I
A STRANGELY BEAUTIFUL INTERIOR
How a young German physicist
arrived at an idea that was very
strange indeed, but described
the world remarkably well-and
the great confusion that followed.
The Absurd Idea of the Young Heisenberg: Observables
It was around three o'clock in the morning when the final results of my calculations were before me. I felt profoundly shaken. I was so agitated that I could not sleep. I left the house and began walking slowly in the dark. I climbed on a rock overlooking the sea at the tip of the island, and waited for the sun to come up . . .
I have often wondered what the thoughts and emotions of the young Heisenberg must have been as he clambered over that rock overlooking the sea, on the barren and windswept North Sea island of Helgoland, facing the vastness of the waves and awaiting the sunrise, after having been the first to glimpse one of the most vertiginous of Nature's secrets ever looked upon by humankind. He was twenty-three.
He was on the island seeking relief from the allergy that afflicted him. Helgoland-the name means Sacred Island-has virtually no trees, and very little pollen. ("Heligoland with its one tree," as James Joyce has it in Ulysses.) Perhaps the legends of the dreadful pirate Stšrtebeker hiding on the island, which Heisenberg loved as a boy, were in his mind as well. But Heisenberg's main reason for being there was to immerse himself in the problem with which he was obsessed, the burning issue handed to him by Niels Bohr. He slept little and spent his time in solitude, trying to calculate something that would justify Bohr's incomprehensible rules. Every so often, he would take a break to climb over the island's rocks or learn by heart poetry from Goethe's West-Eastern Divan, the collection in which Germany's greatest poet sings his love for Islam.
Niels Bohr was already a renowned scientist. He had written formulas, simple but strange, that predicted the properties of chemical elements even before measuring them. They predicted, for instance, the frequency of light emitted by elements when heated: the color they assume. This was a remarkable achievement. The formulas, however, were incomplete: they did not give, for instance, the intensity of the emitted light.
But above all, these formulas had about them something that was truly absurd. They assumed, for no good reason, that the electrons in atoms orbited around the nucleus only on certain precise orbits, at certain precise distances from the nucleus, with certain precise energies-before magically "leaping" from one orbit to another. The first quantum leaps. Why only these orbits? Why these incongruous "leaps" from one orbit to another? What force could possibly cause such bizarre behavior as this?
The atom is the building block of everything. How does it work? How do the electrons move inside it? The scientists of the beginning of the century had been pondering these questions for more than a decade, without getting anywhere.
Like a Renaissance master painter in his studio, Bohr had gathered around him in Copenhagen the very best young physicists he could find, to work together on the mysteries of the atom. Among them was the brilliant Wolfgang Pauli-Heisenberg's extremely intelligent, pretty arrogant friend and former classmate. But Pauli had recommended Heisenberg to the great Bohr, saying that to make any real progress, he was needed. Bohr had taken the advice, and in the autumn of 1924 had brought Heisenberg to Copenhagen from Gšttingen, where he was working as an assistant to the physicist Max Born. Heisenberg had spent a few months in long discussions with Bohr, in Copenhagen, in front of blackboards covered with formulas. The young apprentice and the master had taken long walks together in the mountains, talking about the enigmas of the atom; about physics and philosophy.
Heisenberg had steeped himself in the problem. It had become his obsession. Like the others, he had tried everything. Nothing worked. There seemed to be no reasonable force capable of guiding the electrons on Bohr's strange orbits, and in his peculiar leaps. And yet those orbits and those leaps really did lead to good predictions of atomic phenomena. Confusion.
Desperation pushes us to look for extreme solutions. On that island in the North Sea, in complete solitude, Heisenberg resolved to explore radical ideas.
It was with radical ideas, after all, that twenty years earlier Einstein had astonished the world. Einstein's radicalism had worked. Pauli and Heisenberg were enamored of his physics. Einstein for them was a legend. Had the time perhaps come, they asked themselves, to hazard as radical a step, to escape from the impasse regarding electrons in atoms? Could they be the ones to take it? In your twenties, you can dream freely.
Einstein had shown that even our most rooted convictions can be wrong. What seems most obvious to us now might turn out not to be correct. Abandoning assumptions that seem self-evident can lead to greater understanding. Einstein had taught that everything should be based on what we see, not on what we assume to exist.
Pauli repeated these ideas to Heisenberg. The two young men had drunk deep of this poisoned honey. They had been following the discussions on the relation between reality and experience that ran through Austrian and German philosophy at the beginning of the century. Ernst Mach, who had exerted a decisive influence on Einstein, insisted that knowledge had to be based solely on observations, freed of any implicit "metaphysical" assumption. These were the ingredients coming together in the young Heisenberg's thinking, like the chemical components of an explosive, as he isolated himself on Helgoland in the summer of 1925.
And here he had the idea. An idea that could only be had with the unfettered radicalism of the young. The idea that would transform physics in its entirety-together with the whole of science and our very conception of the world. An idea, I believe, that humanity has not yet fully absorbed.
