Posted on 09/14/2006 10:27:24 PM PDT by snarks_when_bored
Why Quantum Mechanics Is Not So Weird after All
Richard Feynman's "least-action" approach to quantum physics in effect shows that it is just classical physics constrained by a simple mechanism. When the complicated mathematics is left aside, valuable insights are gained.
The birth of quantum mechanics can be dated to 1925, when physicists such as Werner Heisenberg and Erwin Schrödinger invented mathematical procedures that accurately replicated many of the observed properties of atoms. The change from earlier types of physics was dramatic, and pre-quantum physics was soon called classical physics in a kind of nostalgia for the days when waves were waves, particles were particles, and everything knew its place in the world. Since 1925, quantum mechanics has never looked back. It soon became clear that the new methods were not just good at accounting for the properties of atoms, they were absolutely central to explaining why atoms did not collapse, how solids can be rigid, and how different atoms combine together in what we call chemistry and biology. The rules of classical physics, far from being a reliable description of the everyday world that breaks down at the scale of the atom, turned out to be incapable of explaining anything much more complicated than how planets orbit the sun, unless they used either the results of quantum mechanics or a lot of ad hoc assumptions. But this triumph of quantum mechanics came with an unexpected problem-when you stepped outside of the mathematics and tried to explain what was going on, it didn't seem to make any sense. Elementary particles such as electrons behave like waves, apparently moving like ripples on a pond; they also seem to be instantaneously aware of distant objects and to be in different places at the same time. It seemed that any weird idea could gain respectability by finding similarities with some of the weird features of quantum mechanics. It has become almost obligatory to declare that quantum physics, in contrast to classical physics, cannot be understood, and that we should admire its ability to give the right answers without thinking about it too hard. And yet, eighty years and unprecedented numbers of physicists later, naked quantum weirdness remains elusive. There are plenty of quantum phenomena, from the magnetism of iron and the superconductivity of lead to lasers and electronics, but none of them really qualifies as truly bizarre in the way we might expect. The greatest mystery of quantum mechanics is how its ideas have remained so weird while it explained more and more about the world around us. Perhaps it is time to revisit the ideas with the benefit of hindsight, to see if either quantum mechanics is less weird than we usually think it is or the world around us is more so. Classical Mechanics in ActionWhen we think of planets orbiting the sun, we usually adopt Newton's view that they are constantly accelerating-in this case changing direction-in response to gravitational forces. From this, we can calculate the motions precisely, and the impressive accuracy of predictions for total solar eclipses shows how well it works. There is, however, another way of thinking about what is happening that gives exactly the same results. Instead of the Principle of Acceleration by Forces, as we might call it, there is an alternative called the Principle of Least Action, or more correctly, Hamilton's Principle. It is a principle that was first put forward about fifty years after Newton's, in its earliest form by the Frenchman Pierre Maupertuis, and in its ultimate form by the Irishman William Rowan Hamilton. The general idea is that when a planet travels through space, or a ball travels through the air, the path that is followed is the one that minimizes something called the action between the start and end points. Action, for our purposes here, is just something that can be measured out for some particular object moving along a particular path. It is exactly defined and is measured in units of energy multiplied by time. The details are not important unless you need to make calculations. We therefore have two quite different ways of describing situations in classical physics that are equally good in terms of giving the right answer. To give the simplest possible example, we can think of a golf ball travelling across an idealized, frictionless, flat green. In Newton's view (figure 1), the ball moves in a straight line at constant speed, because that is what Newton's Law says it must do. In Maupertuis' view (figure 2), the ball does this because this path is the one that has the least action between the start and end points. This trivial example can be made more interesting by making the green have humps and dips, which are like having forces acting on the ball, but the principles stay the same.
