The Uncertainty Principle

The other day, Dilip D’Souza, a writer whom I enjoy reading, wrote an article that I did not enjoy reading. And it was not for the usual reasons, that it exposed some uncomfortable truths and made me question my assumptions on the society we live in. No, it was on a very familiar topic — the Heisenberg uncertainty principle and I found it not only disappointingly superficial but significantly misleading. So, even though we had a brief email exchange on this, I hope he will not mind my effort to set things straight.

The Heisenberg uncertainty principle is a principle in quantum mechanics, but that has not prevented its being misused and misquoted in popular culture. It says that, for any particle, the position and momentum cannot simultaneously have exact values (and puts a bound on the inexactness).

Now, it is natural to assume (and, in the early days, physicists did assume) that this is an incompleteness in our ability to observe these values: the particle has a position and a momentum, but we cannot know both of them. Why not? Because, if, for example, we determine its position by scattering off another particle, the impact changes its momentum. And this interpretation has invaded popular culture, as in Dilip’s (not original) example of an anthropologist who changes the society that he studies.

But this is really beside the point. It is not that we “cannot know” the particle’s position or momentum exactly: it has no exact position or momentum, and the more we try to define one, the less definite the other becomes.

Here’s the simplest analogy I can think of. Any extended macroscopic object — a slice of pizza, say — lacks a position, too. It has an average position, but how do you define that? The geometric average? The centre of mass? So the idea that a property of an object is not precisely defined should not be a total surprise.

Now suppose you are satisfied with a centimetre-scale definition of position, and your object is a metre rod, metre long and a millimetre thick, which you are forced to lay on the floor (you are not allowed to stand it on its end). If you line up this rod north-south, it does in fact have an exact east-west position (to the nearest centimetre, which is all you want). But it occupies a hundred centimetres in the north-south direction.

If you want it to have an exact north-south position, you can rotate it 90 degrees. Then it occupies a hundred centimetres in the east-west direction.

So this is a sort of uncertainty principle of the metre rod: you can give it a definite north-south position, or a definite east-west position, but not both. This is not a limitation of your ability to measure its position. But imagine, perhaps, that you believe that this rod is a point particle, and you cannot see it directly but only using sophisticated instruments. Then you may notice something odd: the more you localise this object in the east-west axis, the less it is localised in the north-south axis. You may think it is because of your measuring apparatus: the object is so light that squeezing it east-west makes it move about north-south. But the apparatus has nothing to do with it. The problem is your assumption that it is a point particle, when it is in fact a rod.

This is an analogy and not exact (perhaps some of my colleagues will think it very misleading). But I think it is useful. Classical particles are in states with exact positions and momenta. Quantum particles are in a different sort of state. Fundamentally, in those states, position and momentum are complementary. Paradoxes only arise when one tries to think of those states as “classical” states.(*)

Then Dilip brings in Schroedinger’s cat. It was, I believe, Stephen Hawking who said “When I hear of Schroedinger’s cat, I reach for my gun.” Einstein pioneered the use of “thought experiments”, but this cat is probably the most (in)famous thought-experiment of all.

In quantum mechanics, particles are not always in one state or another, but are in a “superposition of states”. Suppose you put a cat in a box with a radioactive atom. If the atom decays, the resulting gamma ray triggers a hammer that breaks a vial of poisonous gas, killing the cat. But a quantum mechanical description of the unobserved atom requires that, after any length of time, it is in a superposition of states — “decayed” and “not decayed”. Is the cat, too, in a superposition of states — “dead” and “not dead”?

Dilip says that it is, and worse, he confuses it with the obvious statement that we don’t know whether it is dead or alive. That is true but it is not quantum superposition. An unobserved electron really is in a superposition of states. A cat (or any macroscopic object) is not, and no serious physicist would claim that it is.

