Heisenberg in 1926, the year after his invention of matrix mechanics.

Transcription

1 Heisenberg in 1926, the year after his invention of matrix mechanics. 42 OPTICS & PHOTONICS NEWS JULY/AUGUST 2014

2 Werner Heisenberg s Path to Matrix Mechanics Barry R. Masters Courtesy AIP Emilio Segrè Visual Archives In 1925, Werner Heisenberg through a combination of intuition and sometimes brilliant guesswork, and strongly influenced by the previous works of Einstein, Ladenburg and Kramers created his inchoate version of quantum mechanics. JULY/AUGUST 2014 OPTICS & PHOTONICS NEWS /14/07-08/42/8-$15.00 OSA 43

3 I n the public mind, Werner Heisenberg s name is indelibly associated with his famous uncertainty principle. That formulation, published in 1927, was only part of an intensive struggle by Heisenberg beginning in 1925 to move past the old quantum theory of Niels Bohr and Arnold Sommerfeld to a new, mathematically consistent quantum mechanics. In that struggle, Heisenberg, along with Max Born and Pascual Jordan at the University of Göttingen, fashioned a version of quantum mechanics that came to be called matrix mechanics. Heisenberg s path to matrix mechanics was not walked alone, but was guided by key figures in modern physics that shaped his intellectual development and opened opportunities for him. He felt and acknowledged that debt, and once wrote of his teachers: From Sommerfeld I learned optimism, from the Göttingen people mathematics, and from Bohr physics. From Gymnasium to the University of Munich Werner Karl Heisenberg was born in Würzburg on 5 December 1901 the second son of August, a classicist and medievalist who in 1910 became the chair of Byzantine studies at the University of Munich, and Anna, daughter of the director of Munich s prestigious Maximilians gymnasium. Max Planck had previously studied at the Gymnasium, and Werner began his studies there in August Heisenberg encouraged his son s early interest in mathematics and languages. In mid-september 1919, Werner began his last year at the Gymnasium, studying religion, Latin, Greek, French, German, mathematics, physics, history and gymnastics. He passed his Abitur (Gymnasium graduation examination) with excellence. Heisenberg had planned to study mathematics at university. At a meeting with University of Munich mathematics professor and administrator Ferdinand von Lindemann, Heisenberg mentioned that he had read Weyl s 1918 book Space-Time- Matter that introduced him to Einstein s theory of relativity. Lindemann replied, In that case you are lost to mathematics. As a result, when Heisenberg entered the University of Munich in the fall of 1920, his major was theoretical physics, with minors in mathematics and astronomy. Wilhelm Wien taught him experimental physics, but Heisenberg s true interests lay in the theoretical realm. Partly through an introduction from his father, who was a friend of Arnold Sommerfeld, the eminent director of the Institute for Theoretical Physics, Heisenberg was able to take part in Sommerfeld s seminar, which focused on reading and critical discussion of the current physics literature. In his first year at Munich, Heisenberg developed a quantum-theoretical analysis of the anomalous Zeeman effect (the anomalous splitting of spectral lines in a magnetic field), publishing his results in He then turned to problems in hydrodynamics, particularly the onset of turbulence in fluid flow work that he published in 1922 and expanded into his doctoral dissertation in Also, from November 1922 to March 1923, Heisenberg worked with Max Born in Göttingen on the many-body problem in atomic physics. On 23 July 1923, his dissertation completed, Heisenberg defended his work in an oral examination. Sommerfeld and Wien examined him in physics, Oskar Perron in mathematics and Hugo von Seeliger in astronomy. Heisenberg deftly handled all of Sommerfeld s theoretical questions, and performed very well on questions related to mathematics and astronomy. But he failed dismally on Wien s experimental questions, unable to discuss the resolving power of a Fabry Pérot interferometer, to define the resolution of a telescope or a microscope, or even to explain the principle of a storage battery. Wien wished to fail the student, but in the end Heisenberg received the highest mark for physics, the lowest mark for experimental physics, and a cum laude designation on his doctorate. The day after his doctoral examination, Heisenberg came to Göttingen to see Max Born, whom he was to join as an assistant. I wonder whether you still want to have me, he told Born, recounting his near failure on the examination. But Born had confidence in Heisenberg and his outstanding abilities. Göttingen and Copenhagen After becoming Born s assistant in October 1923, Heisenberg modified the Bohr-Sommerfeld 44 OPTICS & PHOTONICS NEWS JULY/AUGUST 2014