Hamilton's Principle is fundamentally equivalent to Newton's Laws, and comes into its own when solving more advanced types of classical problems. But as an explanation, it has a major flaw-it seems to mean that things need to know where they are going before they work out how to get there. Actually, this is where classical mechanics makes its first big step toward quantum mechanics, if only we look at it another way. The mathematics of Hamilton's Principle can be described in words alternatively like this: given its starting points and motion, an object will end up at locations that are connected to its starting point by a path whose action is a minimum compared to neighboring paths. If locations away from the classical path are considered, no such paths exist-there will always be a path with the least action, but this is not a minimum. It is an unfamiliar idea, but well worth a little effort to try and digest. One vital change to note is that, while still being classical physics, the emphasis has moved away from knowing the path that is followed to having a test to check whether possible destinations are on the right track. And the crucial factor is being able to compare the actions of different paths. It leads to a third picture for our moving golf ball, central to the later move to quantum physics, which we can call Feynman's view of classical physics (figure 3).
If we stay within the world of classical physics, we can choose to ignore this strange new description and stick with the more comfortable idea that things are accelerated along paths by forces, but this would be a personal preference rather than a rational one. The new view prompts the question: "How do things work out whether possible destinations are linked to the start by a path of minimal action?" We should appreciate, however, that the old Newtonian view prompts equally difficult questions like: "How do things respond to forces by accelerating just the required amount, instant by instant?" Moreover, as we will see, the action version is the one that the world around us seems to use. Roll on, Quantum MechanicsSuppose we take the action question seriously and give it a rather simple answer: Nature has to check out all possible destinations to see if they are on the right track. It must do this by trying to find out if there is a path of minimal action to each destination. It uses a device that can measure the action along all possible paths to each destination. The device is a simple surveyor's wheel for measuring action-just a wheel with a mark on the rim (figure 4). There isn't literally a type of wheel that measures action, but we can imagine that there is. The mechanism assigns probabilities to each destination according to whether, with just this simple measuring tool, it can find a path of minimal action.
When the actions it is trying to measure are large compared to the size of the wheel, the system typically works just as classical physics requires. But in some situations the mechanism fails to produce classical mechanics and gives us quantum mechanics instead. We call the circumference of the wheel "Planck's constant," after Max Planck, who discovered its importance by an indirect route in 1900. You may be wondering how exactly the wheel can tell us what we need to know, but we don't need to go into the details here-those interested should read Richard Feynman's book, QED: The Strange Theory of Light and Matter, or see the summary given in the box on page 43. Differences from Classical PhysicsAs we might expect, the introduction of a mechanism for carrying out classical mechanics only makes a difference when the mechanism can't do its job properly. Specifically, if we want to check out destinations that are too close to the start, as gauged by the size of the wheel, the mechanism doesn't work. It cannot say where the object should be going, and there is an intrinsic fuzziness associated with it, with a scale set by the amount of action known as Planck's constant. This is otherwise known as the Uncertainty Principle. A second feature arises from the simple circular nature of the measuring device. It cannot tell the difference between paths that differ by an amount of action that is an exact whole number of Planck's constants. This can lead to patterns of probabilities that look just like classical waves, because the mathematics of waves is very similar to the mathematics of circular motion. The most important change comes when we consider objects in very small orbits, like electrons around nuclei. The mechanism gives zero probability unless the orbit (or more correctly the state) has an action that is an exact multiple of Planck's constant. This crude mechanism explains why atoms can only shrink to a certain point, to a state with an action of Planck's constant, where they become stable. With one extra idea, which we will mention later, the mechanism seems to explain the workings of chemistry, biology, and all the other successes of quantum mechanics, without ever really stopping being classical mechanics. Three Conceptual Problems with Quantum MechanicsThe way it is normally introduced, quantum mechanics is something quite baffling, and certainly stranger than just classical mechanics with a mechanism. It is worth addressing the three most obvious difficulties directly: 1) Quantum mechanics gives answers that are a set of probabilities all existing at the same time. This is totally unreal. As Schrödinger pointed out, quantum mechanics seems to say that you could create a situation where a cat was both alive and dead at the same time, and we never see this. But this is in fact a very curious piece of ammunition to use against quantum mechanics. We already have a very good nontechnical word for a mixture of possibilities coexisting at the same time-we call it the future. Unless we believe that all events are predetermined, which would be a very dismal view of the world, this is what the future must be like. Of course, we never experience it until it becomes the present, when only one of the possibilities takes place, but the actual future-as opposed to our prediction of one version of it-must be something much like what quantum mechanics describes. This is a great triumph for quantum mechanics over classical mechanics, which by describing all events as inevitable, effectively deprived us of a future. Of course, there is now a new big question of how one of the possibilities in the future is selected to form what we see as the present and what becomes the past, but we should not see the lack of a ready answer as a fault of quantum mechanics. This is a question that is large enough, encompassing such ideas as fate and free will, to be set aside for another time. The headline "Physics Cannot Predict the Future in Detail" should be no great embarrassment. 