Bohr's Magic Wand

Apparition (from Merriam-Webster):

1.     a) an unusual or unexpected sight

b) a ghostly figure

2.     the act of becoming visible

In Harry Potter’s magical world, you need a license to apparate, at least in Great Britain. You’re not supposed to disappear like a ghost and reappear unexpectedly, scaring the jeepers out of anyone around you. How having a license changes the jeepers remains unclear.

In the real world, apparition happens all the time. We just don’t notice.

This strange phenomenon was proposed to be commonplace over a century ago by a physicist named Niels Bohr. All atoms did it. Specifically, all electrons in atoms did it. Fellow physicists were aghast at the idea. While no one accused Bohr of wizardcraft (and he has been lauded as a visionary wizard of physics over the years), he was at least accused of using a ‘magic wand’.

The tale begins in Chapter 5 of Helen Kragh’s Niels Bohr and the Quantum Atom. The chapter is appropriately titled A Magic Wand. The issue at hand is the Correspondence Principle. The accuser who was also a close collaborator, the equally visionary Arnold Sommerfeld, wrote that “Bohr has found a magic wand in his analogy principle, which without clearing up the conceptual difficulties allows him to make the results of the classical wave theory directly useful for the quantum theory.”

Before explaining what this means, first let’s agree on one thing. Quantum mechanics is strange.

The world of human beings, shiny marbles, and Amazon boxes, is the macroscopic world. Hypothetically if a marble was allowed to randomly rattle on the floor of a box (never to escape it), eventually it would explore each speck of floor space equally over a long period of time. The probability of finding the marble is equal at any spot along the floor of the box. It doesn’t matter how fast the marble is moving, or how much kinetic energy it has. This is the ‘classical’ result.

But in the micromicroscopic world, much smaller than a micron, the quantum marble behaves strangely. (See illustration below from McQuarrie’s Quantum Chemistry, the text I use when teaching Quantum.) In a one-dimensional box, at its lowest energy level (n = 1), the particle spends most of its time in the middle of the box. But at the second lowest level (n = 2), it prefers to be at the quarter or three-quarter mark, but avoids the middle and the edges. Now that’s very strange behavior. A crest indicates high probability, while a trough on the box floor indicates zero probability.

As n increases, the probabilities get more and more wiggly until you can’t tell the difference between a crest and a trough, or the wiggle for that matter. At this point, I might as well draw a straight horizontal line to represent the probability distribution – the marble can be found equally anywhere in the box, the classical result. In my Quantum class, this is where students first encounter the Correspondence Principle. It’s weird down low, but fine up high. What exactly happens in the middle, and how it happens, well, that’s magic. (The adabiatic principle is a story for another time.)

But even if you avoided Quantum, you might have learned the Bohr model of the atom in a chemistry class somewhere, sometime. For example, here’s the sodium atom with its three rings (non-elven, unfortunately) and eleven electrons.

The beauty Bohr’s work was devising a model of the atom that explained the spectral lines of hydrogen. No, the lines aren’t spectral in a ghostly sense. They come from light absorbed and emitted when the electron in a hydrogen atom changes energy levels. These energy levels are very specific. Therefore, the change in energy between lines is very specific, and hence the line colors are also very specific. In Bohr’s model the rings correspond to energy levels, with inner rings being lower energies (more stable) than outer rings (less stable).

Thus, an electron is typically most stable when located in the innermost ring. However, if it receives energy in the form of ultraviolet light at 103 nm, the electron can absorb that energy and apparate to the third ring. At some point, in a bid to become more stable, it will emit that energy to apparate back to a smaller ring. For example, moving from the third to the second ring will emit red light at 656 nm. Why do we observe this? Conservation of energy.

Here’s the kicker. Only certain orbits are allowed. (We see very specific lines rather than a continuous rainbow spectrum of color.) How then does the electron know “where to stop” when it moves between levels? It seems to know ahead of the jump where to land, and there was no mechanism to explain any of this. Both acknowledging the power of Bohr’s model and its strangeness, Kragh quotes several physicists writing at the time: “the weak side of the theory consists in the heavy sacrifices it requires at the very outset… notwithstanding its somewhat magically arithmetical character.” “I consider it horrible that this success will help the preliminary, but still completely monstrous, Bohr model on to new triumphs.”

A selection from Kramers and Holst frames the paradox clearly. “On the whole it is very difficult to understand how a hydrogen atom, where the electron makes a transition from orbit 6 to 4, can during the entire transition emit a radiation with a frequency different from that when the electron goes from orbit 6 to 5… Even from the very beginning the electron seems to arrange its conduct according to the goal of its motion and also according to future events. But such a gift is wont to be the privilege of thinking beings that can anticipate certain future occurrences. The inanimate objects of physics should observe causal laws in a more direct manner, i.e., allow their conduct to be determined by their previous states and the contemporaneous influences on them.”

This sounds very much like the three D’s of apparition. Destination, Determination, Deliberation. “One must be completely determined to reach one’s destination, and move without haste, but with deliberation.” So says Wilkie Twycross, Ministry of Magic apparition instructor, in Harry Potter and the Order of the Phoenix. In the books, apparition is sometimes accompanied by sound, ranging from a loud crack to a faint pop. However, the movies stress the visual accompaniments over sound. (Here’s a YouTube collection of such scenes.)

It seems reasonable to assume that casting a magical spell to apparate requires energy. This energy must be drawn from the surroundings. If part of the energy absorbed comes from the visible part of the electromagnetic spectrum, then we should expect to see ‘black’. When an electron absorbs energy, that specific line disappears (or goes ‘black’ in the electromagnetic spectrum). Conversely, when the electron emits energy, it releases light corresponding to the same specific lines.

So, I think many of the scenes in the movies get it right. And I like the choice of black smoke to represent the Death Eaters apparating, although the line of smoke suggests poor apparating ability, absorbing all sorts of energy en route, akin to traveling rather than disappearing and reappearing. In other instances, some bright light accompanies the apparition. I could interpret this as drawing too much energy, and then having to release some of it, particularly in the act of reappearing. The more subtle the apparition (perhaps without even a faint pop), the more the wizard or witch exerts fine control over exactly how to optimally and efficiently perform the spell. Another reason why a magic-user aspiring to the top ranks should learn some chemistry and physics!

Want more of Bohr? See here, my dear.