4 Arnold Sommerfeld AIP Emilio Segrè Visual Archives Niels Bohr Wikimedia Commons Werner Heisenberg in Göttingen, Friedrich Hund /Wikimedia Commons Heisenberg once wrote of his teachers: From Sommerfeld I learned optimism, from the Göttingen people mathematics, and from Bohr physics. quantum rules and was able to advance the understanding of the anomalous Zeeman effect. He attended lectures by David Hilbert and a seminar by Richard Courant and Carl Ludwig Siegel on difference equations. These lectures helped him when he worked with Born on converting the differential quotients that arose in atomic physics into difference quotients analogous to those of the Bohr frequency condition, which described the interaction of light and matter. Heisenberg also joined Hilbert and Born s seminar on the structure of matter, where he learned Henri Poincaré and Carl Ludwig Charlier s methods of perturbation theory of celestial mechanics and their application to atomic systems. At the age of only 22, a mere year after the near-disaster at his Munich doctoral examination, Heisenberg obtained his Habilitation (the right to teach at a German university) and became a Privatedozent (a university lecturer) at Göttingen. Heisenberg made several visits to Copenhagen, where he worked with Niels Bohr and his collaborators. He learned Danish and English and published papers on resonance fluorescence, the dispersion of light by atoms, and the structure of complex spectra and their Zeeman effects. He spent the winter semester with Bohr in Copenhagen, and during that visit he became familiar with radiation and dispersion theory as well as applications of the correspondence principle the assumption that quantum theory contains classical theory as a limiting case. This knowledge prepared him for his invention of quantum mechanics in the following months. The problem of hydrogen line spectra intensities One form of the correspondence principle related the intensities of electronic transitions in atomic states. There is a correspondence between the classical measure of intensities, given by the squares of the Fourier coefficients of the dipole moment, and the transition probability coefficients derived in 1916 by Einstein. In 1919 Hans Kramers used the correspondence principle to calculate the intensities and the polarizations of the hydrogen line spectra, achieving good agreement with experimental results. In 1924 Bohr, Kramers and John Slater published their BKS theory of radiation, which connected discontinuous JULY/AUGUST 2014 OPTICS & PHOTONICS NEWS 45

5 Hans Kramers Wikimedia Commons Werner Heisenberg Wikimedia Commons Credit processes in atoms (line spectra) and the continuous nature of radiation (classical electrodynamics). They replaced the conservation of energy and momentum in the absorption, emission and scattering of radiation by an individual atom with a statistical conservation that is valid for a large number of atoms. Einstein argued against the statistical conservation laws, stating that they were not compatible with Kirchhoff s law of emission and absorption of radiation. Born agreed with him. In the fall of 1924 Heisenberg arrived at Bohr s Institute for Theoretical Physics at the University of Copenhagen, where Bohr and Kramers were investigating dispersion theory and atomic structure. They wished to apply the BKS theory to Einstein s 1916 treatment of radiation. Kramers extended a theory first used in 1921 by Rudolf Ladenburg which modeled atoms as a collection of virtual oscillators, each having the frequency of an atomic transition to include both the absorption and the emission of radiation by atoms. His derivation contained two important points subsequently used by Heisenberg in his invention of matrix mechanics. First, Kramers showed that he could write the classical expression for the polarizability as a differential quotient, and then replace it with the difference of two terms. Second, he stated that his equations contain only such quantities as allow of a direct physical interpretation. Born had, in 1924, published a paper with these two concepts. Upon publication in 1925, the Kramers-Heisenberg paper, which derived Adolf Smekal s (1923) incoherent scattering of light (quanta) by atoms and its frequency shift and two-photon processes, seemed a success. But in April 1925 Walther Bothe and Hans Geiger performed a seminal experiment on the Compton effect that demonstrated energy and momentum conservation in single atomic events. Their study contradicted a fundamental assumption of the Copenhagen theory of radiation, which stated that energy and momentum were conserved only statistically in atomic processes. The Fourier components were the reality Presciently, Heisenberg concluded: One could see the Fourier components were the reality, and not the orbits. So one had to look for those connections between the Fourier components, and to see whether or not similar connections were also true in quantum mechanics i.e., if one took, instead of the Fourier components, the transition amplitudes of the real lines of atoms. Heisenberg accepted that the electron orbits, which could not be observed, could not to be taken literally. He was greatly influenced by his reading of Einstein s 1905 paper on special relativity, which advocated the consideration only of observable quantities. Since 1919, Born and Pauli advocated this observability principle, which stated that only energies, frequency of radiation, intensity and phase could be expressed in the laws of quantum theory. As an alternative, the quantum theory amplitudes should represent the electron coordinates in a new quantum theory in terms of observables, such as the energies of the Bohr stationary states W n and the frequency of the transition between them, v, which depends on two indices: 1 v( nn, 2a)5 { W( n) 2W( n2 a) }. " 46 OPTICS & PHOTONICS NEWS JULY/AUGUST 2014