2) Quantum mechanics means that there is a kind of instant awareness between everything. This is quite true, but by introducing quantum mechanics in the way that we have, the "awareness" is of a very limited kind-limited to the awareness gained through the action-measuring mechanism as it checks all possible destinations. It is very hard to see how the only result of this-a probability associated with each destination-could be used to send a signal faster than light or violate any other cherished principle. It is rather revealing that one of the few novel quantum phenomena is a means of cryptography-a way of concealing a signal rather than sending one. 3) Quantum mechanics doesn't allow us to say where everything is, every instant of the time. This is the most interesting "fault" of quantum mechanics, and it can be expressed in many ways: particles need to be in more than one place at a time; their positions are not defined until they are "observed"; they behave like waves. We will summarize this as an inability to say exactly where particles are all the time. The "classic" illustration of this is the experiment of passing a steady stream of electrons through two slits (figure 5). Instead of the simple shadows we would expect if the particles were just particles, we see an interference pattern, as if the electrons have dematerialized into a wave and passed through both slits at the same time.
There are several ways of coming to terms with this. The first thing to note is that the lack of complete information is not really a problem that arose in quantum mechanics-it originates in the third version of classical mechanics. In the Feynman version, the essence of motion is a process of determining if a destination is on or off the right track. Before the move to quantum mechanics, we can do this as often as we like, so that we can fill in the gaps as closely as we like, but the precedent has been set: physics is about testing discrete locations rather than calculating continuous trajectories. If it is inherent in old-fashioned classical physics, not just "weird" quantum physics, perhaps we can relax a little. The second point is to clarify what the problem is. To take the two-slit example, we never see electrons dematerialize, or rippling through something, we just find it necessary to think that they do to explain the pattern that we see on the screen. If we deliberately try to observe where the electrons go, we see them as particles somewhere else, but the interference pattern disappears. In effect, the problem is that we cannot say what the particles look like only when they cannot be seen. Now this is an uncomfortable thought, because all our instincts tell us that particles must be somewhere, even when we cannot see them. But if quantum mechanics can accurately describe all the information we can ever obtain about the outside world, perhaps we are simply being greedy to ask for anything more. The headline "Physics Fails to Describe Events That Cannot Be Observed" is, again, rather lacking in impact. The final point is a little vague but more fundamental. If we accept that the future is not fixed, we expect it to contain surprises. Crudely speaking, this is not very plausible in a world where particles have continuous trajectories and an infinite amount of information is freely available. It is much more plausible in a world that is in some way discontinuous, where the available information is limited. Even though we have set aside the question of how a future full of possibilities turns into an unchanging past, it must involve something that seems pretty weird compared to our normal experience. Perhaps this example of physics not conforming to our expectations is weirdness of the right sort. The Addition of SpinIt was mentioned earlier that another new idea is needed before the classical physics of electrons and nuclei properly turns into chemistry. That idea is spin, a third property of electrons and nuclei alongside mass and electrical charge. Paul Dirac showed that spin is a natural property of charged particles within quantum mechanics. Wolfgang Pauli showed that the spin of the electron prevents more than one electron occupying the same state at the same time-the Exclusion Principle-a fact responsible for the whole of chemistry. The details are not important here, but quantum mechanics with spin seems to account for pretty much all the world we see around us. Quantum Mechanics-Bringer of StabilityOne of the benefits of viewing the quantum world as not fundamentally different from the classical world is that we can imagine how one changes into the other. With a few simple assumptions, a classical world of point-like electrons and nuclei is blindingly chaotic. Atoms are continually trying to collapse, but are prevented from doing so by the huge amount of electromagnetic radiation that is released in the process. It is not the comfortable place that the word classical implies. As we imagine moving to the quantum realm by increasing the size of Planck's constant from zero, something remarkable happens. At some point, the blinding light disappears to reveal stable atoms, capable of forming molecules. Far from making everything go weird, quantum mechanics makes it go normal. To be sure, if Planck's constant increases too far, the atoms fall apart and a different form of chaos takes over, but that just makes the story even more interesting. So it seems that quantum physics is not weird and incomprehensible because it describes something completely different from everyday reality. It is weird and incomprehensible precisely because it describes the world we see around us-past, present, and future.
