochastic, I agree), to say something is deterministic is not the same as to say it is stochastic. Chaotic dynamical systems are a mathematical concept, not a physical one (and are much more than sensitive dependence). So your objection to a typical creationist distortion is correct, it seems you might have stated it a wee bit strongly. No?

This was a nice post, but I find your choice of terminology a bit confusing. In my language, stochastic is the opposite of deterministic, or at least it is contradictory with it. It is true, as you say, that QM is deterministic about the probabilities, but that is very diferent from being deterministic, period. It is a lawful indeterminism, one that allows predictions, one consistent with materialism, physicalism and all the things that terrify tender-hearted creationists and mysticists... but an indeterminism nonetheless.

To put it diferently, the way I would refute the creationist mistake is by explaining that materialism is not commited to determinism, that their association is an old-fashioned leftover from the Newtonian period, and that our best scientific worldview includes both materialism and a lawful kind of indeterminism that allows prediction of probabilities. This is the same that you try to do, but keeping the normal terminology, which both Revere and I understand as equating "stochastic" with "non-deterministic".

I'm ashamed that I don't understand this better, being a physics major and all. I get that the total number of electrons that go up and the total number of electrons that go down must be consistent with the initial angular momentum of the system. For example, say you fire a million electrons, with a total angular momentum of zero, and the first 500,000 of them all go up. Then, without measuring the rest of the electrons, you know the rest of them will go down. In a sense, when each electron goes either up or down, that result is "broadcast" instantly to the rest of the universe. (and no, you can not use this for faster-than-light communication, for reasons too complicated to explain) In other words, if you know the total angular momentum of the system in advance, you know exactly how many of the electrons will go up and how many will go down, even if you don't know which electrons go up and which go down (and that doesn't really matter because all electrons are identical). More generally, we can't observe QM with our eyes because the probabilities always average out over time.

Now, is it possible a construct a system such that the total angular momentum of the electrons is probabilistic? In this case, it would be an unknown how many electrons go up and how many go down. Now, let's say two scientists gamble their life savings over whether more electrons go up or down. A creationist would state that God might like one scientist better than another, and being omnipotent, would be able to "nudge" a certain number of electrons up or down. If God did this, we would never, ever figure it out. I believe this is the Creationist argument of "QM proves God!" and is really difficult to refute in terms a non-scientist would understand.

Did I get my facts straight, here? And what exactly is the best way to refute someone who makes that claim?

Excellent post. Many postmodernists as well as creationists would do well to absorb this lesson, I think. ;-)

I'm greatly looking forward to your lecture at Hypericon this year.

I'm with revere here, as QM combines discrete and continous states seamlessly, it also combines stochasticity with determinism as you describe.

A creationist would state that God might like one scientist better than another, and being omnipotent, would be able to "nudge" a certain number of electrons up or down. If God did this, we would never, ever figure it out.

The problem with that is that AFAIK we could presumably figure it out if it were a localized attempt of inserting a correlation, by the Bell test or worse. (I suspect it would be rather visible at the border of the affected volume.)

The only way it would be unmeasurable would be if the creationist gods would be faking our physics wholesale, I think.

By Torbjörn Larsson, OM (not verified) on 08 Jun 2007 #permalink

The Stern Gerlach thing reminds me of the surprising properties of classically polarized light. Take two polarizing filters, one vertically oriented, and one horizontally, and all the photons are absorbed. Now into the middle of this perfect photon barrier insert a third polarizer, if it's orientation is not parallel to one of the other two it allows some photons to transit the system. Very mysterious to most people.

The way that QM mysteriousness has been used in entertainment only feeds into it's misuse as a tool of theists.

but it certainly means "not deterministic" as your final sentence suggests.

I disagree.

Stochastic means probabilistic. It is not deterministic-- but there are other things that are not deterministic. "Done at my whim" is also not deterministic -- there's no fundamental law that you can apply to figure out what will happen -- but it's also not purely probabilistic. It's Robalistic.

The misinterpretations of quantum mechanics come from taking stochastic to mean "not deterministic," and then projecting into that anything, however batty, that isn't deterministic. Thus, the reason I think that we should be more careful with language, and the reason that I think it incorrect or, at best, misleading to state that stochastic is the opposite of deterministic.