6 Heisenberg had found the goal that would lead to the new quantum mechanics rewrite the classical expressions of electrons in orbits in a form based on Bohr s correspondence principle and the notion of virtual oscillators. Heisenberg had found the goal that would lead to the new quantum mechanics rewrite the classical expressions of electrons in orbits, from the early descriptions of atomic processes, in a form based on two ideas: (1) Bohr s correspondence principle, and (2) the notion of virtual oscillators, which expressed the quantum analog in terms of frequencies and probabilities of transitions between energy states, not of unobservable electron orbits. In April 1925, he returned to Göttingen to accomplish the task. The struggle: Searching for classical analogs In his investigation of atomic spectra, Heisenberg proposed that the quantum modeling should follow a strict analogy to classical behavior. In classical physics the intensity of the radiation at a given frequency was given by square of the amplitude of the particular Fourier components. Therefore, he asked: What quantumtheoretic quantity will take the place of x(t) 2 in the Fourier domain? A periodic motion with the position as a function of time, x(t) was classically expressed as the sum of an infinite number of Fourier v(n) components: a51 ` S ( ) = ( ) ( ) x n t A ne i v, n a t a52 ` a Instead of orbits, Heisenberg used the concept of virtual oscillators, and for a single oscillator instead of writing the amplitude of the n th component A n, he wrote A(n, n a). Instead of writing the frequency of the n th component as v n, he wrote v(n, n a), which is consistent with every frequency being the result of a transition between two stationary states. The infinite number of terms was in a square array where the numbers n and n a give the position in the array. Heisenberg found that the combination of frequencies for the quantum case differs from the classical-harmonics case; in the quantum case, the frequencies combine following the combination principle developed by Walther Ritz in 1908, which states that every spectral line of each element can be expressed as the difference of two spectral terms : ( ) ( 2 ) ( ) v nn, 2a 1 v n2a, n b 5 vnn, 2b. He searched for the quantum analog, using the Bohr correspondence principle and systematic guessing, for the hydrogen atom. He could not make any progress, so he sought an analog in a less complicated physical system. He rejected the simple harmonic oscillator, because its frequency was the mechanical frequency, and that restriction had led to failure for the previous dispersion theory (though the introduction of a damping term resolved the difficulty). He selected instead the anharmonic (nonlinear) oscillator. Although it is not periodic, it can be modeled by sums of periodic functions i.e., by a Fourier series and he was able to integrate the equations of motion. This involved the use of a new multiplication rule for physical quantities and was consistent with the conservation of energy. Hay fever, Helgoland, and a Hamiltonian In a quagmire with these calculations, Heisenberg finally reverted to a more fundamental question: If he could get the amplitudes of a coordinate x and of another coordinate y, how could he get the amplitude of the product xy? The quantities x and y could be quantum amplitudes that correspond in classical physics to a Fourier series. He reasoned that the quantum analog to squaring the position expression was two successive transitions. The square of the transition amplitudes determines the transition probabilities between two states and equals Einstein s transition probability. The first transition was from n to n a, and the second transition was from n a to n b. This yielded the following expression for the multiplication formula for two successive transition amplitudes: a51` 5 S a52` ( ) ( ) ( ) C n, n2b A nn, 2a B n2a, n2b In the middle of these attempts to work out the quantum theory for the anharmonic oscillator, JULY/AUGUST 2014 OPTICS & PHOTONICS NEWS 47

