Saturday, February 20, 2010

The Uncertainty Principle Metaphor

Posted by Jeff Lipshaw

As I mentioned to an e-mail correspondent who found a typo in The Venn Diagram of Business Lawyering Judgments:  Toward a Theory of Practical Disciplinarity, I am a serial reviser, so I was fiddling today with the section that deals with the appropriateness of using metaphors from science or mathematics in either social Unknown sciences, humanities or daily life.  One of those metaphors often is to the Heisenberg Uncertainty Principle (see Heisenberg pictured, left).  For what it's worth, here's the little bit I inserted today in a footnote.  I've eliminated the citation, but it is to Roger Penrose's book, The Emperor's New Mind, at pages 291-390 (the chapter entitled "Quantum Magic and Quantum Mystery"):

To the extent that people use the phrase "uncertainty principle" as a metaphor, I believe one meaning is that the act of measuring impacts the system being measured. So, for example, peer evaluation of classroom teaching is subject to a kind of "uncertainty principle" because the act of observation affects the teacher being observed.

Quantum mechanics involves the insight that microscopic bits of matter and energy, by virtue of their microscopic size, exhibit the tendencies of both particles and waves. A wave "spreads out" in a field and a particle is somewhere, but an electron or a photon appears to do both. The mathematics of quantum mechanics precisely and deterministically describes quantum states as weightings of possible positions of the particle in relation to the momentum space associated with the wave. At this point, any further explanation gets into far more detail than I want to provide, except to say that a problem arises when we, macroscopic beings, want to measure the quantum world, and must therefore translate a quantum concept of "amplitude" into a macroscopic concept of probability, which seem analogous (and, indeed, have a mathematical relationship) but are not the same.

Is it an effective metaphor or analogy? The answer is probably not. Position and momentum are complementary in quantum mechanics, meaning that the act of measuring imposes an absolute limit to our ability to know both precisely at the same time (the mathematical relation is ∆x + ∆p > h, where ∆x is the range of possible positions of the particle, ∆p is the range of possible momenta, and h is Planck's constant). While the "knowability floor" of Planck's constant is itself a tiny number (6.6 x 10-34 Joule seconds), it still puts real limits on our ability to know x and p at the same time. If, for example, we measure x so that we know the particle's position exactly (so ∆x is zero), then the value for ∆p will have to be at least h, and that will likely be enough not to know anything about what p actually is at the same time.

The candidate for an analogy here is MEASURING A QUANTUM STATE=OBSERVING A CLASS [or MEASURING A TEACHER]. Does the analogy help us understand why observing the class is problematic? Perhaps, but it is only in the broadest sense that measurement affects the thing being measured. The reasons for the effect in each case are very much different. For quantum mechanics, it is something about the mathematical relationships of micro-amplitudes to macro-probabilities; for classrooms, it is about the psychology of the teacher. Moreover, a quantum state must be made random by a measurement; a teacher need not be affected by observation at all!  Finally, even macroscopic objects are subject to ∆x + ∆p > h, but because h is so small in relation to the size of any object we can actually observe, we can be very, very certain of both position and momentum.

Notwithstanding whether the metaphor works, we nevertheless have come attribute a meaning to "uncertainty principle" outside of quantum mechanics along the lines of the one I have suggested.

(Heisenberg's involvement with nuclear research under the Third Reich is another subject, and his meeting with Niels Bohr in occupied Denmark in 1941 was the basis of Michael Frayn's play Copenhagen.)  I invite comments whether I have correctly (in broad terms) explained the quantum mechanics!

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Posted by: steve | Sep 27, 2010 11:16:32 AM

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