1. Quantum Mechanics: The Other Great Revolution of the 20 th Century – Part III Michael Bass, Professor Emeritus CREOL, The College of Optics and Photonics University of Central Florida © M. Bass

2. Quantum Mechanics The formulations of W. Heisenberg and E. Schrödinger. How to interpret quantum phenomena and what was the meaning of the things Heisenberg and Schrödinger calculated? Prior to 1925 quantum physics was a “hodgepodge” of hypotheses, principles, theorems and recipes. It was not a logically consistent theory.

3. © M. Bass Prior to 1925 Every single quantum theory problem had to be solved first, in terms of classical physics using the Correspondence Principle, and then translated into the language of quantum physics. First you described such classical things as position and momentum and then converted them to their quantum physics analogues.

4. © M. Bass 1925 Correspondence had enabled one to retain such classical ideas as energy, momentum, angular momentum, spin and so on. Then you had to tailor the results to fit the experimental data. Werner Heisenberg recognized that there was an alternative to this cumbersome approach and published it in the paper “On a quantum theoretical interpretation of kinematical and mechanical relations” Within months of Heisenberg’s paper, Schrödinger published his paper with the now famous “wave” equation for quantum mechanics.

5. © M. Bass “On a quantum theoretical interpretation of kinematical and mechanical relations” To formulate a proper quantum theory you must abandon any classical description of the motion and instead describe nature by an analysis of observable magnitudes. Such observables were optical frequencies and intensities. (Heisenberg actually referred to dipole amplitudes.) He struggled to present a formulation of quantum phenomena that today we know as the Matrix Approach. (More on this later.) We will see that his struggle was the result of not yet knowing what a matrix was.

6. © M. Bass Heisenberg the person Born in 1901 and so he was just 24 years old when he revolutionized quantum mechanics. During WWI he left school for farm work as his contribution to the war effort. The elements that formed his personal values: rebellion against non-ideal values, such as capitalism, materialism, hypocrisy and moral decadence; adherence to a close circle of likeminded friends united in a "harmony of souls"; a love of nature; and a deeply felt affinity with German culture.

7. © M. Bass More history He graduated with his Ph. D. in 1923 working for Arnold Sommerfeld at the University of Munich. "It may be that you know something; it may be that you know nothing. We shall see." At his thesis defense he could not answer experimentalist Wilhelm Wien's questions about: the resolving power of optical instruments and how a storage battery works. Wien only let him pass after Sommerfeld vigorously defended his pupil. In 1925 he was working with Max Born at Gottingen where both he and Pauli attended and were influenced by Bohr’s lectures. In 1926 he went to Coppenhagen to become Bohr’s assistant.

8. © M. Bass Under the Nazis In 1932 (the same year the Nazi’s came to power in Germany) Heisenberg won the Nobel prize for his 1925 work. Must emphasize no one claims Heisenberg was in fact a Nazi though he participated in the Nazi scientific community during the period from 1932-1945. Other German scientists were Nazis – e.g., Johannes Stark and Phillip Lenard Heisenberg stayed in Germany and worked in the Nazi scientific establishment rising to be the “Oppenheimer” of the Nazi atomic bomb project. See the play “Coppenhagen” to try to understand the trip he made in 1941 to try to recruit Bohr to work with the Germans and the ambiguity surrounding his circumstances.

9. © M. Bass At Farm Hill in 1945 The Alsos mission led by Samuel Goudsmit scoured Europe just behind the Allied troops to find German scientists, to de-brief them, and to secure them from the Soviets. The nuclear scientists were sent to Farm Hill in England where they were when the Hiroshima and Nagasaki bombs were dropped ending the hostilities. Their rooms were bugged and they were overheard discussing: How impossible it was for US scientists to have done it. After all the great Heisenberg had shown that it would require using heavy water to moderate the reaction. Besides, weren’t they, the Germans, the greatest scientists of all.

10. © M. Bass More things overheard We, the German scientists, were so moral and ethical that we were only working on the heavy water concept to develop nuclear power plants in fuel short Germany. They didn’t see the contradiction in this. In fact, this idea was being suggested by a group of unrepentant Nazi scientists as a cover story. Later, Heisenberg tried to claim that he had been trying to lead the “maniacs” away from nuclear weaponry by his wrong calculations and his insistence on heavy water. (He died in 1976 holding to this fairy tale.) There is little to support this idea. The Nazis would have summarily killed a deceiver and there were rabid Nazi scientists like Stark who would have turned in the culprit. I suspect Heisenberg was ashamed of himself and his colleagues and wanted to improve his place in history.

