A Brief History of Quantum Mechanics

Appendix A of
The Strange World of Quantum Mechanics

This World Wide Web page written by Dan Styer, Oberlin College Physics Department;
copyright © Daniel F. Styer 1999

Up to now this book has focused on the behavior of nature. I could say more: more about measurement, more about the classical limit, more about different rules for assigning amplitudes, and so forth, but the main points have been made. So instead of talking more about nature I'm going to talk about people -- about how people discovered quantum mechanics.


I am not a historian of science. The history of science is a very difficult field. A historian of science must be just as proficient at science as a scientist is, but must also have a good understanding of personalities, and a good knowledge of the social and political background that affects developments in science and that is in turn affected by those developments. He or she has to know not only the outcome of the historical process, namely the science that is generally accepted today, but also the many false turns and blind alleys that scientists tripped across in the process of discovering what we believe today. He or she must understand not only the cleanest and most direct experimental evidence supporting our current theories (like the evidence presented in this book), but must understand also how those theories came to be accepted through a tightly interconnected web of many experiments, no one of which was completely convincing but which taken together presented an overwhelming argument.

Thus a full history of quantum mechanics would have to discuss Schrödinger's many mistresses, Ehrenfest's suicide, and Heisenberg's involvement with Nazism. It would have to treat the First World War's effect on the development of science. It would need to mention "the Thomson model" of the atom, which was once the major competing theory to quantum mechanics. It would have to give appropriate weight to both theoretical and experimental developments. Needless to say, such a complete history will never be written, and this brief appendix will not even broach most of these topics. The references in the printed book will lead you to further information.

The historian of science has problems beyond even these. The work of government is generally carried out through the exchange of written memos, and when verbal arguments are used (as in Congressional hearings) detailed written transcripts are maintained. These records are stored in archives to insure that historians interested in government decisions will have access to them. Historians of science do not have such advantages. Much of the work of science is done through informal conversations, and the resulting written record is often sanitized to avoid offending competing scientists. The invaluable oral record is passed down from professor to student repeatedly before anyone ever records it on paper. Naturally, the stories tend to become better and better as they are transmitted over and over. In addition, there is a tendency for the exciting stories to be repeated and the dull ones to be forgotten, leading to a Darwinian "survival of the funniest" -- rather than of the most accurate.

Finally, once all the historical records have been sifted and analyzed, there remains the problem of overall synthesis and presentation. Many scientific historians (and even more scientists) like to tell a story in which each step follows naturally from the one preceding it, scientists always work cooperatively and selflessly, and where harmony rules. Such stories infuriate me. They remind me of the stock market analysts who come onto television every evening and explain in detail the cause of every dip and curve in the Dow for the preceding day. If they know the stock market so well, why do they wait until evening to tell me about it? Why don't they tell me in the morning so that it can do me some good? For that matter, why are they on television at all, rather than out relaxing on their million-dollar yachts? The fact is that scientific history, like the stock market and like everyday life, does not proceed in an orderly, coherent pattern. The story of quantum mechanics is a story full of serendipity, personal squabbles, opportunities missed and taken, and of luck both good and bad.

Because I find the sugar-sweet stories of the harmonious development of science to be so offensive, when I tell the story I emphasize the conflicts, the contingencies, and the unpredictablities. Hence the story I tell is no more accurate than the sweet talk, because I go too far in the opposite direction. Keep in mind, as you read the story that follows, that I suffer from this overreaction as well as all the other difficulties mentioned in this section.

Status of physics: January 1900

In January 1900 the atomic hypothesis was widely but not universally accepted. Atoms were considered point particles, and it wasn't clear how atoms of different elements differed. The electron had just been discovered (1897) and it wasn't clear where (or even whether) electrons were located within atoms. One important outstanding problem concerned the colors emitted by atoms in a discharge tube (familiar today as the light from a fluorescent tube or from a neon sign). No one could understand why different gas atoms glowed in different colors. Another outstanding problem concerned the amount of heat required to change the temperature of a diatomic gas such as oxygen: the measured amounts were well below the value predicted by theory. Because quantum mechanics is important when applied to atomic phenomena, you might guess that investigations into questions like these would give rise to the discovery of quantum mechanics. Instead it came from a study of heat radiation.

