A little while ago, I noticed a book cover that had a major problem. The book was the UK edition of Dr. Chad Orzel's How to Teach Quantum Physics to Your Dog, which sports a couple of scientific formulas on its cover... and one of them didn't look right.
"Ah-Ha!" I said in an accusatory manner to the imaginary publisher sitting in front of me. "Did you think your mistake would go unnoticed? Ghosting the equation back to a 30% gray has failed to mask your egregious error, you cretinous lackwit!"
But before transmitting a scathing missive to the real publisher like an annoying little know-it-all douche, I decided to check and make sure I was remembering things correctly. As a reflective, self-aware, highly insecure individual, I felt it was important to make sure *I* wasn't the one who was "Wrong, wrong, wrong!"
The equation in question was for the position-momentum form of the Heisenberg uncertainty principle. It's named after Werner Karl Heisenberg who developed the theory in the late 1920s/early 1930s. I've said it before and I'll say it again: When your peers are naming stuff after you, you know you're good. Heisenberg died in 1976 with a Nobel Prize on his shelf and a place in history for helping invent the field of quantum mechanics.
In retrospect, the uncertainty principle seems pretty obvious: Light is energy and we need light to see things, but at the subatomic level, you can't shine a light on something without giving it energy thus changing the nature of the subatomic thing you're trying to look at. One way of expressing this idea is with the equation "delta-x times delta-p greater-than-or-equal-to h divided by two pi", or
This equation describes what we think is probably happening to some subatomic particle we're interested in. Delta-x and Delta-p can be thought of as measures of probability. Delta-x represents how sure we are a particle is in a certain position. Delta-p represents how sure we are about a particle's momentum. But momentum (p) is equal to mass (m) times velocity (v), typically written as p = mv. So, since the mass of the particle doesn't change, you can say that Δp is really a measure of how sure we are about a particle's velocity.
That "h" is Planck's constant, a number related to the energy and frequency with which a particle oscillates. It was discovered by Max Karl Ernst Ludwig Planck around 1900. (When people start naming stuff after you...) In the uncertainty equation,
h = 6.626 x 10-34 J•s
The units for Planck's constant are "Joule-seconds" (sometimes I will willingly stress over detailed explanations of what the units mean, but not today). The quantity [h/(2π)] turns up so often in physics that it was given a special symbol (called h-bar) which is why you sometimes see the uncertainty relation written as
In Germany, there was even a Heisenberg stamp with the equation on it, fer chrissakes! So that's what I thought the uncertainty equation was, meaning the equation I saw on the book cover was wrong. Except it wasn't.
According to several web sites and the author of Quantum Mechanics for Applied Physics and Engineering, the uncertainty equation is actually
meaning ΔxΔp could be an even smaller fraction of h and what I saw on the book cover wasn't a typo. So what was going on?
One clue was the reference to the [h/(4π)] version in Quantum Mechanics as the "rigorous" form of the equation. I figured it would be okay to write to Dr. Orzel himself and ask him if he could help me understand this. "There are a couple of quick-and-dirty ways to get the idea of the uncertainty across," he responded, "and some of them give you just h-bar for the uncertainty. To get the extra factor of 1/2... you need to consider a true minimum uncertainty wavepacket, which is a Gaussian (the same mathematical function as the classic 'bell curve' probability distribution), and in the process, you wind up with an extra 1/2. Most popular treatments just sort of hand-wave past that... it's a little fiddly, and doesn't provide any extra insight for people who aren't physicists."
Physics classes aren't supposed to be math classes as much as it sometimes feels like they are. You can get the gist of the uncertainty concept with ΔxΔp ≥ [h/(2π)] -- and a more intuitive derivation of the equation is good enough. But the fact is, you can get a more precise relation if you're willing to do more math... just not ordinary math. We're talking he-man, heavy-liftin', kill-a-grizzly-with-your-bare-hands-and-wear-its-skin math involving commutators and Schwarz inequalities and Hermitian operators and other things that even some engineering students don't go near. Seriously, anyone who understands this stuff is worth a million sorry-ass real estate developers or Wall Street shit-bags.
At least I can say this: Learning I was the one who was wrong before assaulting an innocent book publisher prevented me from becoming one of the shrill, tyrannical, pedants that I so thoroughly despise. And I would have sounded like an idiot.
|Idiocy doesn't look good on anybody.|
And yet, I still find it somewhat shaming. It's like the difference between wearing a regular "big boy" tie and a clip-on. The college I attended had "Physics for Non-Engineers" classes for anyone not studying science or engineering. I thought, "Who could take such a class and retain an ounce of self-respect? Weaklings!" I would see them in the library -- pre-law students and lit majors -- struggling to master simple vector math. I'd hear their fine whines as they wrestled with even the most trivial trigonometric manipulations and mechanical concepts.
I used to laugh
and call them names
and not let them play my reindeer games.
I have come to learn
that I too
was taught physics
with fucking training wheels.
Je suis désolé!
- Fundamentals of Quantum Mechanics for Solid State Electronics and Optics, C. L. Tang. © 2005, published by Cambridge University Press.
- Quantum Mechanics for Applied Physics and Engineering by Albert Thomas Fromhold, Jr. © 1981, published by Dover Publications, Inc.
- Sears and Zemansky's University Physics, 12th Ed. by Hugh D. Young and Roger A. Freedman. © 2008, published by Pearson Addison-Wesley.
- The Strange Story of the Quantum by Banesh Hoffmann. © 1947, published by Dover Publications, Inc.
- The Uncertainty Principle and Foundations of Quantum Mechanics: A Fifty Years' Survey, William C. Price and Seymour S. Chissick, Eds. © 1977, published by John Wiley & Sons, Ltd.
- "The Nobel Prize in Physics 1932". Nobelprize.org. 4 May 2011 http://nobelprize.org/nobel_prizes/physics/laureates/1932/