Heisenberg's leap is as daring as it is simple. No one has been able to find the force capable of causing the bizarre behavior of electrons? Fine, let's stop searching for this new force. Let's use instead the force we are familiar with: the electric force that binds the electron to the nucleus. We cannot find new laws of motion to account for Bohr's orbits and his "leaps"? Fine, let's stick with the laws of motion that we're familiar with, without altering them.
Let's change, instead, our way of thinking about the electron. Let's give up describing its movement. Let's describe only what we can observe: the light it emits. Let's base everything on quantities that are observable. This is the idea.
Heisenberg attempts to recalculate the behavior of the electron using quantities we observe: the frequency and amplitude of emitted light.
We can observe the effects of the electron's leaps from one of Bohr's orbits to another. Heisenberg replaces the physical variables (numbers) with tables of numbers that have the orbits of departure in their rows and the orbits of arrival in their columns. Each entry of the table stands in a row and in a column: it describes the leap from one orbit to another. He spends his time on the island trying to use these tables to calculate something that could justify Bohr's rules. He doesn't get much sleep. But he fails to do the math for the electron in the atom: too difficult. He tries to account for a simpler system instead, choosing a pendulum, and looks for Bohr's rules in this simpler case.
On June 7, something begins to click:
When the first terms seemed to come right [giving Bohr's rules], I became excited, making one mathematical error after another. As a consequence, it was around three o'clock in the morning when the result of my calculations lay before me. It was correct in all terms.
Suddenly I no longer had any doubts about the consistency of the new "quantum" mechanics that my calculation described.
At first, I was deeply alarmed. I had the feeling that I had gone beyond the surface of things and was beginning to see a strangely beautiful interior, and felt dizzy at the thought that now I had to investigate this wealth of mathematical structures that Nature had so generously spread out before me.
It takes our breath away. Beyond the surface of things, "a strangely beautiful interior." Heisenberg's words resonate with those written by Galileo on first seeing the mathematical regularity appear in his measurements of the fall of objects along an inclined plane: the first mathematical law describing the motion of objects on Earth ever discovered by humankind. Nothing is like the emotion of seeing a mathematical law behind the disorder of appearances.
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.
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. 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.
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. 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. 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." He completes it, with acrobatic technicality, in the space of a few weeks.
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." 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."
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 felt great admiration for Heisenberg when, at just twenty-three, he managed it in one swoop."
Except for Born, who is in his forties, Heisenberg, Jordan, Dirac and Pauli are all twentysomethings. In Gšttingen they call their physics Knabenphysik, or "boys' physics."
I
A STRANGELY BEAUTIFUL INTERIOR
How a young German physicist
arrived at an idea that was very
strange indeed, but described
the world remarkably well-and
the great confusion that followed.
The Absurd Idea of the Young Heisenberg: Observables
It was around three o'clock in the morning when the final results of my calculations were before me. I felt profoundly shaken. I was so agitated that I could not sleep. I left the house and began walking slowly in the dark. I climbed on a rock overlooking the sea at the tip of the island, and waited for the sun to come up . . .
I have often wondered what the thoughts and emotions of the young Heisenberg must have been as he clambered over that rock overlooking the sea, on the barren and windswept North Sea island of Helgoland, facing the vastness of the waves and awaiting the sunrise, after having been the first to glimpse one of the most vertiginous of Nature's secrets ever looked upon by humankind. He was twenty-three.
He was on the island seeking relief from the allergy that afflicted him. Helgoland-the name means Sacred Island-has virtually no trees, and very little pollen. ("Heligoland with its one tree," as James Joyce has it in Ulysses.) Perhaps the legends of the dreadful pirate Stšrtebeker hiding on the island, which Heisenberg loved as a boy, were in his mind as well. But Heisenberg's main reason for being there was to immerse himself in the problem with which he was obsessed, the burning issue handed to him by Niels Bohr. He slept little and spent his time in solitude, trying to calculate something that would justify Bohr's incomprehensible rules. Every so often, he would take a break to climb over the island's rocks or learn by heart poetry from Goethe's West-Eastern Divan, the collection in which Germany's greatest poet sings his love for Islam.
Niels Bohr was already a renowned scientist. He had written formulas, simple but strange, that predicted the properties of chemical elements even before measuring them. They predicted, for instance, the frequency of light emitted by elements when heated: the color they assume. This was a remarkable achievement. The formulas, however, were incomplete: they did not give, for instance, the intensity of the emitted light.
But above all, these formulas had about them something that was truly absurd. They assumed, for no good reason, that the electrons in atoms orbited around the nucleus only on certain precise orbits, at certain precise distances from the nucleus, with certain precise energies-before magically "leaping" from one orbit to another. The first quantum leaps. Why only these orbits? Why these incongruous "leaps" from one orbit to another? What force could possibly cause such bizarre behavior as this?
The atom is the building block of everything. How does it work? How do the electrons move inside it? The scientists of the beginning of the century had been pondering these questions for more than a decade, without getting anywhere.