11. © M. Bass Back to 1925 Heisenberg did not throw out correspondence. He modified it into the foundations of his approach. Quantum mechancs demands correspondence as one of its fundamental tenets. He insisted that all that matters is what can be observed. In this Heisenberg, as was Einstein and Schrödinger, strongly influenced by the philosophy of Ernst Mach. In fact, Heisenberg attributes to Einstein his concern with the observable and nothing else.

12. © M. Bass Matrices Heisenberg analyzed the Fourier components of time dependent quantities and through a strange multiplication method identified them with such measurables as energy and dipole moment. The “strange multiplication” was matrix multiplication but no one yet knew that. We will see that Pascual Jordan was the one to explain it to Born and then Heisenberg. It was this barrier, the lack of familiarity with matrices, that prevented most physicists from taking to Heisenberg’s formulation and led to the widespread pleasure they took in Schrödinger’s yet to be published wave formulation.

13. © M. Bass Matrix Mathematics Isn’t it uncanny how mathematics would be prepared to support the new physics? It was only in 1907, when Heisenberg was 6, that M. Bocher published what was to become the standard text on matrix theory in English in New York. It was 1910 before it was translated into German and that was only one year after G. Kowalewski’s treatise on determinants. The second edition of Bocher’s book made it into German in 1924 just in time for those few who had read it to connect matrices to Heisenberg’s ideas.

14. © M. Bass Reluctance Keep in mind that most physicists in 1925 did not know what matrices were much less how to manipulate them. Just consider that Pauli was to reject Born’s offer to “work towards a logically consistent foundation of matrix mechanics” because he was (along with others) reluctant to apply them to theoretical problems. Of course, this refusal allowed Pauli the time to explore the statistics of systems of electrons, identify the exclusion principle, and win a Nobel prize. Pauli was not alone. Heisenberg’s paper was not reviewed in the Phisikalishe Berichte, the official abstracts of the German Physical Society. It was given only one sentence. Similarly in the Physical Society of London/Institute of Electrical Engineers Common Science Abstracts and in the American Physical Society publications.

15. © M. Bass Even Fermi Emilio Segre describes Enrico Fermi’s attitude in 1926, when leaving Born’s group in Gottingen to work in Leyden, “Heisenberg’s great paper on matrix mechanics of 1925 did not appear sufficiently clear to Fermi, who reached a full understanding of quantum mechanics only later through Schrödinger’s wave mechanics. I want to emphasize that this attitude of Fermi was certainly not due to the mathematical difficulties and novelty of matrix algebra (for Fermi such difficulties were minor obstacles) but rather the physical ideas underlying this paper that were alien to him.” Born had to search further for someone to put matrix mathematics on a firm footing. Quite by accident he found the someone in Pascual Jordan while on a train trip.

16. © M. Bass The fateful train trip Shortly after Heisenberg’s paper appeared there was to be a meeting in Hanover, Germany. On the train Born mentioned Heisenberg’s work to a colleague. Sitting in the same compartment was a very young Pascual Jordan who was going to the same conference. Jordan overheard the conversation and, as they left the train, had the temerity to approach the great professor, Born, and comment that he, Jordan had some skills in mathematics and recognized Born’s description as that of matrix manipulation. Exactly 60 days later, on September 27, 1925 the Born-Jordan paper was received at Zeitschrift fur Physik. It contained the pertinent theorems of matrix math and the proof of the matrix equation

17. © M. Bass More Collaboration Born and Jordan got together with Heisenberg and only 2 months later another paper arrived at Zeitschrift fur Physik. This was entitled “On Quantum Mechanics II” This was a vast generalization of the two previous papers including extension to systems of many degrees of freedom, canonical transformations, and quantum mechanical treatments of time- dependent and time-independent perturbations.

18. © M. Bass Uncertainty From these analyses Heisenberg began to realize that there were problems. Pairs of canonical variables were connected to Planck’s constant in such a manner that you could not know both simultaneously with unlimited accuracy. If you determined one with great accuracy, you lost information about the other. This, the Uncertainty Principle, has become a fundamental feature of the quantum world and since the whole universe was once very, very small, a governing feature of what we can know about everything. Think about it!

19. © M. Bass Another conceptual development Start from light, not the mechanics of particles. After all light is our main means to perceive the universe. Light seemed to be both wave and particle like. It made its choice depending on how we observed it. As far back as Hamilton, efforts had been under way to establish the “mechanics of light” (remember Newton and his corpuscles of light). De Broglie had shown that the things we thought were particles could also be considered waves. Then came Erwin Schrödinger who, followed a deliberate mathematical path to the theory of wave mechanics. A synthesis of the wave-particle duality.