The question of how “superpositions of states” cease to occur as you go to larger objects is thorny but, today, fairly well understood. I won’t go into it here, but you could, if you like, think of a cat as a measuring device: if it dies (and we can detect this easily enough from outside the box), the fate of the atom is known too. But it does not matter whether we are observing the cat or not. (Parenthetically, a cat has zillions of states available to it, not just two; so even if you wanted to prepare a cat in a dead-alive superposition state, how would you isolate it to just two states of all those zillions?)

These are topics that confused physicists three generations ago. It is understandable that they confuse lay people today — but that does not, in my opinion, excuse journalists who write confused articles on the matter.

(*)For electrical engineers and others familiar with waves, I can give a much better analogy: the relation between the spatial “spread” of a wavepacket and the component wavelengths that it contains. A wave with only a single wavelength is infinitely long. If you want to compress it into a localised packet, you have to add together many different wavelengths, which will tend to interfere destructively over most of space, except in one particular region where they add up. The smaller and sharper that region, the greater the “spread” of wavelengths you need. The position of the wavepacket and the wavelengths (or their inverse, wavenumbers) are precisely analogous to position and momentum in quantum mechanics. For a slightly more technical explanation that I wrote many years ago for Resonance, the science education magazine (it requires only high school mathematics), go here.

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7 Comments

  1. gaddeswarup

     /  September 20, 2011

    What I read in the fifties was similar to what dilip described but i have not really kept in touch since then. I wonder what you think of this article:
    http://astro.temple.edu/~powersmr/vol7no3.pdf

    Reply
    • Rahul Siddharthan

       /  September 20, 2011

      swarup – I will think about that. To me, the Heisenberg uncertainty principle is a result in linear algebra. If you have two non-commuting operators, they cannot be simultaneously diagonalised. And if the commutator is a constant a, then in any state, the lower bound for the product of variances of the two operators for that state is a/2.

      Reply
  2. An unobserved electron really is in a superposition of states. A cat
    (or any macroscopic object) is not, and no serious physicist would
    claim that it is.

    How do you decide when an object is “macroscopic” and so the laws of quantum mechanics and, in particular, superposition, no longer apply?

    A probability, no matter how small — is a possibility.

    Moreover, once you approach problems in physics by invoking “scales”, the “laws of physics” loose part of their universality. This is not a point of view that I oppose. However, I know many physicists who would be unhappy if the universe were divided into regimes of applicability of mathematical models.

    Reply
    • To clarify this a bit further. I think the problem is with the macroscopic state of dead/alive. This is not a measurable quantity in the sense of quantum mechanics. The “state” of the cat is the combination of the state of all the particles of which it is made. That is indeed a superposition of “pure” states.

      Reply
      • Rahul Siddharthan

         /  September 20, 2011

        Kapil — your second comment is spot-on. We are all in superpositions of gazillions of states (in fact, we are not even in pure states, but in mixed states, since we cannot be isolated from our environments). But these states are all very nearby, in some sense. The question is whether a superposition of widely separated states in a macroscopic particle can occur. I have further thoughts on this but will put it in a new blog post.

        Regarding your first point, I am afraid only a mathematician (and very few mathematicians, perhaps) would say that. Yes, any non-zero number is mathematically different from zero. But if it is not likely to occur in the age of the universe, or in 10100 times the age of the universe, it is not a possibility. The probability that the cat will tunnel out of the box, coherently and intact, is mathematically non-zero too, but it is not a possibility. Quantum mechanics works at macroscopic scales, but it is indistinguishable from classical mechanics at such scales.

        But there is a problem here, of what constitutes the “collapse” of the wavefunction. I made a comment on that here (on Dilip’s blog).

        Reply
        • Rahul Siddharthan

           /  September 20, 2011

          (ps — a pedantic clarification: the probability that the cat will tunnel out of the box in a dismembered or disintegrated state is vanishingly small too, but still non-zero.)

          Reply
  1. The “single-slit experiment”: A thought experiment on superposing macroscopic states | E's flat, ah's flat too

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