Heat radiation

You know that the coals of a campfire, or the coils of an electric stove, glow red. You probably don't know that even hotter objects glow white, but this fact is well known to blacksmiths. When objects are hotter still they glow blue. (This is why a gas stove should be adjusted to make a blue flame.) Indeed, objects at room temperature also glow (radiate), but the radiation they emit is infrared, which is not detectable by the eye. (The military has developed -- for use in night warfare -- special eye sets that convert infrared radiation to optical radiation.)

These observations can be explained qualitatively by thinking of heat as a jiggling of atoms: like jello, but on a smaller scale so that you can't see the vibrations due to heat. At higher temperatures the atoms jiggle both farther and faster. The increased distance of jiggling accounts for the brighter radiation from hotter bodies, while the increased speed accounts for the change in color.

In the year 1900 several scientists were trying to turn these observations into a detailed explanation of and a quantitatively accurate formula for the color of heat radiation as a function of temperature. On 19 October 1900 the Berliner Max Planck (age 42) announced a formula that fit the experimental results perfectly, yet he had no explanation for the formula -- it just happened to fit. He worked to find an explanation through the late fall and finally was able to derive his formula by assuming that the atomic jigglers could not take on any possible energy, but only certain special "allowed" values. He announced this result on 14 December 1900. This date is now considered the birthday of quantum mechanics (and there is certain to be a big celebration on its one hundredth anniversary) but at the time no one found it particularly significant. We know this not only from contemporary reports, but also because the assumption of allowed energy values raises certain obvious questions that no one bothered to follow up. For example, how does the jiggler change from one allowed energy to another if the intermediate energies are prohibited? Again, if a jiggling atom can only assume certain allowed values of energy, then there must also be restrictions on the positions and speeds that the atom can have. What are they? Planck never tried to find out.

Thirty-one years after his discovery Planck wrote:

I can characterize the whole procedure as an act of desperation, since, by nature I am peaceable and opposed to doubtful adventures. However, I had already fought for six years (since 1894) with the problem of equilibrium between radiation and matter without arriving at any successful result. I was aware that this problem was of fundamental importance in physics, and I knew the formula describing the energy distribution . . . hence a theoretical interpretation had to be found at any price, however high it might be.
It should be clear from what I have already said that this is just a beautiful and romantic story that was developed with good thirty-year hindsight. Here is another wonderful story, this one related by Werner Heisenberg:

In a period of most intensive work during the summer of 1900 [Planck] finally convinced himself that there was no way of escaping from this conclusion [of "allowed" energies]. It was told by Planck's son that his father spoke to him about his new ideas on a long walk through the Grunewald, the wood in the suburbs of Berlin. On this walk he explained that he felt he had possibly made a discovery of the first rank, comparable perhaps only to the discoveries of Newton.
As much as I would like for this beautiful story to be true, the intensive work took place during the late fall, not the summer, of 1900. If Planck did indeed take his son for a long walk on the afternoon that he discovered quantum mechanics, the son would probably remember the nasty cold he caught better than any remarks his father made.

The old quantum theory

Although the ideas of Planck did not take the world by storm, they did develop a growing following and were applied to more and more situations. The resulting ideas, now called "old quantum theory", were all of the same type: Classical mechanics was assumed to hold, but with the additional assumption that only certain values of a physical quantity (the energy, say, or the projection of a magnetic arrow) were allowed. Any such quantity was said to be "quantized". The trick seemed to be to guess the right quantization rules for the situation under study, or to find a general set of quantization rules that would work for all situations.

For example, in 1905 Albert Einstein (age 26) postulated that the total energy of a beam of light is quantized. Just one year later he used quantization ideas to explain the heat/temperature puzzle for diatomic gases. Five years after that, in 1911, Arnold Sommerfeld (age 43) at Munich began working on the implications of energy quantization for position and speed.

In the same year Ernest Rutherford (age 40), a New Zealander doing experiments in Manchester, England, discovered the atomic nucleus -- only at this relatively late stage in the development of quantum mechanics did physicists have even a qualitatively correct picture of the atom! In 1913, Niels Bohr (age 28), a Dane who had recently worked in Rutherford's laboratory, introduced quantization ideas for the hydrogen atom. His theory was remarkably successful in explaining the colors emitted by hydrogen glowing in a discharge tube, and it sparked enormous interest in developing and extending the old quantum theory.

This development was hindered but not halted completely by the start of the First World War in 1914. During the war (in 1915) William Wilson (age 40, a native of Cumberland, England, working at King's College in London) made progress on the implications of energy quantization for position and speed, and Sommerfeld also continued his work in that direction.

