3. The Development of Quantum Mechanics (1925 – 1927) |
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The early 1920s witnessed fundamental difficulties in atomic physics. The quantum theory of atomic structure, founded by Bohr and largely developed by Bohr and Sommerfeld, did not describe the properties of complicated atoms and molecules. Moreover, the discovery of the Compton effect at the end of 1922 focussed attention on the problem of the nature of radiation. Its interpretation in the light-quantum hypothesis contradicted classical radiation theory, and the radical attempt by Bohr, Kramers and Slater in early 1924 to resolve the difficulty by assuming only statistical conservation of energy and momentum was refuted by the experiment of Walther Bothe and Hans Geiger in April 1925.
Heisenberg was growing ever more concerned with these and other difficulties in atomic theory. His works on the anomalous Zeeman effect, only successful in part, and his unsuccessful calculation of the helium states with Born had sensitized him by early 1925 to the “crisis” of current theory. Nevertheless, his latest calculations in Copenhagen on dispersion theory and on complex spectra, especially the principle of “sharpened” correspondence applied in these works, seemed to point toward a future satisfactory theory.
With characteristic optimism the Göttingen Privatdozent took on a new and difficult problem at the beginning of May 1925, the calculation of the line intensities in the hydrogen spectrum. Heisenberg began with a Fourier analysis of the classical hydrogen orbits, intending to translate them into a quantum theoretical scheme – just as he had done with Kramers for the dispersion of light by atoms. But the hydrogen problem proved much too difficult, and he replaced it with the simpler one of an anharmonic oscillator. With the help of a new multiplication rule for a quantum-theoretical Fourier series he succeeded in writing down a solution for the equations of motion for this system. On 7 June 1925 he went to the island of Helgoland to recover from a severe attack of hay fever. There he completed the calculation of the anharmonic oscillator, determining all the constants of the motion. He made use, in particular, of a modified quantum condition that was later called by Born, Pascual Jordan and himself a “commutation relation”, and he proved that the new theory yielded stationary states (conservation of energy). Returning to Göttingen on 19 June 1925 Heisenberg composed his fundamental paper “Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen” (“On a Quantum Theoretical Reinterpretation of Kinematic and Mechanical Relations”), which was completed on 9 July 1925. In this paper, the starting point for a new quantum mechanics, Heisenberg announced as the leading philosophical principle of quantum mechanics that only observable quantities are allowed in the theoretical description of atoms. Heisenberg reported his new results during visits shortly thereafter with Paul Ehrenfest in Leiden and with Ralph Fowler in Cambridge.
After Born and Jordan managed in August and September 1925 to develop the mathematical content of Heisenberg's work into a consistent theory with the help of infinite Hermitian matrices (Z. Phys. 34, 858, 1925), Heisenberg participated, starting in September 1925, in the completion and application of the new “matrix mechanics”, culminating in the long “three-man-paper”, by Born, Heisenberg and Jordan, submitted on 16 November 1925. Further developments followed rapidly: Pauli calculated the stationary states of the hydrogen atom in October 1925; Cornelius Lanczos in Frankfurt and Born and Norbert Wiener in the USA extended the method of operator mechanics to describe continuous motions (December 1925); and Paul Adrien Maurice Dirac in Cambridge developed independently of the Göttingen school a different scheme based upon Heisenberg's July paper, the method of q-numbers (November 1925), in which many-electron atoms and the relativistic Compton effect could be handled successfully (spring 1926). In addition Heisenberg and Jordan utilized electron spin and matrix mechanics to solve the old problems of hydrogen fine structure and the anomalous Zeeman effect (April 1926); and finally Heisenberg discovered the phenomenon of quantummechanical resonance (June 1926), which played a decisive role in his subsequent calculation of the term system of the helium atom (July 1926).
In May 1926 Niels Bohr offered Heisenberg a position at his institute in Copenhagen as Lector and successor to his assistant Kramers. There Heisenberg delivered lectures at the university (in Danish) on contemporary physical theories, directed beginning students, helped guest researchers with their problems, and discussed with Bohr the most important results of quantum mechanics. In the summer and fall of 1926 the main topic of discussion was wave mechanics, the quantum atomic theory that Erwin Schrödinger began introducing in January 1926. The complete mathematical equivalence between Göttingen's matrix mechanics or Dirac's q-number scheme and Schrödinger's wave mechanics was proved by Jordan and Dirac in December 1926, after preparatory work by Schrödinger (March 1926), Pauli (April 1926), and Carl Eckart (June 1926). However, Schrödinger's physical interpretation of the square of the wave amplitude as the continuously distributed charge density of the electron was rejected by Born, Bohr and Heisenberg and replaced on Born's proposal by the interpretation that it is the probability for finding the electron at each location (June 1926). In close contact with Pauli, and in intense discussion with Bohr, Heisenberg analyzed what he termed the “perceptual content of the quantum-theoretical kinematics and mechanics”. As a result of the analysis he presented in March 1927 the so-called “indeterminacy” or “uncertainty relations”, which limit the simultaneous measurement of canonically conjugate variables, such as the position and momentum of a particle. Bohr, on the other hand, pondered the simultaneous use of the physical pictures of particles and waves, which resulted in his general principle of “complementarity” announced in fall of 1927.