-Rob

Much of what Rob says in his discussion can also be found in an article by Ernest Nagel, "The Causal Character of Modern Physical Theory," published more than 50 years ago. Some lessons take a long time to "sink in."

By bob koepp (not verified) on 09 Jun 2007 #permalink

Schwinger, in the prologue to his quantum mechanics book, puts it this way:

By causal, one means...that when the state of the system is given at a particular time...then the state is completely determined at any other time. ...What makes it deterministic is that the knowledge of the state all determines all possible physical phenomenon precisely.

This distinction between causal and deterministic may not seem very important until we come up to the rather different situation that appears in quantum mechanics...

So the key point is that quantum mechanics is causal, but not deterministic. Semantics, yes, but if two people are disagreeing on something like this, there's probably an instructive hair to split.

Brandon, angular momentum has nothing to do with it. The standard source for Stern-Gerlach style experiments is an ion with a net spin 1/2 in a furnace, allowed to escape from a pinhole. From statistical mechanics, we get a uniform distribution of the spin states coming out, and these states are totally uncorrelated from one ion to another.

I personally like to look at this a different way: the silver ion comes out with some wave function, including spin. Go ahead and think of it as a wave, not a particle. You don't lose anything, and the mathematics is the same. The particle reaches the neighborhood of the magnet, and its magnetic moment (spin) interacts with the electrons of the magnet by exchanging a photon. Photons are just another kind of quanta, so if we take the ion and the magnet as boundary conditions, we get some finite number of allowable states for the photon. The ion/magnet system excites one of these photons out of the vacuum. Photons, however, must carry momentum, and the photon states are all symmetric perpendicular to the line through the magnet. Thus its momentum vector is in that plane, and it will be either up or down. Since the photon's momentum has to come from somewhere (when it is emitted) and go somewhere (when it is absorbed), at some point the ion is giving or receiving momentum, which causes the deflection.

This is the sketch. There's more detail as you go from the cartoon system to any real, experimental apparatus.

Fred -- indeed, if you take the viewpoint you sketch and continue to sketch, it's in many ways a much more accurate description of quantum mechanics than the one I've sketched.

The waves are the "probabilities" I've talked about, although really they're probability *densities*, and really really they aren't even that, they're something from which you can straightforwardly calculate probability densities. (That latter distinction does make difference when it comes to things like interference.)

Needless to say I haven't read Schwinger's book :)

Causal is a good word to describe it. It's something that both classical and quantum physics share.

To be clear, I was never trying to assert that quantum mechanics was deterministic; I was merely trying to assert that stochastic has a rigorous and materialistic definition, and that it's different from "anything I can think of that isn't deterministic."

-Rob

"Clearly the two pictures the deterministic picture of classical physics and the stochastic picture of QM are completely different. However, it is a vast misinterpretation of QM to take its stochastic nature as evidence that reality works in any other sort of non-deterministic way you can think of."
I get you and make the same precise KIND of point in dealing with poststructuralist vis-a-vis Enlightenment rationism. It's not that, with the rigorous critique of Modernity and its assumptions (e.g. Newtonian science), now, "anything goes." It's that the situation is different than we thought in precise, nuanced, and highly determinate ways. What a marvelous post this is on QM!
But I notice that even in this thread the postmodernists have to get bagged on and thrown in with the "creationists".... Now I understand why; there's this very worthy desire to dump on slipshod, merely trendy, merely political, or downright non-rigorous claims to rigor (creation science), but I am going to keep on pointing out that the genuine poststructuralist thinkers are not equivalent to their wildly misinterpreted "supposed" positions. (And we in our fields can tell the sham postmodernists from the rigorous ones, just as you can tell the difference between sham and excellence in the scientific disciplines.) Sokal's hoax was a great (and well-deserved) spoof, but his follow-up book on Lacan et alia was terribly unfair. It makes me as upset (and grieved at heart) as all these unsupportable mis-uses of "quantum indeterminacy" make you science folks!