With the coming of the armistice in 1918, work in quantum mechanics expanded rapidly. Many theories were suggested and many experiments performed. To cite just one example, in 1922 Otto Stern and his graduate student Walther Gerlach (ages 34 and 23) performed their important experiment that is so essential to the way this book presents quantum mechanics. Jagdish Mehra and Helmut Rechenberg, in their monumental history of quantum mechanics, describe the situation at this juncture well:

At the turn of the year from 1922 to 1923, the physicists looked forward with enormous enthusiasm towards detailed solutions of the outstanding problems, such as the helium problem and the problem of the anomalous Zeeman effects. However, within less than a year, the investigation of these problems revealed an almost complete failure of Bohr's atomic theory.

The matrix formulation of quantum mechanics

As more and more situations were encountered, more and more recipes for allowed values were required. This development took place mostly at Niels Bohr's Institute for Theoretical Physics in Copenhagen, and at the University of Göttingen in northern Germany. The most important actors at Göttingen were Max Born (age 43, an established professor) and Werner Heisenberg (age 23, a freshly minted Ph.D. from Sommerfeld in Munich). According to Born "At Göttingen we also took part in the attempts to distill the unknown mechanics of the atom out of the experimental results. . . . The art of guessing correct formulas . . . was brought to considerable perfection."

Heisenberg particularly was interested in general methods for making guesses. He began to develop systematic tables of allowed physical quantities, be they energies, or positions, or speeds. Born looked at these tables and saw that they could be interpreted as mathematical matrices. Fifty years later matrix mathematics would be taught even in high schools. But in 1925 it was an advanced and abstract technique, and Heisenberg struggled with it. His work was cut short in June 1925.

It was late spring in Göttingen, and Heisenberg suffered from an allergy attack so severe that he could hardly work. He asked his research director, Max Born, for a vacation, and spent it on the rocky North Sea island of Helgoland. At first he was so ill that could only stay in his rented room and admire the view of the sea. As his condition improved he began to take walks and to swim. With further improvement he began also to read Goethe and to work on physics. With nothing to distract him, he concentrated intensely on the problems that had faced him in Göttingen.

Heisenberg reproduced his earlier work, cleaning up the mathematics and simplifying the formulation. He worried that the mathematical scheme he invented might prove to be inconsistent, and in particular that it might violate the principle of the conservation of energy. In Heisenberg's own words:

One evening I reached the point where I was ready to determine the individual terms in the energy table, or, as we put it today, in the energy matrix, by what would now be considered an extremely clumsy series of calculations. When the first terms seemed to accord with the energy principle, I became rather excited, and I began to make countless arithmetical errors. As a result, it was almost three o'clock in the morning before the final result of my computations lay before me. The energy principle had held for all the terms, and I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. I was far too excited to sleep, and so, as a new day dawned, I made for the southern tip of the island, where I had been longing to climb a rock jutting out into the sea. I now did so without too much trouble, and waited for the sun to rise.

By the end of the summer Heisenberg, Born, and Pascual Jordan (age 22) had developed a complete and consistent theory of quantum mechanics. (Jordan had entered the collaboration when he overheard Born discussing quantum mechanics with a colleague on a train.)

This theory, called "matrix mechanics" or "the matrix formulation of quantum mechanics", is not the theory I have presented in this book. It is extremely and intrinsically mathematical, and even for master mathematicians it was difficult to work with. Although we now know it to be complete and consistent, this wasn't clear until much later. Heisenberg had been keeping Wolfgang Pauli apprised of his progress. (Pauli, age 25, was Heisenberg's friend from graduate student days, when they studied together under Sommerfeld.) Pauli found the work too mathematical for his tastes, and called it "Göttingen's deluge of formal learning". On 12 October 1925 Heisenberg could stand Pauli's biting criticism no longer. He wrote to Pauli:

With respect to both of your last letters I must preach you a sermon, and beg your pardon for proceeding in Bavarian: It is really a pigsty that you cannot stop indulging in a slanging match. Your eternal reviling of Copenhagen and Göttingen is a shrieking scandal. You will have to allow that, in any case, we are not seeking to ruin physics out of malicious intent. When you reproach us that we are such big donkeys that we have never produced anything new in physics, it may well be true. But then, you are also an equally big jackass because you have not accomplished it either . . . . . . (The dots denote a curse of about two-minute duration!) Do not think badly of me and many greetings.

