Abstract
In classical mechanics it is possible to calculate, for example, the vibrational modes of a string, membrane or resonator by solving a wave equation, subject to certain boundary conditions. At the very beginning of the development of quantum mechanics, one was faced with the problem of finding a differential equation describing discrete states of an atom. It was not possible to deduce exactly such an equation from old and well-known physical principles; Instead, one had to search for parallels in classical mechanics and try to deduce the desired equation on the basis of plausible arguments. Such an equation, not derived but guessed at intuitively, would then be a postulate of the new theory, and its validity would have to be checked by experiment. This equation for the calculation of quantum-mechanical states is called the Schrödinger equation; let us now “derive” it.
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Biographical Notes
Schrödinger, Erwin, Austrian physicist, * Vienna 12.8.1887, †Alpbach (Tirol) 4.1.1961, was a student of F. Hasenöhrl. As a professor in Zurich, S. worked on statistical thermodynamics, the theory of general relativity and the theory of colour vision. Excited by L. De Broglie’s Ph.D thesis and A. Einstein’s publications concerning Bose statistics, S. created wave mechanics. In December, 1925 he defined the Klein-Gordon equation and later, in January 1926, he invented the Schrödinger equation, which describes, in nonrelativistic approximation, the atomic eigenvalues. In March 1926, S. proved the mathematical equivalence of his theory with matrix mechanics (M. Born, W. Heisenberg and P. Jordan). S. always attacked the statistical interpretation of quantum theory (as did Einstein, von Laue and de Broglie), especially the “Copenhagen interpretation”. In 1927 S. went to Berlin as Planck’s successor and emigrated in 1933, as a convinced liberal, to Oxford. In the same year he was awarded, together with P.A.M. Dirac, the Nobel Prize in Physics. In 1936 S. went to the University of Graz, Austria, emigrating a second time when Austria was annexed. The Institute for Advanced Studies was founded in Dublin for him and others. In 1956, S. returned to Austria.
Bose, Satyendra Nath, Indian physicist, * 1.1.1894, †4.2.1974 Calcutta. Together with Einstein, he set up a theory of quantum statistics (Bose-Einstein statistics) that differs from the classical Boltzmann statistics and from Fermi statistics, too. B. invented this statistics for photons; Einstein extended it to massive particles. B. was a professor in Dacca and Calcutta from 1926 to 1956.
Boltzmann, Ludwig, Austrian physicist, * Vienna 1844, †Duino near Trieste 1906. He studied physics at the university of Vienna where he was an assistant of Josef Stefan. B. became professor of mathematical physics at the university of Graz in 1869. He also taught at Vienna, Munich and Leipzig. Among his students were S. Arrhenius, W. Nernst, F. Hasenöhrl and L. Meitner. The young B. worked successfully on experimental physics (he proved the relationship between the refractive index and the dielectric constant for sulphur, which was required by Maxwell). Near the end of his life he occupied his mind with philosophical issues, but his main interest was always theoretical physics. The central problem of his life’s theoretical work was the reduction of thermodynamics to mechanics, requiring the solution of the contradiction between the reversibility of mechanical processes and the irreversibility of thermodynamical processes. He showed the relationship between the entropy S and the probability of a state W with the formula S = klnW (k: Boltzmann’s constant). This was the starting point of quantum theory both in the formulation of Max Planck in 1900 and in the expanded version of Albert Einstein (1905). Other important achievements of B. are the formulas for the energy distribution of atoms moving freely or in force fields (Maxwell-Boltzmann distribution) and the theoretical explanation of the law of the radiation power of a black body (Stefan-Boltzmann law, 1884).
B. was an exponent of the atom theory. The small response and even rejection that he received for it from many contemporary physicists disappointed him throughout his lifetime. He did not live to see the final victory of the atom theory introduced in 1905 by Einstein’s theory of Brownian motion.
B. committed suicide at the age of 62.
Fermi, Enrico, Italian physicist, * 29.9.1901, Rome, †28.11.1954, Chicago. F. was a professor in Florence and Rome before going to Columbia University in New York in 1939. There he stayed until 1946, when he went to Chicago. F. was mainly engaged in quantum mechanics. He discovered the conversion of nuclei by the bombardment of neutrons, and, beginning in 1934, was thus able to create many new synthetic radioactive substances that he thought were transuranic. F. formulated the statistics named after him (Fermi statistics) in his treatise “Sulla Quantiz-zazione del gas perfetto monatomico” (Lincei Rendiconti 1935; Zeitschrift für Physik 1936). In 1938 he was awarded the Nobel Prize in Physics. During World War II, F. was substantially engaged in projects devoted to making use of atomic energy. Under his guidance the first nuclear chain reaction was performed at the Chicago nuclear reactor on 2.12.1942. In memory of F., the Enrico Fermi Prize was established in the United States.
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© 1993 Springer-Verlag Berlin Heidelberg
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Greiner, W. (1993). The Schrödinger Equation. In: Quantum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-30374-0_6
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DOI: https://doi.org/10.1007/978-3-662-30374-0_6
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