Like a Renaissance master painter in his studio, Bohr had gathered around him in Copenhagen the very best young physicists he could find, to work together on the mysteries of the atom. Among them was the brilliant Wolfgang Pauli-Heisenberg's extremely intelligent, pretty arrogant friend and former classmate. But Pauli had recommended Heisenberg to the great Bohr, saying that to make any real progress, he was needed. Bohr had taken the advice, and in the autumn of 1924 had brought Heisenberg to Copenhagen from Gšttingen, where he was working as an assistant to the physicist Max Born. Heisenberg had spent a few months in long discussions with Bohr, in Copenhagen, in front of blackboards covered with formulas. The young apprentice and the master had taken long walks together in the mountains, talking about the enigmas of the atom; about physics and philosophy.
Heisenberg had steeped himself in the problem. It had become his obsession. Like the others, he had tried everything. Nothing worked. There seemed to be no reasonable force capable of guiding the electrons on Bohr's strange orbits, and in his peculiar leaps. And yet those orbits and those leaps really did lead to good predictions of atomic phenomena. Confusion.
Desperation pushes us to look for extreme solutions. On that island in the North Sea, in complete solitude, Heisenberg resolved to explore radical ideas.
It was with radical ideas, after all, that twenty years earlier Einstein had astonished the world. Einstein's radicalism had worked. Pauli and Heisenberg were enamored of his physics. Einstein for them was a legend. Had the time perhaps come, they asked themselves, to hazard as radical a step, to escape from the impasse regarding electrons in atoms? Could they be the ones to take it? In your twenties, you can dream freely.
Einstein had shown that even our most rooted convictions can be wrong. What seems most obvious to us now might turn out not to be correct. Abandoning assumptions that seem self-evident can lead to greater understanding. Einstein had taught that everything should be based on what we see, not on what we assume to exist.
Pauli repeated these ideas to Heisenberg. The two young men had drunk deep of this poisoned honey. They had been following the discussions on the relation between reality and experience that ran through Austrian and German philosophy at the beginning of the century. Ernst Mach, who had exerted a decisive influence on Einstein, insisted that knowledge had to be based solely on observations, freed of any implicit "metaphysical" assumption. These were the ingredients coming together in the young Heisenberg's thinking, like the chemical components of an explosive, as he isolated himself on Helgoland in the summer of 1925.
And here he had the idea. An idea that could only be had with the unfettered radicalism of the young. The idea that would transform physics in its entirety-together with the whole of science and our very conception of the world. An idea, I believe, that humanity has not yet fully absorbed.
Heisenberg's leap is as daring as it is simple. No one has been able to find the force capable of causing the bizarre behavior of electrons? Fine, let's stop searching for this new force. Let's use instead the force we are familiar with: the electric force that binds the electron to the nucleus. We cannot find new laws of motion to account for Bohr's orbits and his "leaps"? Fine, let's stick with the laws of motion that we're familiar with, without altering them.
Let's change, instead, our way of thinking about the electron. Let's give up describing its movement. Let's describe only what we can observe: the light it emits. Let's base everything on quantities that are observable. This is the idea.
Heisenberg attempts to recalculate the behavior of the electron using quantities we observe: the frequency and amplitude of emitted light.
We can observe the effects of the electron's leaps from one of Bohr's orbits to another. Heisenberg replaces the physical variables (numbers) with tables of numbers that have the orbits of departure in their rows and the orbits of arrival in their columns. Each entry of the table stands in a row and in a column: it describes the leap from one orbit to another. He spends his time on the island trying to use these tables to calculate something that could justify Bohr's rules. He doesn't get much sleep. But he fails to do the math for the electron in the atom: too difficult. He tries to account for a simpler system instead, choosing a pendulum, and looks for Bohr's rules in this simpler case.
On June 7, something begins to click:
When the first terms seemed to come right [giving Bohr's rules], I became excited, making one mathematical error after another. As a consequence, it was around three o'clock in the morning when the result of my calculations lay before me. It was correct in all terms.
Suddenly I no longer had any doubts about the consistency of the new "quantum" mechanics that my calculation described.
At first, I was deeply alarmed. I had the feeling that I had gone beyond the surface of things and was beginning to see a strangely beautiful interior, and felt dizzy at the thought that now I had to investigate this wealth of mathematical structures that Nature had so generously spread out before me.
It takes our breath away. Beyond the surface of things, "a strangely beautiful interior." Heisenberg's words resonate with those written by Galileo on first seeing the mathematical regularity appear in his measurements of the fall of objects along an inclined plane: the first mathematical law describing the motion of objects on Earth ever discovered by humankind. Nothing is like the emotion of seeing a mathematical law behind the disorder of appearances.
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.
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. 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.
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. 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. 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." He completes it, with acrobatic technicality, in the space of a few weeks.
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." 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."
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 felt great admiration for Heisenberg when, at just twenty-three, he managed it in one swoop."
Except for Born, who is in his forties, Heisenberg, Jordan, Dirac and Pauli are all twentysomethings. In Gšttingen they call their physics Knabenphysik, or "boys' physics."
