# Mr. Heisenberg Goes to Copenhagen

/A 1941 meeting between Werner Heisenberg and Niels Bohr is the subject of Michael Frayn's *Copenhagen*. The link takes you to a PBS production of the play, starring ~~James Bond~~ Daniel Craig as Heisenberg. The central question is why? Why did Heisenberg go to Copenhagen to meet Bohr?

The historical context is that Denmark was under Nazi occupation at the time. Heisenberg was in charge of the nascent German nuclear program (well, *everyone's* nuclear program was nascent then) and naturally he would want Bohr's opinion. Since the Gestapo was escorting Heisenberg and Bohr's home was surely wired, they took a walk. What was said? No one knows. In the play, Heisenberg asks "does a physicist have a moral right to work on fission?" Bohr responds by refusing to answer and walking away.

Oh, I forgot to mention that this is being told via flashback; you see, the only three characters in the play are Heisenberg, Bohr, and Bohr's wife Margrethe and they are dead. Their ghosts are having a conversation about the conversation. Memory is a funny thing and they can't quite agree on what happened. And why didn't Heisenberg succeed in building a bomb? That's the really interesting aspect and he comes off as a rather sympathetic character. In reality, other physicists refused to even shake Heisenberg's hand after the war since they assumed he had tried to build a bomb. Did he? Frayn leads us to believe that his failure was intentional.

So, where's the math here? Two things. First, of course, is Heisenberg's Uncertainty Principle. This isn't math as much as it is physics, but there is a precise mathematical statement which is fairly easy to understand. Suppose a particle is moving along a path. Its position \(X\) is a random variable whose probability density function is \( f(x)\) as \(x\) varies over some interval. The momentum of the particle is another random variable \(P\). The statement of the uncertainty principle is then \[\sigma_X\sigma_P \ge \frac{\hslash}{2},\] where \(\sigma_X\) and \(\sigma_P\) are the standard deviations of the random variables \(X\) and \(P\) and \(\hslash\) is the reduced Planck constant. This is a very small number (\(1.054\times 10^{-34}\)), but it is positive. What this means is that if we want to increase the precision of one of the measurements (shrink its deviation), we necessarily lose precision of the other (its deviation increases).

Of course, this only applies at the quantum scale. On a macroscopic level, I can obviously look out my window, see my car parked in the driveway, and know its precise position and momentum (zero mo, of course). This quantum uncertainty, where everything is expressed as probabilities, takes some getting used to, but once it sinks in it becomes a natural way of thinking. Einstein rather famously did not like this idea at first, leading him to quip that "God does not play dice."

The other interesting bit of math in the play is an instance of the Prisoner's Dilemma. During one scene, Heisenberg asks Bohr if the Allies have a nuclear program and, if so, how far along they are. Bohr claims he doesn't know (no reason not to believe him--he was in occupied Denmark, after all). Here is the dilemma: if the Allies aren't working on a bomb, then perhaps Germany has no need to (Heisenberg hints), but of course if the Allies are building one then Germany should as well. This is the classic Cold War MAD theory (Mutual Assured Destruction) in its infancy. Here's the payoff matrix:

Germany doesn't build | Germany builds | |

Allies don't build | no risk | Germany dominates |

Allies build | Allies dominate | tense stalemate |

Created with the HTML Table Generator

The lower right corner, which is what happened ultimately, is a Nash equilibrium; that is, if either party changes strategy unilaterally it results in a worse payoff. The best strategy is the upper left corner, but purely rational actors will choose the Nash equilibrium.

*Rational* has a fairly precise mathematical meaning that isn't exactly how real people operate. Like all mathematical models, two-person games are a simplification of reality, useful on some level but not the whole story. *Copenhagen* is much the same: we don't know the whole story and we never will, but it gives us a lens through which to examine history, uncertain as it is.