The wavefunction formulation of quantum mechanics

While this work was going on at Göttingen and Helgoland, others were busy as well. In 1923 Louis de Broglie (age 31), associated an "internal periodic phenomenon" -- a wave -- with a particle. He was never very precise about just what that meant. (De Broglie is sometimes called "Prince de Broglie" because his family descended from the French nobility. To be strictly correct, however, only his eldest brother could claim the title.)

It fell to Erwin Schr&oum;ldinger, an Austrian working in Zürich, to build this vague idea into a theory of wave mechanics. He did so during the Christmas season of 1925 (at age 38), at the alpine resort of Arosa, Switzerland, in the company of "an old girlfriend [from] Vienna", while his wife stayed home in Zürich.

In short, just twenty-five years after Planck glimpsed the first sight of a new physics, there was not one, but two competing versions of that new physics! The two versions seemed utterly different and there was an acrimonious debate over which one was correct. In a footnote to a 1926 paper Schrödinger claimed to be "discouraged, if not repelled" by matrix mechanics. Meanwhile, Heisenberg wrote to Pauli (8 June 1926) that

The more I think of the physical part of the Schrödinger theory, the more detestable I find it. What Schrödinger writes about visualization makes scarcely any sense, in other words I think it is shit. The greatest result of his theory is the calculation of matrix elements.
Fortunately the debate was soon stilled: in 1926 Schrödinger and, independently, Carl Eckert (age 24) of Caltech proved that the two new mechanics, although very different in superficial appearance, were equivalent to each other. [Very much as the process of adding arabic numerals is quite different from the process of adding roman numerals, but the two processes nevertheless always give the same result.] (Pauli also proved this, but never published the result.)


With not just one, but two complete formulations of quantum mechanics in hand, the quantum theory grew explosively. It was applied to atoms, molecules, and solids. It solved with ease the problem of helium that had defeated the old quantum theory. It resolved questions concerning the structure of stars, the nature of superconductors, and the properties of magnets. One particularly important contributor was P.A.M. Dirac, who in 1926 (at age 22) extended the theory to relativistic and field-theoretic situations. Another was Linus Pauling, who in 1931 (at age 30) developed quantum mechanical ideas to explain chemical bonding, which previously had been understood only on empirical grounds. Even today quantum mechanics is being applied to new problems and new situations. It would be impossible to mention all of them. All I can say is that quantum mechanics, strange though it may be, has been tremendously successful.

The Bohr-Einstein debate

The extraordinary success of quantum mechanics in applications did not overwhelm everyone. A number of scientists, including Schrödinger, de Broglie, and -- most prominently -- Einstein, remained unhappy with the standard probabilistic interpretation of quantum mechanics. In a letter to Max Born (4 December 1926), Einstein made his famous statement that

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.
In concrete terms, Einstein's "inner voice" led him, until his death, to issue occasional detailed critiques of quantum mechanics and its probabilistic interpretation. Niels Bohr undertook to reply to these critiques, and the resulting exchange is now called the "Bohr-Einstein debate". At one memorable stage of the debate (Fifth Solvay Congress, 1927), Einstein made an objection similar to the one quoted above and Bohr

replied by pointing out the great caution, already called for by ancient thinkers, in ascribing attributes to Providence in every-day language.
These two statements are often paraphrased as, Einstein to Bohr: "God does not play dice with the universe." Bohr to Einstein: "Stop telling God how to behave!" While the actual exchange was not quite so dramatic and quick as the paraphrase would have it, there was nevertheless a wonderful rejoinder from what must have been a severely exasperated Bohr.

The Bohr-Einstein debate had the benefit of forcing the creators of quantum mechanics to sharpen their reasoning and face the consequences of their theory in its most starkly non-intuitive situations. It also had (in my opinion) one disastrous consequence: because Einstein phrased his objections in purely classical terms, Bohr was compelled to reply in nearly classical terms, giving the impression that in quantum mechanics, an electron is "really classical" but that somehow nature puts limits on how well we can determine those classical properties. I have tried in this book to convince you that this is a misconception: the reason we cannot measure simultaneously the exact position and speed of an electron is because an electron does not have simultaneously an exact position and speed. It is no defect in our measuring instruments that they cannot measure what does not exist. This is simply the character of an electron -- an electron is not just a smaller, harder edition of a marble. This misconception -- this picture of a classical world underlying the quantum world -- poisoned my own understanding of quantum mechanics for years. I hope that you will be able to avoid it.