20. © M. Bass Erwin Schrödinger Schrödinger was schooled in the classical “gymnasium” learning both Latin and Greek and seriously studying philosophy. It took him just 4 years, 1906-1910, to receive his Ph. D. in Vienna from Fritz Hazornel in theoretical matters, Franz Exner in experimental work and Wilhelm Wirtinger in mathematics. In 1911 Exner took him on as an assistant. Schrödinger became involved in the controversy between atomicity and continuity of matter coming down strongly on the side of atomicity. This was in part because Ludwig Boltzmann was his scientific hero and pointed the way to an atomistic view of matter. Along the way he acquired a mastery of eigenvalue problems that would serve him well later.

21. © M. Bass A philosophical viewpoint Schrödinger was strongly influenced by the philosophical writings of Spinoza, Schopenhauer, Mach and others who formed the realist school of philosophy. Like Einstein and Heisenberg, Schrödinger believed that what was important in physics was what could be observed (e.g.: measured). It didn’t hurt that this theoretician had had an experimentalist’s aspect of his Ph. D. studies and an experimentalist’s assistantship. This concern with observables is clear in his effort to interpret his wave functions in terms of probability for measurable event.

22. © M. Bass Schrödinger's journey – part 1 By 1920 he was unhappy in Vienna and went, first to work with Max Wien in Jena, and then took an assistant professorship at the Technische Hochschule in Stuttgart. Both of these are engineering schools. Then he went to Breslau and finally settled at the University of Zurich in 1921 as the successor to Peter Debye. He stayed in Zurich until 1927 during which time he developed wave mechanics.

23. © M. Bass Schrödinger’s journey – part 2 In 1927 Schrödinger moved to the University of Berlin as the successor to Max Planck. This was the most prestigious position in German science. Schrödinger was recognized and honored. However, five years later, deeply repelled by the Nazis and their rise to power in Germany, Schrödinger gave it up. He left Germany for the Institute for Advanced Study in Dublin and stayed there until 1956. He returned to Vienna where he died in 1958.

24. © M. Bass Debye’s choice When Debye identified Schrödinger as his replacement it resulted in part from the fact that neither thought they could understand some seminars given by de Broglie. Debye asked Schrödinger to present some seminars on the subject that greatly impressed Debye. Schrödinger was chosen but he considered the seminars as the starting point of his work on wave mechanics. In fact, he wrote later in his life that he had needed the “pushing” of Debye before he realized that the subject was attractive.

25. © M. Bass Wave mechanics shaky start The seminars gave rise to an effort to generalize de Broglie’s waves for bound particles. Schrödinger finally found what he called a “neat” solution that gave the energy levels as eigenvalues of a certain operator (the Hamiltonian). Then he applied this concept to the hydrogen atom and got the wrong results. We now know that his error was due to not including spin. Of course, you can’t fault him as spin had not yet been discovered. Schrödinger was so disappointed that he abandoned his method as inadequate. A few months later he returned to it and noticed that if he treated the electron non relativistically he got agreement with observation in the non relativistic limit.

26. © M. Bass Schrödinger's triumph In January 1926 he sent his manuscript containing a complex diffusion equation that he showed was a wave equation for a complex function to Annalen der Physik. This manuscript included his results for the hydrogen atom. Between January and June he submitted all four parts of his wave mechanics including both the time independent and time dependent Schrodinger equations, their solutions for the spherical hydrogen atom, and Schrödinger's comments on the meaning of the wave function and calculations of measurable quantities. The rest, as they say, is history.

27. © M. Bass Go with what you know Physicists immediately realized that they were familiar with Schrödinger's equation. They could follow his math and his statistical interpretation of the meaning of the wave function fit with the non causal concepts first put forward by Einstein. The problem remained, however, how do you define an acceptable wave function. In time it became clear that by insisting that measurables were all that mattered then only single valued, finite and continuous wavefunctions could exist wherever the potential energy was finite.

28. © M. Bass Equivalence Eventually wave and matrix mechanics were shown to give the same results and were, in fact, completely equivalent. In fact Korel Lanczos, Schrodinger and Pauli were to do this independently. Lanczos was a colleague of Einstein and gave the singularly most impenetrable lecture I ever heard. Paul Adrian Maurice Dirac eventually, with knowledge of spin, was able to develop a fully relativistic form of wave mechanics.

29. © M. Bass Philosophical difficulties It is still hard to accept that the most we can know about something is the probability of its happening. Nevertheless, it is beyond doubt that quantum mechanics works – it is the most precisely tested theory we have and it works as advertised. In fact, Richard Feynman and others demonstrated that quantum concepts can remove singularities from other theories making quantum phenomena essential to the foundations of physics.

30. © M. Bass Afterthoughts By the 1950s quantum mechanics had been shown to be a fundamental feature of the universe. It was to be proven in ever more exquisite experiments over the next near half century. Optics played a critical role in this just as in the beginning it provided the data that demanded quantum phenomena. We all casually talk of energy levels, photon energy, transitions between stable states and the like but we rarely realize the incredible intellectual journey we took to get here.