7 From Heisenberg s Reinterpretation to Matrix Mechanics Max Planck Institute Pascual Jordan Before 1924 physicists typically did not use matrix mathematics, although there were notable exceptions. Hermann Minkowski s 1908 theory of the electrodynamics of moving bodies was formulated in terms of matrices, as well as Mie s 1912 nonlinear electrodynamics and Born and Theodore von Kármán s 1921 lattice theory of crystals. When Heisenberg derived his noncommutative multiplication rule first described for finite square matrices in 1850 by Arthur Cayley he was ignorant of matrices. It was left to Pascual Jordan and Max Born, who had learned many of the mathematical methods including matrices in earlier years, to reformulate Heisenberg s intuition into a formal infinite matrix mathematics. Initially, Born asked Pauli to be his assistant on this project, but Pauli refused. Then Born turned to Jordan to work on the problem and he accepted. They showed a proof for energy conservation and the Bohr frequency condition, hn nm = W n W m. They used infinite Hermitian matrices that follow from the infinite dimensions of Hilbert space. These matrices are square matrices with complex entries, in which the matrix is equal to its conjugate transpose (i.e., the matrix formed by interchanging the rows and the columns and taking the complex conjugate of each element). Charles Hermite had shown that such a matrix has real eigenvalues. The paper On Quantum Mechanics, published by Born and Jordan in Zeitschrift für Physik in 1925, showed that Heisenberg s law of multiplication was the well-known rule of matrix multiplication, and formulated matrix mechanics for systems with one degree of freedom the harmonic oscillator and the anharmonic oscillator. Born and Jordan used infinite square matrices arrays of numbers characterized by two integral indices m and n. An atom making a transition from state m to state n will emit the frequency n mn. Similar matrices can be constructed for the generalized coordinates, a set of N independent quantities that is used to define a system of N degrees of freedom, and for the momentum. The square matrix for the classical Hamiltonian is a function of generalized coordinates and momentum H(p,q). The diagonal elements E n are the eigenvalues of that matrix, and the off-diagonal elements and the Einstein transition probabilities give the intensities of transitions from the rates of absorption, stimulated emission and spontaneous emission of radiation. Thus, the solution of the eigenvalue problem of Hermitian matrices in linear algebra yields the energy values E n derived by Heisenberg at Helgoland. Born and Jordan also derived the following noncommutability relation for a dynamic system described by the spatial coordinate matrix q and the momentum matrix p: pq -qp5 h 1 2pi Max Born Wikimedia Commons where 1 is the unit matrix. This commutation relation was guessed at by Born and is inscribed on his tombstone, but later Jordan proved it. Heisenberg suffered a severe case of hay fever and decided to spend two weeks on the pollen-free island of Helgoland in the North Sea. There, he had the remarkable insight that transition amplitude matrix multiplication did not commute i.e., that AB BA. Order is crucial, because the first transition must occur before the second transition can occur. Heisenberg made two other assumptions: that the equations of motion should have the same form as Newton s second law, and a modified version of the Bohr-Sommerfeld quantization rule to a form that relates the difference between two cyclic integrals, Rpdq, evaluated at the initial and the final states of a transition. He then used brilliant guesswork, and always seeking the quantum analogs to the classical expressions calculated the anharmonic oscillator Hamiltonian as a diagonal matrix with the elements 1 En = n+ hn, n5 012,,,... 2 which yielded the zero-point energy. Returning to Göttingen after the hay fever had passed, Heisenberg fully solved, as an approximation, the anharmonic oscillator. He confirmed his result independently through comparison with Born s perturbation calculation, and both methods gave similar results. Heisenberg also derived the Kramers dispersion relation from the use of virtual oscillators, perturbative techniques and difference equations, and discovered the important quantum mechanical rule that the transition amplitude for a transition from one state to another is calculated from the sum over all possible intermediate states. The result of this intense work in 1925 was Heisenberg s inchoate and 48 OPTICS & PHOTONICS NEWS JULY/AUGUST 2014