Born's statistical interpretation of Schrödinger's wave function, Heisenberg's uncertainty relations, and Bohr's complementary principle formed the basis of the physical interpretation of the new quantum mechanics, as explicated by Bohr in his lectures at the Volta Conference in Como (September 1927) and at the Solvay Congress in Brussels (24-29 October 1927). This “Copenhagen Interpretation” of quantum mechanics, as it was later called, found acceptance by most physicists, but not by all: Albert Einstein in particular raised serious objections to it at the 1927 and 1930 Solvay conferences and later, for example, in his paper with Boris Podolsky and Nathan Rosen (Phys. Rev. 47, 777, 1935).
Heisenberg was growing ever more concerned with these and other difficulties in atomic theory. His works on the anomalous Zeeman effect, only successful in part, and his unsuccessful calculation of the helium states with Born had sensitized him by early 1925 to the “crisis” of current theory. Nevertheless, his latest calculations in Copenhagen on dispersion theory and on complex spectra, especially the principle of “sharpened” correspondence applied in these works, seemed to point toward a future satisfactory theory.
With characteristic optimism the Göttingen Privatdozent took on a new and difficult problem at the beginning of May 1925, the calculation of the line intensities in the hydrogen spectrum. Heisenberg began with a Fourier analysis of the classical hydrogen orbits, intending to translate them into a quantum theoretical scheme – just as he had done with Kramers for the dispersion of light by atoms. But the hydrogen problem proved much too difficult, and he replaced it with the simpler one of an anharmonic oscillator. With the help of a new multiplication rule for a quantum-theoretical Fourier series he succeeded in writing down a solution for the equations of motion for this system. On 7 June 1925 he went to the island of Helgoland to recover from a severe attack of hay fever. There he completed the calculation of the anharmonic oscillator, determining all the constants of the motion. He made use, in particular, of a modified quantum condition that was later called by Born, Pascual Jordan and himself a “commutation relation”, and he proved that the new theory yielded stationary states (conservation of energy). Returning to Göttingen on 19 June 1925 Heisenberg composed his fundamental paper “Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen” (“On a Quantum Theoretical Reinterpretation of Kinematic and Mechanical Relations”), which was completed on 9 July 1925. In this paper, the starting point for a new quantum mechanics, Heisenberg announced as the leading philosophical principle of quantum mechanics that only observable quantities are allowed in the theoretical description of atoms. Heisenberg reported his new results during visits shortly thereafter with Paul Ehrenfest in Leiden and with Ralph Fowler in Cambridge.
After Born and Jordan managed in August and September 1925 to develop the mathematical content of Heisenberg's work into a consistent theory with the help of infinite Hermitian matrices (Z. Phys. 34, 858, 1925), Heisenberg participated, starting in September 1925, in the completion and application of the new “matrix mechanics”, culminating in the long “three-man-paper”, by Born, Heisenberg and Jordan, submitted on 16 November 1925. Further developments followed rapidly: Pauli calculated the stationary states of the hydrogen atom in October 1925; Cornelius Lanczos in Frankfurt and Born and Norbert Wiener in the USA extended the method of operator mechanics to describe continuous motions (December 1925); and Paul Adrien Maurice Dirac in Cambridge developed independently of the Göttingen school a different scheme based upon Heisenberg's July paper, the method of q-numbers (November 1925), in which many-electron atoms and the relativistic Compton effect could be handled successfully (spring 1926). In addition Heisenberg and Jordan utilized electron spin and matrix mechanics to solve the old problems of hydrogen fine structure and the anomalous Zeeman effect (April 1926); and finally Heisenberg discovered the phenomenon of quantummechanical resonance (June 1926), which played a decisive role in his subsequent calculation of the term system of the helium atom (July 1926).
In May 1926 Niels Bohr offered Heisenberg a position at his institute in Copenhagen as Lector and successor to his assistant Kramers. There Heisenberg delivered lectures at the university (in Danish) on contemporary physical theories, directed beginning students, helped guest researchers with their problems, and discussed with Bohr the most important results of quantum mechanics. In the summer and fall of 1926 the main topic of discussion was wave mechanics, the quantum atomic theory that Erwin Schrödinger began introducing in January 1926. The complete mathematical equivalence between Göttingen's matrix mechanics or Dirac's q-number scheme and Schrödinger's wave mechanics was proved by Jordan and Dirac in December 1926, after preparatory work by Schrödinger (March 1926), Pauli (April 1926), and Carl Eckart (June 1926). However, Schrödinger's physical interpretation of the square of the wave amplitude as the continuously distributed charge density of the electron was rejected by Born, Bohr and Heisenberg and replaced on Born's proposal by the interpretation that it is the probability for finding the electron at each location (June 1926). In close contact with Pauli, and in intense discussion with Bohr, Heisenberg analyzed what he termed the “perceptual content of the quantum-theoretical kinematics and mechanics”. As a result of the analysis he presented in March 1927 the so-called “indeterminacy” or “uncertainty relations”, which limit the simultaneous measurement of canonically conjugate variables, such as the position and momentum of a particle. Bohr, on the other hand, pondered the simultaneous use of the physical pictures of particles and waves, which resulted in his general principle of “complementarity” announced in fall of 1927.
Born's statistical interpretation of Schrödinger's wave function, Heisenberg's uncertainty relations, and Bohr's complementary principle formed the basis of the physical interpretation of the new quantum mechanics, as explicated by Bohr in his lectures at the Volta Conference in Como (September 1927) and at the Solvay Congress in Brussels (24-29 October 1927). This “Copenhagen Interpretation” of quantum mechanics, as it was later called, found acceptance by most physicists, but not by all: Albert Einstein in particular raised serious objections to it at the 1927 and 1930 Solvay conferences and later, for example, in his paper with Boris Podolsky and Nathan Rosen (Phys. Rev. 47, 777, 1935).