ReferenceFeynman, Richard P. 1985. QED: The Strange Theory of Light and Matter. Princeton, N.J.: Princeton University Press.
About the AuthorPaul Quincey is a physicist at the National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, United Kingdom. E-mail: paul.quincey@npl.co.uk.
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Quantum Mechanics: The Dreams Stuff Is Made Of
ping
You're wrong. It's a statement about reality.
You're being distracted by the fact that in the case of a cat, the "dead" state does not interfere with the "alive" state. But in the real world, the superposition is one of simultaneous realities and not simply a statement about our ignorance of the state of the system.
Two examples that spring immediately to mind are double-slit diffraction and the Bohm-Aharonov experiment. In each case, there's a measurable interference term caused by the simultaneous realities.
In the case of double-slit diffraction, you get an interference pattern caused by the simultaneous superposition of Schrödinger waves from each slit, which is very different from a mere sum of wave amplitudes from two individual slit sources.
Let's be more specific. Suppose we label one slit the "left" slit and the other the "right" slit. We project electrons through the slits and towards a screen, but we don't know much about their exact trajectories. Classically, they might go through one slit or the other. Quantum mechanically, the wave function is a superposition of the state where it goes through the left slit AND the state where it goes through the right slit.
If the superposition were merely a statement of our ignorance, our mere knowledge of whether each electron went through the left slit ("alive", if you prefer) or the right slit ("dead", if you prefer) wouldn't affect the final distribution of electrons from the slits when projected onto the screen. But the experimental reality--and the mathematical prediction--is that that knowledge matters a great deal. If we add up the cases where we remain in ignorance, we see a series of light and dark bands on the screen: interference fringes. If we add up the cases where we know which path the electron took, we see the sum of two Gaussians. No fringes.
Quantum mechanically, what is happening is that, by knowing the trajectories, we are collapsing the electron trajectory eigenstates into either "left" or "right". The interference between the slits then disappears; it's as if the slit not used were simply covered up. Classically, there is no analogue.
Now, you might think that perhaps the interference effect has nothing to do with individual wave functions or trajectories. Perhaps if you get a cloud of electrons flying through two slits, they'll bounce off each other, block each other, and knock one another to this or that side in such a way as to produce the interference fringes, even though each individual electron took a definite, well-defined path through one slit or the other. But that's not the case: the interference effect works even if we send the electrons through one at a time, so that each electron can't have any contact or "knowledge" of the rest of the ensemble. Any simple interpretation will have to say that each electron went through both slits, taking both paths simultaneously.
The Aharonov-Bohm experiment is more technical, but it involves two (field-free!!) paths around a magnetic solenoid, and the shifting of interference fringes as the field within the solenoid is altered. The principle is one of gauge symmetry: there's a phase angle associated with the difference in integrated vector potential along each path.
OK, that was a mouthful, but the key is this: BOTH paths have to contribute simultaneously for each electron, or the effect disappears. It cannot be said that each electron passed either to the right or left of the solenoid.
I'm giving special heed to Bohr's remarks. Wikipedia notes that one can speak of the "collpase", or simply say that the theory makes predictions without granting any kind of physical reality to wavefunctions ( or propagators. )
"Niels Bohr emphasized that it is only the results of the experiments that should be predicted, and therefore the additional questions are not scientific but rather philosophical."
Taking this lead, I've always seen the CI as the "bare bones" interpretation, which contents itself with the predictions made using the theory. Bohr emphasized that we are constrained to speak and understand in everyday terms, and the theory itself is part of our everyday experience of pencils and paper and apparatus.