8 Heisenberg did not know it, but his inspired insight on noncommutative mathematics pointed directly to a matrix formulation of quantum mechanics. abstruse paper On a quantum-mechanical reinterpretation of kinematic and mechanical relations, published in Zeitschrift für Physik on 29 July The paper marked the birth of the new quantum mechanics. Matrix mechanics and beyond Heisenberg did not know it, but his inspired insight on noncommutative mathematics pointed directly to a matrix formulation of quantum mechanics, which was subsequently developed by Born and Jordan (see sidebar on facing page). On Quantum Mechanics II, published by Born, Heisenberg and Jordan in 1925, extended Heisenberg s methods to periodic systems with an arbitrary number of degrees of freedom, nondegenerate and degenerate systems, and time-independent and time-dependent perturbation theory. At the end of the paper, they derived Einstein s 1909 equation for the fluctuations in black-body radiation. In the fall of 1925 Pauli published On the hydrogen spectrum from the standpoint of the new quantum mechanics, in which he showed that the Balmer terms of the hydrogen atom could be correctly calculated, as well as the effect of crossed electric and magnetic fields on the spectrum. Thus, Pauli completed a derivation of the Stark effect based on matrix mechanics. His seminal paper convinced many physicists of the validity of quantum mechanics. Leipzig, the Third Reich, and postwar years After the 1927 publication of his celebrated paper on uncertainty, Heisenberg became a professor and chair at Leipzig University starting in 1 February In 1933 the year Hitler came to power Heisenberg was awarded the Nobel Prize in Physics for 1932 for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen. During the Third Reich, Heisenberg remained in Germany, in spite of his many offers of positions abroad. He performed his army reserve duties and directed his Institute on Leipzig. He also was involved in the German effort to explore the building of a fission bomb activities that damaged his reputation after the war. What is unarguable is that Heisenberg did nothing to help Jewish Holocaust victims for example, the parents of his colleague Samuel Abraham Goudsmit, who died in the Auschwitz gas chamber. In April 1937, he married Elisabeth Schumacher the daughter of Hermann Schumacher, a professor of economics at the University of Berlin. In 1938 their twins were born and they subsequently had five additional children. After World War II, Heisenberg became director of the Kaiser Wilhelm Institute for Physics (which subsequently became the Max Planck Institute), and in subsequent decades he worked on a wide variety of problems in particle physics. On 1 February 1976, he died of cancer in Munich. OPN Barry R. Masters (brmail2001@yahoo.com) is a Fellow of AAAS, OSA and SPIE. References and Resources c W. Heisenberg. The Physical Principles of the Quantum Theory, University of Chicago Press, Chicago (1930). Dover Publications, New York (1949). c B.L. van der Waerden, Ed. Sources of Quantum Mechanics, Dover Publications, New York (1967). c E. MacKinnin. Heisenberg, models, and the rise of quantum mechanics, Hist. Stud. Phys. Sci. 8, 137 (1977). c A.D. Beyerchen. Scientists under Hitler. Politics and the Physics Community in the Third Reich, Yale Univ. Press, New Haven (1979). c W. Heisenberg. Collected Works, Series A: Original Scientific Papers; Series B: Scientific Review Papers, Talks, and Books; Series C: Philosophical and Popular Writings. W. Blum, H.-P. Dürr and H. Rechenberg, eds., Piper, Munich and Springer-Verlag, Berlin (1984). c J. Mehra and H. Rechenberg. The Historical Development of Quantum Theory, Vol. 2 and Vol. 3, Springer-Verlag, New York (1987). c D.C. Cassidy. Uncertainty. The Life and Science of Werner Heisenberg, W.H. Freeman, New York (1992). c M. Weinreich. Hitler s Professors. The Part of Scholarship in Germany s Crimes Against the Jewish People, Yale University Press, New Haven (1999). c I.J.R. Aitchison, D.A. MacManus and T.M. Snyder. Understanding Heisenberg s magical paper of July 1925: A new look at the calculational details, Am. J. Phys. 72, 1370 (2004). c C. Carson. Heisenberg in the Atomic Age. Science and the Public Sphere, Cambridge University Press, New York (2010). JULY/AUGUST 2014 OPTICS & PHOTONICS NEWS 49