When Schrödinger boarded his train out of Copenhagen, neither he nor Bohr seemed to have ceded any ground to the other, but both were shaken. Each had argued passionately for days without winning the other over, and also without finding a reply to every challenge put by the other. Both men could see that they did not fully understand their own ground. Max Born could proclaim himself “entirely satisfied” physicswise, but perhaps that revealed an undemanding nature. Bohr was not satisfied, and after Schrödinger’s departure he and Heisenberg worked intently together in an effort to gain a practical understanding of quantum physics through and through.
As Denmark’s nights lengthened, the two spent more and more time talking about how the physics worked. They struggled logically through one experiment after another, figuring out its math and what the solution suggested about the physics of the event. Colleagues though they were, and evident as the respect in which each held the other, their imaginations worked very differently. Heisenberg’s skill and thinking ran to mathematics. If the equations covered the ground, he would be satisfied.
Bohr wanted to find coherent, unambiguous language that described the logical nature of the case.
Heisenberg thought mathematics provided all the coherence and precision needed.
Mathematics, however, was not the language Bohr wanted to use, or at least formal math was not sufficient. He wanted to know what the numbers were about. The 2 × 5 = 10 kind of counting was too empty for him. He preferred the sort that says two sandwiches for each of five schoolchildren yields 10 empty sandwich wrappers. So how are we to understand that p times q does not equal q times p? Max Born was satisfied with a purely mathematical explanation, but Bohr wondered, and he wondered, too, about how we are to talk about quanta if sometimes it is a particle and sometimes a wave? The triumph of the Compton effect had established the correctness of Einstein’s idea, but where was its sense?
Heisenberg and Bohr thrashed over these questions, but their styles were too incompatible. It was like trying to compose an opera with music by a formalist like J.S. Bach and libretto by an anti-formalist like Ezra Pound. Each might admire the other’s genius, but together they were not going to advance far. At the same time, they stimulated so many thoughts in the other that the moment the two separated, their minds erupted with fruitful ideas.
Bohr left Heisenberg alone at the end of February 1927, when he set off to vacation in Norway, and immediately ideas began to blossom. Heisenberg continued thinking about quantum experiments and how to understand them, but with Bohr gone, Heisenberg suddenly recalled Einstein saying that “Theory determines what we observe,” and he decided to see what his theory observed.
Einstein could have told him he was stepping into a logical circle. Heisenberg had said “Here is what we can observe,” and he developed a theory to match that. Now he was asking, “Here is the theory, what does it say we can observe?” Not surprisingly, this ride let him off right where he got on board. But the journey was not sterile because it led him to look more deeply into the implications of what he had wrought.
The most important insight was one that has become known variously as the uncertainty principle or the indeterminacy principle. The “uncertainty” term is the less radical sounding of the two. Uncertainty
is part of everybody’s life, and nobody will be surprised to learn that there are details of an experience that are beyond discovery. “Indeterminacy,” however, radically implies there is something in the event outside physical causes. The confusion over terms reflects the deeper confusion over the principle’s meaning. Most readers are probably more familiar with the uncertainty term, which was the word Heisenberg used, although many physicists, including Bohr, preferred indeterminacy. This history will take a neutral course and refer to it as Heisenberg’s principle.
Such quarreling over interpretation was one of the things science had seemed to abolish. Philosophers had once argued interminably about the interpretation of the most elementary facts, but science after Galileo and Newton made tremendous strides by somehow limiting that tendency. Galileo, like philosophers before him, wrote dialogues in which people disputed the meaning of facts, but once the science of natural law took hold, mathematics and meaning went together. But with quantum physics, from the start, the meaning was elusive. Its laws predicted experience without explaining it.
Planck introduced his hυ. What was that?
An element of energy.
Okay, but what exactly is it?
Maybe it is a particle, maybe a wave, maybe a wave packet. Whatever it is, it is omnipresent.
In its mathematical form, Heisenberg’s principle states that when you subtract qp from pq you will get, as a minimum, Planck’s h. The quantum thus becomes an expression of something—uncertainty, indeterminacy, we cannot quite agree on what it means, but plainly it lies at the heart of the universe.