Stochastic is synonymous with "random." The word is of Greek origin and means "pertaining to chance" (Parzen 1962, p. 7). It is used to indicate that a particular subject is seen from point of view of randomness. Stochastic is often used as counterpart of the word "deterministic," which means that random phenomena are not involved. Therefore, stochastic models are based on random trials, while deterministic models always produce the same output for a given starting condition.

SEE ALSO: Deterministic, Random, Random Variable, Stochastic Approximation, Stochastic Calculus, Stochastic Geometry, Stochastic Group, Stochastic Matrix, Stochastic Optimization, Stochastic Process, Stochastic Resonance.

This entry contributed by Vincenzo Origlio

REFERENCES:

Doob, J. L. Stochastic Processes. New York: Wiley, p. 46, 1953.

Gnedenko, B. V. and Khinchin, A. Ya. An Elementary Introduction to the Theory of Probability. New York: Dover, pp. 60 and 121, 1962.

Lawler, G. F. Introduction to Stochastic Processes. Boca Raton, FL: Chapman & Hall/CRC, p. 1, 1995.

Neelamkavil, F. Computer Simulation and Modelling. New York: Wiley, p. 4, 1987.

Parzen, E. Stochastic Processes. Oakland CA: Holden Day, p. 7, 1962.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 588 and 968, 2002.

LAST MODIFIED: June 7, 2002

CITE THIS AS:

Origlio, Vincenzo. "Stochastic." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Stochastic.html

� 1999 CRC Press LLC, � 1999-2007 Wolfram Research, Inc.

The fact that stupid people with contrary agendas mistake one's meaning does not give one leave to deny the literal facts of the English language. Stochastic and deterministic are in opposition. That does not justify mystical nonsense -- but neither does mystical nonsense justify asserting the opposition does not exist. Sophistry will not do; the problem must be addressed at source.

Stochastic doesn't mean simply random; it's probabilistic. In that sense, they are not opposites in the way that -1 is the opposite of 1. The meanings are a bit more subtle.

but I am going to keep on pointing out that the genuine poststructuralist thinkers are not equivalent to their wildly misinterpreted "supposed" positions.

I must have the same problem that non-scientists have when seeing seemingly deep but ultimately empty descriptions of the mysticism of quantum mechanics. It's hard to tell the crap from the real stuff.

As you saw on my other post, my first real introduction to the whole notion was some humanities-type professor glibly waving off the whole notion of the Big Bang as nothing but result of a Judeo-Christian cultural predisposition. It was a bad starting point for me; and, as such, what I've understood of post-modernism has by and large struck me as either so obfuscated as to be incomprehensible, or pseudoprofound musings that are directly at odds with any world-view compatible with the enterprise of science. Given that, it's been very difficult for me to develop a positive view of it... but I *DO* realize that I don't fully understand it, and that even though I *know* there is crap flying the post-modernist banner, I admit the possibility that there is real and good stuff in there that I simply don't understand.

-Rob

I don't agree that stochastic is not the opposite of deterministic but, as you've pointed out, the probabilities in quantum mechanics are determined by the structure of the problem. It isn't as though new age woo will let us manifest wealth by thinking really hard about it, and if quantum mechanics does have bearing on the "debate" between materialism and spooky mumbo jumbo, it doesn't come down on the side of Deepak Chopra -- that's for sure.

Rudolf Carnap wrote a couple of essays on distinguishing between causality and determinism, two often conflated but perfectly distinct things -- I think that might be the better approach.

what I've understood of post-modernism has by and large struck me as either so obfuscated as to be incomprehensible, or pseudoprofound musings that are directly at odds with any world-view compatible with the enterprise of science.

If I'm ever in a position where I have to write another philosophy paper for some bead wearing New Ager, I'm going to quote that.

Great post.

Just a quick (and anonymous) note:

As far as I remember, the term postmodernism was most effectively described by fredric jameson in his 'cultural logic of late capitalism' essay, and as such is a derogatory term used to describe the effect that commodification has had in the destabilisation of modernity, materiality and enlightenment.

Jameson is no fan of post-modernism

He's a post-structuralist. THE post structuralist.

This is not the sort of thing I can say in meetings, but it is nevertheless true.

By anonymous (not verified) on 10 Jun 2007 #permalink