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Quantum Mechanics in 1926-27 |
Werner Heisenberg |
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Werner Heisenberg was often labeled as the wunderkind (wonder child) of physics in the 1920's. Some thought he looked like a farm boy, but he was anything but a country bumpkin. His father was an academic scholar on the faculty of Munich University. Werner's grandfather was the head of a prestigious gymnasium (secondary school). Werner and his older brother attended that school. After graduation Werner Heisenberg wanted to study mathematics at Munich University but his interview in the mathematics department there did not go well and he was not accepted. His father called upon his old friend, the physicist Arnold Sommerfeld, to accept Werner into the physics program. Sommerfeld was one of the top physicists in the world at that time and Werner was indeed fortunate to undertake his education at Sommerfeld's institution. In happened that Wolfgang Pauli was also studying there and Werner made his acquaintance.
Niels Bohr gave a series of lectures at Göttingen and Sommerfeld and a contingent of his students, including Heisenberg, journeyed there to attend those lectures. Bohr sought out Heisenberg and invited him to come to Copenhagen for a term. Sommerfeld however wanted Heisenberg to study first with Max Born at Göttingen. Heisenberg did so and a lifelong collaboration with Born commenced. After his term at Göttingen Heisenberg went back to complete a thesis for his doctorate. In the defense of his thesis Heisenberg did poorly in explaining some topics unrelated to his thesis and one examiner wanted to fail him, but Sommerfeld negotiated a compromise. Heisenberg was given his doctorate but with the lowest possible grade. In disgrace, Heisenberg fled to Göttingen. After completing some research on the anomalous Zeeman Effect Heisenberg contacted Bohr to tell him of his work. Bohr invited him to visit his institute in Copenhagen for a few weeks. At Bohr's Institute initially Bohr was too busy to spend any time with Heisenberg, but then Bohr took time out from his research to spend several days on a hiking tour with Heisenberg. After becoming acquainted with the depth of Heisenberg's talent Bohr invited him to stay at Bohr's institute for an extended period of time. The time was 1924 and Heisenberg was only 22 years old.
In 1925 Heisenberg returned to Göttingen and was disappointed with the lower intellectual level there compared to Copenhagen. He set about trying to explain the intensity of the spectral lines of hydrogen. Bohr's model gave accurate predictions of the frequency and wavelength of those spectral lines for hydrogen but had nothing to say about why some were brighter than others. He did not make much progress in this task and was despondent. Then in June of 1925 he had a sever attack of hay fever because of allergies to pollen. His eyes swelled shut. In desperation he decided to go to the Baltic coast to get away from the pollen. He had a two week leave from Göttingen and decided to spend it on the island of Helgoland.
Freed from his hay fever Heisenberg began to work on the problem of the intensity of the spectral lines of hydrogen. He conceptualized the problem as there being a set of successively higher energy levels for an electron in a hydrogen atom, say {E_{k}: k=0, 1, 2, …}. When an electron moved from a higher level k to a lower level j radiation of energy (E_{k}−E_{j}) is emitted. The frequency of this radiation is such that
where h is Planck's constant. This much was the standard perception and in it there no reason for one frequency to give any brighter line than another frequency.
His real insight came with the realization that the intensity depended upon the probability of the transition. When the probability is higher more atoms are involved in the transitions and hence the spectral line looks brighter.
Somehow, while Heisenberg was musing on rock overlooking the Baltic Sea, this led to a revelation that he needed a type of quantities such that their product depended upon their order in the multiplication. In other words, there are terms, say P and Q, such that P×Q is different from Q×P and hence P×Q−Q×P is not zero.
When Heisenberg left Helgoland he went immediately to Hamburg to see Wolfgang Pauli. Pauli was encouraging, which was not typical of Pauli for strange ideas. Pauli once remarked of someone's theory that it was so far afield that it wasn't even wrong. With this encouragement from Pauli, Heisenberg decided to write up his ideas as an article. At Göttingen he gave a draft of the article to Max Born to read in a short period of time. When Born got to read the article he was enthusiastic about it. The odd sort of multiplication puzzled Born until he realized that the multiplication of square matrices had that property.
Max Born |
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Born asked Pauli to help Heisenberg with the mathematics for the article, but Pauli declined because he felt the mathematics would get in the way of the physics. Born then chose Pascual Jordan, a 22 year old graduate student in physics at Göttingen who had prior training in mathematics. Jordan was familiar with the algebra of matrices so he was the perfect choice.
Pascual Jordan |
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Born and Jordan then worked on extending Heisenberg's ideas. They came up with the formulation that if Q represented the state of a quantum system and P its momenta then
where juxtaposition represents multiplication, i is the square root of negative one, h is Planck's constant and I is the identity matrix. An identity matrix is one with ones on the principal diagonal and zeroes everywhere else. It serves as the equivalent of unity in matrix multiplications; i.e., for any matrix X, IX=X and XI=X. And, oh yes, the matrices in the above relationship are of infinite order so one can only see a middle portion of them. It was heady mathematics.
The body of theory that was developed by Heisenberg, Born and Jordan came to be known as Matrix Mechanics. It was difficult but useful methodology. Pauli used it to compute the spectrum of hydrogen. People had only praise for matrix mechanics.