But what is that heart doing? What does it mean? Philosophically, it puts an end to the hope for a full accounting of quantum events. Heisenberg’s principle works like a balance on the scales of justice in which a jury must always find reasonable doubt. If an investigator gets one side of the scale absolutely right, the other side becomes so hazy that the jury cannot know what happened there.
Well, ponders a juror, why not solve one side and get it right and then solve the other side and get it right. Then the investigator will
have solved both sides. But that common-sense effort misses the power of Heisenberg’s principle. We cannot solve both sides at the same time. Examining the one side permanently obscures events on the other side. Heisenberg said that if we design an experiment to measure both sides of the scale at once, neither answer will be exact. The balance will swing like a doctor’s scale gone mad in which the pointer flits up and down and makes it impossible for anybody to determine where reality lies.
Colorful though this image is, it does not get at the physics of the problem. Why must the scale swing so wildly? Heisenberg’s paper attempted to give reasons. His essay is generally considered revolutionary. Compared with Newtonian doctrines, it was. From the perspective of the papers in the early days of the quantum revolution, however, Heisenberg was beating a retreat toward normalcy. His most radical early idea—the absence of space and time from quantum mechanics—was now abandoned, replaced by his new principle. In this new paper, things happened at specific times and in definite locations, although we might never be able to discover those exact times and places. Heisenberg also backed away from Max Born’s radical probabilities and did not propose to explain his uncertainties in terms of pure chance.
A mathematical view might be that we can know only the probable values of qp, not its exact value, but Heisenberg did look for underlying physical reasons. His most famous example envisioned an experiment in which a physicist tries to look at an electron through a gamma-ray microscope. (Gamma rays are similar to X-rays, but have more energy.) Picking a microscope was Heisenberg’s way of thumbing his nose at his theory’s vocal enemy, Wilhelm Wien. During the oral exam for Heisenberg’s doctorate, Wien had asked Heisenberg about the precision of a microscope’s optics? What were the elements that determined how clearly an object could be studied? Heisenberg answered so miserably that Wien wanted to fail him. So Heisenberg was getting a bit of his own back when he asked how you can use a microscope to determine precisely where an electron is. You can peer through the gamma-ray microscope; however, thanks to the Compton effect, the ray that finds the electron and shows its location also knock
the electron, scattering it and changing its momentum. Finding out one thing, Heisenberg said, changes and obscures something else.
Heisenberg also used uncertainty to settle the long-standing enigma of the quantum jump. His solution was to say that when energy levels were known very exactly, the time could be known only inexactly. An experiment could say only that the energy change took place over a time of uncertain duration. Thus, the change in electron position no longer looked quite so miraculous.
Finally, Heisenberg’s paper beat a retreat from Max Born’s radical insistence on probabilities. Heisenberg ended his paper with the remark, “Quantum mechanics establishes the final failure of causality.” That sounds plenty radical, but a few sentences earlier Heisenberg had undermined this stance by explaining that the error lies in believing that “When we know the present precisely, we can predict the future.” We cannot predict the future because we cannot know the present precisely. That is much tamer than Max Born’s doctrine that we cannot predict the future because the present is free to act as it chooses. This backing away from quantum’s radicality made the whole revolution much easier for many physicists to accept. But Einstein, who from the beginning said the quantum was “revolutionary,” was not to be assuaged by still more cooling. As long as Ψ, the key element of the new quantum mechanics, had no real-world meaning, he was not going to quiet down. Einstein wanted to press on with the revolution and find new concepts that would explain what psi really was.
When he finished his paper, Heisenberg dashed a copy off to his comrade in calculations, Wolfgang Pauli. Not surprisingly, Pauli thought it was great. After all, it was Pauli who has pointed out that uncertainty is implicit in the math of quantum mechanics. Then Bohr returned from the chills of his Norwegian winter, excited by the thoughts he had developed beneath the northern lights. He was startled to learn that his assistant had been busy, too, and had already submitted an article to the leading journal of quantum physics, Zeitschrift für Physik.
To Heisenberg’s dismay, Bohr was displeased with the new paper, finding it wrong in some of its physics, often superficial, and insufficiently radical. Bohr in Norway had concluded that quantum physics
stands in perpetual contradiction to classical physics, and he was most unhappy to see a paper from his institute hide this decisive contradiction behind a screen of uncertainty. A new, ferocious argument erupted between Bohr and Heisenberg. Its centerpiece was Bohr’s insistence that Heisenberg withdraw his article so that it could be revised to agree with Bohr’s own pending article. Heisenberg refused to change a word.
