I’m reading the Michael Freyn play Copenhagen, and there is something in there that is so brilliant I have to write about it.

(OK, truthfully, it gives me an excuse to try a new way of teaching something, and that’s really why I’m writing about it. But I do think it’s brilliant. Really.)

A couple years ago, I convinced my wife Julie to go and see the play Copenhagen with me. OK, actually we made a deal. She would go see Copenhagen with me, and I would go to the Rick Springfield concert with her. We’re an interesting pair, we are.

I LOVED it (no, sorry, pop culture fans. I loved Copenhagen. Rick Springfield . . . well, I love Julie).

Why did I love it? Because that’s just the insufferable know-it-all science geek that I really am at heart. I loved seeing a play about two people I knew a lot about, but probably many in the audience didn’t. I loved laughing out loud when the physics was funny, and having the rest of the audience look at me like, “what’s he laughing at, the insufferable know-it-all?” I loved being indignant when in the play Hahn and Strassmann are credited with the discovery of fission, when I knew it was Lise Meitner who made the all-important, but historically neglected, cognitive leap. I loved watching the passion and the excitement for science that I’ve felt myself and seen (on a much grander scale) in real working physicists portrayed so honestly and so well by two actors on stage. I’m that kind of insufferable know-it-all science geek.

For my birthday somebody gave me a Barnes and Noble gift card and I used it to buy Copenhagen the play and Copenhagen the PBS movie. Yes, I do know how to party, thank you very much. In case you don’t know, the play is about German physicist Werner Heisenberg’s visit to Danish (and half-Jewish) physicist Niels Bohr in occupied Denmark in 1941. That real-life meeting is the stuff of legend and controversy. It resulted in an argument that kept the teacher (Bohr) and the student (Heisenberg) apart for the rest of their lives. Heisenberg was head of the German atomic bomb effort. Bohr was the world’s expert on fission. What had Heisenberg said that had so angered his mentor that they never spoke civilly again? Why had Heisenberg come to Copenhagen? What had he hoped to accomplish?

That’s the subject of the play, and it’s a beautiful (if probably non-historical) piece of theater, with a moment that literally made me jump out of my skin, I was so enraptured.

But that’s not what I want to write about.  

Reading and watching the play again, I’ve come across an idea that flew by too quickly in the live play, but that now has fully caught my attention and utterly thrilled me.

Heisenberg of course (he says in his insufferable know-it-all science geek way) is known for the Uncertainty Principle, and his first description of it is described in the play. Imagine Margrethe Bohr (Niels’ wife) as the nucleus of an atom. Imagine Niels as an electron in orbit. We can’t see electrons, we can’t measure them in any way, so we have no idea where Niels is. On this scale, Niels could be anywhere in Copenhagen, so much larger is the atom than its nucleus.

But Heisenberg, playing the part of a photon, collides with Niels, detecting him. Photons have energy, and this act of collision knocks Niels from his orbit. We know where Niels was, but now we don’t know where he is, because the act of measuring has altered him forever. That, says Heisenberg, is uncertainty.

Not so fast, says Bohr. It is true that the collision has affected me. But, my dear Heisenberg, the collision has also affected you. If we simply measure you before and after, we can know exactly what happened to me. We can still know where the electron is and where it is going. The problem, Heisenberg, is that we can’t measure YOU.

This is exactly the discussion Bohr and Heisenberg had when Heisenberg first introduced the uncertainty principle. Heisenberg wanted to define it entirely in terms of measurement messing up our ability to know everything. Bohr recognized the fallacy. Uncertainty isn’t an emergent property, happening only when two particles interact. Instead, uncertainty is inherent, the result of the fact that both interacting particles themselves possess an intrinsic uncertainty.

Huh? OK, here’s my try at teaching the Uncertainty Principle in a different way, as best I understand it, anyway. I’ve been wanting to do this for a long time, so bear with me.

Einstein showed that light is lumpy. It comes in waves, but also particles called photons. Depending on what it’s doing, light can be wave-like or particle-like. De Broglie suggested that not just light, but matter, too, has this duality. An electron, in other words, can behave like a wave.

OK, imagine an electron as a wave. It is what we call a wave pulse. In most locations the amplitude (up and down disturbance) of the wave is zero. In a particular location the amplitude gets large. That’s the electron!


Ignore the “time” caption. It’s really spread in space, not time, but that’s the best image I could find. What’s waving? Wait for it . . . it’s a probability wave. The peaks show a high probability, the zeros show a low probability that the electron is there.

Where the peaks are big, there’s a better chance of finding the electron. Where the peaks are small, there’s less chance of finding it.

Now here’s the thing. To really know the electron, we have to know where it is (its position) and what it’s doing (its momentum). Momentum is related to the wavelength of the electron, the number De Broglie said all electrons have. Now if we know its position exactly, that means at one spot in space the peak is “1” and everyplace else it’s zero. So what’s the wavelength? We can’t know! We have no information on wavelength (and therefore on momentum) if we know its position exactly.

What if we know the wavelength exactly? Well, it turns out that for a wave pulse, to know the wavelength exactly you have to have an infinite number of crests and troughs – who’s to say it won’t change somewhere along the line, after all? If we have an infinite number of crests and troughs, then our wave pulse has become an ordinary infinite sine (or cosine) wave, like this:


So now we know the wavelength exactly. We have no information whatsoever on the position!

For anything in-between, we know something about the position (but not exactly) and something about the wavelength (but not exactly). We can never know both exactly at the same time. And that is the Uncertainty Principle. Notice that there’s nothing weirdly quantum mechanical in here at all, except for the notion that matter has a wavelength. Once we let that idea into physics, uncertainty just pops right out.

I owe this explanation to my quantum mechanics prof at Ohio State, Bill Reay. When he first presented it to me as a sophomore back in 1987 (yes, really), I remember thinking this must be really important, but I didn’t really get it. Since then I’ve gone back to it again and again, and I think I’ve got enough of it now to make it a deep part of how I understand and look at quantum mechanics. I hope I’ve taught it successfully.

OK, if you’re still with me after all that, either you’re an insufferable know-it-all science geek like me, or else you really have nothing to do. So what about Copenhagen? The brilliant part, I think, is that Freyn takes this very esoteric argument about the nature of uncertainty, an argument that Bohr and Heisenberg really had, and turns it into a literary device that reveals the whole point of his play. Where does uncertainty come from? Not from our interactions, that’s just a side effect. Uncertainty comes from each of us. Why don’t we know what really happened in that fateful meeting? Because Bohr didn’t know himself. Because Heisenberg didn’t know himself. There is intrinsic uncertainty within all of us.

Bohr: But Heisenberg! Heisenberg! You also have been deflected! If people can see what’s happened to you, to their piece of light, then they can work out what must have happened to me! The trouble is knowing what’s happened to you!

Though it happens on  p 69 of a 94-page play, that for me is the climax, the critical moment when it all comes clear – or, ironically, intrinsically unclear. God, how I wish I had written something a tenth as brilliant as that analogy.