Helgoland by Rovelli, Erica (ebook reader below 3000 .txt) 📗

On June 9, Heisenberg leaves Helgoland and returns to his university in Göttingen. He sends a copy of his results to his friend Pauli, with the comment “Everything is still very vague and unclear to me, but it seems that electrons no longer move in orbits.”

On July 9, he sends a copy of his work to Max Born, the professor he was assisting, with a note saying: “I have written a crazy paper and do not have the courage to submit it anywhere for publication.” He asks Born to read it and to advise.

On July 25, Max Born himself sends Heisenberg’s work to the scientific journal Zeitschrift für Physik.3

Born has seen the importance of the step taken by his young assistant. He seeks to clarify matters. He gets his student Pascual Jordan involved in trying to bring order to Heisenberg’s outlandish results.4 For his part, Heisenberg tries to get Pauli involved, but Pauli is unconvinced: it all seems to him like a mathematical game, far too abstract and abstruse. At first it is just the three of them working on the theory: Heisenberg, Born and Jordan.

They work feverishly, and in just a few months manage to put in place the entire formal structure of a new mechanics. It is very simple: the forces are the same as in classical physics; the equations are the same as those of classical physics (plus one,* which I will talk about later). But the variables are replaced by tables of numbers, or “matrices.”

Why tables of numbers? What we observe of an electron in an atom is the light emitted when, according to Bohr’s hypothesis, it leaps from one orbit to another. A leap involves two orbits: the one the electron leaves and the one it leaps to. Each observation can then be placed, as I have mentioned, in the entries of a table where the orbit of departure determines the row; the orbit of arrival, the column.

Heisenberg’s idea is to write all the quantities which describe the movement of the electron—position, velocity, energy—no longer as numbers, but as tables of numbers. Instead of having a single position x for the electron, we have an entire table of possible positions X: one for every possible leap. The idea is to continue to use the same equations as always, simply replacing the usual quantities (position, velocity, energy and frequency of orbit and so on) with such tables. Intensity and frequency of light emitted in a leap, for example, will be determined by the corresponding box in the table. The table corresponding to energy has numbers only on the diagonal, and these will give the energies of the Bohr orbits.

Is that clear? It is not. It’s as clear as tar.

ORBIT OF ARRIVAL

ORBIT OF DEPARTURE

Orbit 1

Orbit 2

Orbit 3

Orbit 4

. . .

Orbit 1

X11

X12

X13

X14

. . .

Orbit 2

X21

X22

X23

X24

. . .

Orbit 3

X31

X32

X33

X34

. . .

Orbit 4

X41

X42

X43

X44

. . .

. . .

. . .

. . .

. . .

. . .

. . .

A Heisenberg matrix: the table of numbers that “represent” the position of the electron. The number X23, for example, refers to the leap from the second to the third orbit.

And yet this absurd maneuver of substituting variables with tables enables us to compute the correct results, predicting what is observed in experiments.

To the astonishment of the three Göttingen musketeers, before the year is out, Born receives by post a brief essay by a young Englishman in which essentially the same theory as their own is constructed, using a mathematical language even more abstract than the Göttingen matrices.5 Its author is Paul Dirac. In June, Heisenberg had given a lecture in England, at the end of which he had mentioned his ideas about quantum leaps. Dirac was in the audience. But he was tired and understood nothing. Later he had been given Heisenberg’s first paper by his professor, who had received it by post and found it inscrutable. Dirac reads it, decides it is nonsensical, puts it aside. But a couple of weeks later, reflecting on it during a walk in the countryside, he realizes that Heisenberg’s tables resemble something that he has studied in one of his courses. Not remembering what exactly, he has to wait until Monday for the library to open so he can refresh his memory about the ideas in a certain book.6 From there, in brief, he independently constructs the same complete theory as the three wizards of Göttingen.

All that remains to do is to apply the new theory to the electron in the atom and see if it really works. Will it actually yield all of Bohr’s orbits?

The calculation turns out to be difficult, and the three cannot manage to complete it. They seek help from Pauli, the most brilliant as well as the most arrogant of them all. “This is indeed a calculation that is too difficult,” he quips, “. . . for you.”7 He completes it, with acrobatic technicality, in the space of a few weeks.8

The result is perfect. The energy values calculated using the matrices of Heisenberg, Born and Jordan are precisely those hypothesized by Bohr. Bohr’s strange rules for atoms follow from the new scheme. But this is not all. The theory also permits us to compute the intensity of emitted light, as Bohr couldn’t. And these also turn out to accord precisely with those obtained in experiments!

It is a complete triumph.

Einstein writes, in a letter to Born’s wife, Hedi: “The ideas of Heisenberg and Born have everyone in suspense, and are preoccupying everyone with the slightest interest in theory.”9 And in a letter to his old friend Michele Besso: “The most interesting theorization of recent times is that of Heisenberg-Born-Jordan on quantum states: a calculation of real witchery.”10

Bohr, the master, will recall years later: “We had at the time only a vague hope of [being able to arrive at] a reformulation of the theory in which every inappropriate use of classical ideas would be gradually eliminated. Daunted by the difficulty of such a program, we all