Even allowing that the condition of the cat is "undefined", we can take this to mean it is undefined by the theory, which is plain to see since its predictions are probabilistic. To put any more into it is mere mystification, as I see it, and I believe I'm following Bohr in this.
... and yet! ... and yet! One can never be very happy with this view, can one? The mystery of QM is ineluctable.
I would also comment that I put a lot of weight on Feynman's remarks on neutron scattering in a crystal in his Lectures Vol. III, 3-3. He contrasts the case of no-spin-interaction, where the scattering nucleus is indeterminate, with the spin-interaction case, where a particular nucleus is affected by the scattering. Thus the crystal itself "observes" the scattering, and localizes the interaction.
"You may argue, 'I don't care which atom is up.' Perhaps you don't, but nature knows; ..."
This obviates the question of an "observer", since after all why may not the cat be regarded as an observer, or what if we put a physicist in there? Or for that matter a clock, or any kind of recording instrument?
I'm not very intelligent, but does communication play a role in all of this stuff? And, I don't mean what you stated in your posts, but the whole of this quantum stuff.
Something (the sender) must tell something else (receiver) to behave or do something. The message sent must both be efficient, in the sense that is received correctly and understood, and effective, in the sense that the response of the receiver is the desired response of the sender.
K. A. Milton, Julian Schwinger (1918-1994) (PDF)
A paragraph with some relevance to the point you were making:
"Schwinger learned from his competitors, particularly Feynman and Dyson. Just as Feynman had borrowed the idea from Schwinger that henceforward would go by the name of Feynman parameters, Schwinger recognized that the systematic approach of Dyson-Feynman was superior in higher orders. So by 1949 he replaced the Tomonaga-Schwinger approach by a much more powerful engine, the quantum action principle. This was a logical outgrowth of the formulation of Dirac [21], as was Feynmans path integrals; the latter was an integral approach, Schwingers a differential. The formal solution of Schwingers differential equations was Feynmans functional integral; yet while the latter was ill-defined, the former could be given a precise meaning, and for example, required the introduction of fermionic variables, which initially gave Feynman some difficulty. It may be fair to say, at the beginning of the new millennium, that while the path integral formulation of quantum field theory receives all the press, the most precise exegesis of field theory is provided by the functional differential equations of Schwinger resulting from his action principle."
The "sending and receiving" stuff that you see in regards to quantum entanglement experiments is heuristic, or philosophical, or whatever. It is definitely not part of Quantum Mechanics!
A striking feature of QM is what Heisenberg called "quantum kinematics". That is, the framework of the theory itself. Many of its amazingly numerous and varied results stem directly from the workings of this framework, and this includes entanglement, degeneracy pressure, and quantum levels themselves, not to mention Light Amplification by Stimulated Emission of Radiation.
Of course, the theory cannot offer any explanation of its own framework, yet traditional heuristics offer many such quasi-classical, and even anthropomorphic explanations. I try to ignore these as much as possible, but you can't get away from them. Note that the term, "stimulated", in the acronym LASER, is just such a heuristic. The laser phenomenon is a pure implication of quantum statistics, and does not involve any sort of postulated "stimulation".
Thank you.
This stuff is hard for the layperson, but extremely interesting. It's like the more I read, the more I understand how much I didn't know.
"You were not made to live like brutes, but to follow virtue and knowledge." - Dante
Wonderful words, dear Pipe. Thank you so much for including me in the ping to them.
Very true. I'm so glad we can share such things in Christ!
LoL.... !Quite funny until the reality of it is considered...
Fascinating. For a layman, I would love to be able to read something that will delve further into this. Got any specific links?
ping for later reading
You know, this is about as far as it goes for laymen until they start digging into the math. But, if you do that you will be doing the real stuff. These made-up examples aren't of much use except to point out that they are mostly nonsense pointing the way to the actual physics. The math isn't really all that bad once you take that step.
Too many people believe that quantum "weirdness" opens a window for the paranormal, supernatural, mystical uncertainty, etc. when in reality, it makes specific physical predictions with a rigor and accuracy unparalleled by any other scientific theory.
The cat was pregnant.
"Any simple interpretation will have to say that each electron went through both slits, taking both paths simultaneously."
Maybe the electron simply swallowed the grid and spit it back out when finished.
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