On the other hand, the Bohr-Einstein debate also had at least one salutary product. In 1935 Einstein, in collaboration with Boris Podolsky and Nathan Rosen, invented a situation in which the results of quantum mechanics seemed completely at odds with common sense, a situation in which the measurement of a particle at one location could reveal instantly information about a second particle far away. The three scientists published a paper which claimed that "No reasonable definition of reality could be expected to permit this." Bohr produced a recondite response and the issue was forgotten by most physicists, who were justifiably busy with the applications of rather than the foundations of quantum mechanics. But the ideas did not vanish entirely, and they eventually raised the interest of John Bell. In 1964 Bell used the Einstein-Podolsky-Rosen situation to produce a theorem about the results from certain distant measurements for any deterministic scheme, not just classical mechanics. In 1982 Alain Aspect and his collaborators put Bell's theorem to the test and found that nature did indeed behave in the manner that Einstein (and others!) found so counterintuitive.

The amplitude formulation of quantum mechanics

The version of quantum mechanics presented in this book is neither matrix nor wave mechanics. It is yet another formulation, different in approach and outlook, but fundamentally equivalent to the two formulations already mentioned. It is called amplitude mechanics (or "the sum over histories technique", or "the many paths approach", or "the path integral formulation", or "the Lagrangian approach", or "the method of least action"), and it was developed by Richard Feynman in 1941 while he was a graduate student (age 23) at Princeton. Its discovery is well described by Feynman himself in his Nobel lecture:

I went to a beer party in the Nassau Tavern in Princeton. There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think a good place to discuss intellectual matters is a beer party. So he sat by me and asked, "What are you doing" and so on, and I said, "I'm drinking beer." Then I realized that he wanted to know what work I was doing and I told him I was struggling with this problem, and I simply turned to him and said "Listen, do you know any way of doing quantum mechanics starting with action -- where the action integral comes into the quantum mechanics?" "No," he said, "but Dirac has a paper in which the Lagrangian, at least, comes into quantum mechanics. I will show it to you tomorrow."

Next day we went to the Princeton Library (they have little rooms on the side to discuss things) and he showed me this paper.

Dirac's short paper in the Physikalische Zeitschrift der Sowjetunion claimed that a mathematical tool which governs the time development of a quantal system was "analogous" to the classical Lagrangian.

Professor Jehle showed me this; I read it; he explained it to me, and I said, "What does he mean, they are analogous; what does that mean, analogous? What is the use of that?" He said, "You Americans! You always want to find a use for everything!" I said that I thought that Dirac must mean that they were equal. "No," he explained, "he doesn't mean they are equal." "Well," I said, "let's see what happens if we make them equal."

So, I simply put them equal, taking the simplest example . . . but soon found that I had to put a constant of proportionality A in, suitably adjusted. When I substituted . . . and just calculated things out by Taylor-series expansion, out came the Schrödinger equation. So I turned to Professor Jehle, not really understanding, and said, "Well you see Professor Dirac meant that they were proportional." Professor Jehle's eyes were bugging out -- he had taken out a little notebook and was rapidly copying it down from the blackboard and said, "No, no, this is an important discovery."

Feynman's thesis advisor, John Archibald Wheeler (age 30), was equally impressed. He believed that the amplitude formulation of quantum mechanics -- although mathematically equivalent to the matrix and wave formulations -- was so much more natural than the previous formulations that it had a chance of convincing quantum mechanics's most determined critic. Wheeler writes:

Visiting Einstein one day, I could not resist telling him about Feynman's new way to express quantum theory. "Feynman has found a beautiful picture to understand the probability amplitude for a dynamical system to go from one specified configuration at one time to another specified configuration at a later time. He treats on a footing of absolute equality every conceivable history that leads from the initial state to the final one, no matter how crazy the motion in between. The contributions of these histories differ not at all in amplitude, only in phase. . . . This prescription reproduces all of standard quantum theory. How could one ever want a simpler way to see what quantum theory is all about! Doesn't this marvelous discovery make you willing to accept the quantum theory, Professor Einstein?" He replied in a serious voice, "I still cannot believe that God plays dice. But maybe", he smiled, "I have earned the right to make my mistakes."