Erwin Schrödinger |
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Then, seemingly out of nowhere, came a superior methodology. Erwin Schrödinger was a well-respected physicist with somewhat of a specialization in optics. He was in his late thirties in contrast to Heisenberg and Jordan who were in their early to middle twenties. Schrödinger had not been previously involved in the development of quantum theory. What brought him into the field was the idea of Louis de Broglie that particles have a wave aspect. This was an exciting notion for a physicist with an orientation towards optics. Schrödinger sought our de Broglie's work and read it avidly. He then wrote six articles that developed what he initially called undulatory mechanics, but which subsequently came to be known as wave mechanics. The mathematical basis for wave mechanics was partial differential equations, a field more familiar to physicists than matrix algebra. In part, this work came from the insight that Schrödinger had that the discreteness of quantum physics did not have to be assumed as an axiom; it could arise as eigenvalues of solutions to the relevant equations.
Although most physicists praised Schrödinger's wave mechanics Werner Heisenberg was not one of them. In a letter to a colleague he said
The more I think about the physical portion of the Schrödinger theory, the more repulsive I find it. What Schrödinger writes about the visualizability of his theory "is probably not quite right," in other words it's crap.
Schrödinger subsequently wrote an article in which he showed that matrix mechanics and wave mechanics gave the same results in quantum analysis and were therefore equivalent. Later Heisenberg made use of wave mechanics himself in one of his articles because of the greater ease with which problems can be analyzed.
Niels Bohr |
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When Heisenberg was back in Copenhagen with Bohr he began pondering the matter of beta ray (electron) tracks in a cloud chamber. Heisenberg did not use any notion of electron orbits in atoms because he considered them unobservable. The cloud chamber tracks might indicate that he was wrong about the electron orbits not being observable. He went out for a walk in the park even though it was the middle of the night. Cogitating on the fundamentals of physics he realized that it might not be possible, because of the physical interactions, to specify simultaneously the location and velocity of a particle like an electron. Back in his room he used the example of a microscope investigate the possibility of the simultaneous specification of location and velocity. It was then that he realized the limitation on the accuracy of such specifications. He called the concept the indeterminacy principle, but it subsequently became known as the uncertainty principle.
Heisenberg concluded that concepts like path, trajectory and orbit have no meaning at the quantum level. In late February of 1927 Heisenberg wrote a 14 page letter to Pauli describing his uncertainty principle and its basis. Pauli's reaction was favorable and Heisenberg turned the letter into a draft of an article in the early part of March. At the time, Bohr was away in Norway on a long vacation. After Bohr returned he read Heisenberg's article. To Heisenberg's surprise Bohr disagreed with Heisenberg's assessment of the source of the uncertainty. Heisenberg thought the uncertainty stemmed from the discontinuities of particle collisions; Bohr thought it was from the dual nature of particle-waves. They ended their discussion still in disagreement. A few days later they talked again. Bohr did not want Heisenberg to publish his article until he had rewritten it. Heisenberg did not want to change anything and at the rejection of his ideas by his revered mentor. Finally Heisenberg broke out in tears because of the pressure Bohr was putting on him. Bohr had formulated his own new concept while on his skiing vacation in Norway; one that he called complementarity. This was the notion that a particle and its wave were simply manifestation of some more fundamental entity.
Despite Bohr's imploring of him not to do so Heisenberg sent away his article for publication near the end of March in 1927. It appeared in print at the end of May as a 27 page article. He soon received offers of professorships and he accepted the one from Leipzig University. He left Copenhagen in June of 1927. A copy of Heisenberg's article had been sent to Albert Einstein but Einstein did not respond.
Bohr invited Schrödinger to come Copenhagen in October of 1926 to give lectures and for discussions. Bohr made Schrödinger his house guest and the discussions continued from early in the morning until late at night. These discussions were a relentless effort on Bohr's part that his Copenhagen Interpretation of atomic phenomena was the right one. That position involved denying that orbits, trajectories and so forth had any meaning at the quantum level. Instead particles made instantaneous jumps between discrete quantum states. These were known as quantum jumps. On this matter Schrödinger said,
If we have to go on with these damned quantum jumps,
then I am sorry that I ever got involved.
Bohr replied,
But the rest of us are extremely grateful that you did. Your wave mechanics has contributed so much to the mathematical clarity and simplicity that it represents a giant advance over all previous forms of quantum mechanics.
Bohr did not convince Schrödinger but after several days of this intense debate Schrödinger fell ill. While Bohr's wife was trying to nurse Schrödinger back to health
there was Bohr, the indefatigable debater that he was, sitting on the edge of the bed continuing the debate.
Thus this was the drama of the development of quantum physics in the late 1920's. There were three physicists of the genius level involved: Bohr, Heisenberg and Schrödinger. Each had their own version of quantum physics. Schrödinger's wave mechanics prevailed as the preferred methodology. But there still was the question of the interpretation of the results of the analysis by wave mechanics. Bohr promoted the interpretation in which particles were no longer the essential element of reality but instead probability density distribution. Particles no long existed on their own. Instead the process of measurement resulted in the probability density distribution collapsing to a particle. This view of reality as a matter of probability density distributions was called the Copenhagen Interpretation. Heisenberg agreed with the Copenhagen Interpretation in the matter of particles not having a precise location and velocity or trajectories. Einstein on the other hand disagreed with the notion that probability density distributions are the essence of reality. The debate between Einstein and Bohr continued for another couple of decades with the view of Bohr apparently prevailing. Bohr was an indefatigable debater.
There were other geniuses involved with the creation of quantum physics in the 1920's, notably Paul Dirac of England and John von Neumann of Hungary. Dirac provided another proof that matrix mechanics and wave mechanics are equivalent, but he was peripheral to the core development in the 1920's. The brilliant but enigmatic Dirac, who was probably an Asperger's Syndrome person, went on to make the most crucial developments of quantum physics in the 1930's. John von Neumann developed a rigorous mathematical foundation for quantum physics in terms of Hilbert space, but then left physics to conquer other fields.
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