Perhaps it was during this ordeal that Heisenberg developed his sympathy with the harangue Schrödinger had endured. The ceaseless demand for surrender came not merely from a senior colleague or admired genius, but from Heisenberg’s hero and employer. Could he resist such pressure forever?
What was wrong with the paper?
For one thing, the experiment with the gamma-ray microscope was wrong. As he had during his oral examination, Heisenberg again missed the subtleties of the system. The limitations in Heisenberg’s equation were correct, but not for the mechanical reasons Heisenberg had given.
What were the reasons then?
Bohr could not explain himself. His instincts told him that Heisenberg’s physics was too clumsy and that the real reason was deeper, less mechanical. The paper, Bohr insisted, was often superficial.
Heisenberg thought not.
Striking examples of superficiality lay in the article’s many references to Einstein. The name pops up throughout the essay, starting with his second sentence, and Heisenberg specifically sought to compare his work with Einstein’s, boasting that “it is natural [in its uncertainty relation] to compare quantum theory with special relativity.” Heisenberg then went on to compare Einstein’s idea that nothing can be absolutely simultaneous with his own notion that position is uncertain.
Heisenberg liked the claim, but it missed Einstein’s achievement of removing ambiguity from his account of physical phenomena. Bohr had an idea for removing all ambiguity from quantum phenomena as well, and he wanted Heisenberg to hold off publication until that idea could be included.
Heisenberg was revising nothing. Was there nothing in the paper for Bohr to admire?
The moments of untamed radicality remained admirable in Bohr’s eyes.
The radicality of the uncertainty mathematics itself was good. It asserted that you could never see the whole of a phenomenon at once. If p and q referred to Newtonian concepts of position and momentum, you could not get a complete Newtonian picture of quantum behavior. Or if p and q symbolized thermodynamic notions of time and energy, you could not get a complete thermodynamic picture of quantum change. This limited view of reality fit well with Bohr’s epiphany in the Norwegian wood.
Bohr also liked the kind of thinking that led Heisenberg to write “the ‘orbit’ [of an electron] comes into being only when we observe it … [it] can be calculated only statistically from the initial conditions.” This kind of antirealism appealed to Bohr very much, and doubtless he also found plenty to enjoy in Heisenberg’s remark that although a person might argue “that behind the perceived statistical world there still hides a ‘real’ world in which causality holds, … such speculation seem to us, to say it explicitly, fruitless and senseless.”
Einstein would have replied that the real world can be found through new concepts that show the connections between statistical observations, but Bohr did not think that way and wanted no changes in that approach.
Mostly, however, Bohr did want changes in the paper and Heisenberg firmly refused to make them.
This dispute could end only in ugliness or with intervention from the gods. In this case, the gods lowered a rescuer into the scene. Pauli suddenly arrived in Copenhagen for a visit, and both men fell on him with their accounts of the disagreement.
After listening to both sides Pauli told them they were not in dispute.
Then what had they been fighting about?
Heisenberg, Pauli said, had presented an account of the physical meaning of quantum theory that agreed with Bohr, but Bohr had pushed the idea to greater depth. If only each of them could calm
down and understand the other, they would both see that there was no fundamental disagreement between them.
It was the great tactic of conflict resolution—find a point of common ground and declare it to be the central ground.
Heisenberg did not pull his paper from publication, but he did add a passage to its closing. He retreated from a few positions, notably, admitting that the gamma-ray experiment is “not so simple as was assumed.”
Mostly the effect of Heisenberg’s addendum was like the “to be continued” ending in a serialized Charles Dickens episode. Much wonder and amazement was inspired by the chapter in print, and yet it served largely to make readers pace eagerly in anticipation of what would come next. “I owe great thanks to Professor Bohr,” Heisenberg concluded, “for sharing with me at an early stage the results of these more recent investigations of his—to appear soon in a paper on the conceptual structure of quantum theory—and for discussing them with me.”
With these words, Heisenberg put the quantum revolution on hold until Bohr’s paper should “appear soon.” And everybody knew that Bohr was a slow writer.