The New Quantum Mechanics Of Heisenberg, Schrödinger, And Dirac
From 1925 to 1927 three equivalent new versions of quantum mechanics were proposed that extended the Bohr-Sommerfeld theory, cured its main difficulties, and produced an entirely novel view of the microworld. These new theories were the matrix mechanics of Werner Heisenberg (1901–1976), the wave mechanics of Erwin Schrödinger (1887–1951), and the transformation theory of Paul A. M. Dirac (1902–1984), the last being a more general version that includes both of the other versions. The three physicists were awarded Nobel prizes in physics at a single award ceremony in Stockholm in 1933, Heisenberg receiving the prize for 1932 and Dirac and Schrödinger sharing the prize for 1933.
In his Nobel address, Heisenberg stated:
"Quantum mechanics … arose, in its formal content, from the endeavor to expand Bohr's principle of correspondence to a complete mathematical scheme by refining his assertions. The physically new viewpoints that distinguish quantum mechanics from classical physics were prepared … in analyzing the difficulties posed in Bohr's theory of atomic structure and in the radiation theory of light." (Nobel Lectures in Physics, p. 290)
The difficulties were many, including that of calculating the intensities of spectral lines and their frequencies in most cases (hydrogen being an exception) and in deducing the speed of light in various materials. In classical theory an atomic electron would emit radiation of the same frequency with which it orbited a nucleus, but the frequencies of spectral lines are practically unrelated to that orbital frequency and depend equally upon the final and initial atomic states. Heisenberg argued that the electron orbiting the nucleus was not only "un-observable," but also "unvisualizable," and that perhaps such an orbit did not really exist! He resolved to reformulate the theory following Einstein's procedure in formulating the theory of relativity, using only observable quantities. Heisenberg started, therefore, with the spectral lines themselves, not an atomic model, and introduced a transition "amplitude" Aif, depending on an initial state i and a final state f, such that the line intensity would be given by the square of Aif. That is analogous to the fact that light intensity in Maxwell's optical theory is given by the square of the field intensity. (Actually, Aif is a complex number, involving the square root of −1, so in quantum mechanics we use the absolute square, which is a positive real number.)
In calculating the square to produce the light intensities, Heisenberg found it necessary to multiply amplitudes together, and he discovered that they did not behave like ordinary numbers, in that they failed to commute, meaning that in multiplying two different A's the result depended on the order in which they were multiplied. Heisenberg was working in Göttingen under the direction of Max Born, and when he communicated his new result, Born realized that the mathematics involving arrays of numbers such as Aif was a well-known subject known as matrix algebra. Born, together with his student Pascual Jordan (1902–1980) and Heisenberg, then worked out a complete theory of atoms and their transitions, known as matrix mechanics. Born also realized that the matrix Aif was a probability amplitude, whose absolute square was a transition probability. This meant that the law for combining probabilities in quantum mechanics was entirely different from that of classical probability theory.
Heisenberg was only twenty-four years old when he made his major discovery. The son of a professor of classics at the University of Munich, he received his Ph.D. under Sommerfeld at Munich in 1923, after which he went to work with Born and later with Bohr. He had an illustrious career, not unmarked by controversy. The author of wave mechanics, Erwin Schrödinger, on the other hand, was already in 1926 an established professor at the University of Zurich, holder of a chair once held by Einstein, and an expert on thermodynamics and statistical mechanics. He was born in Vienna to a wealthy and cultured family and received his Ph.D. at the University of Vienna in 1910.
Schrödinger's theory was based on an idea that the French physicist Louis de Broglie (1875–1960) put forward in his Ph.D. thesis at the University of Paris in 1924. Einstein had advanced as further evidence of the particle character of the photon that it had not only an energy hν, but also a momentum p = hν/c. The American experimentalist Arthur Compton (1892–1962) showed in 1923 that X-rays striking electrons recoiled as if struck by particles of that momentum. Photons thus have wavelength λ = c/ν = h/p. De Broglie conjectured that all particles have wave properties, their wavelengths being given by the same formula as photons, with p being given by the usual particle expressions (p = mv, nonrelativistically, or p = mvβ in relativity, with β(1 − v2/c2)−1/2). C. J. Davidson and L. H. Germer in America and G. P. Thomson in England in 1927 showed the existence of electron waves in experiments. An important application is the electron microscope.
Schrödinger's theory merged the particle and wave aspects of electrons, stressing the wave property by introducing a "wave function," often denoted by the Greek letter psi (ψ), which is a function of space and time and obeys a differential equation called Schrödinger's equation. Psi has the property that its absolute square at a certain time and place represents the probability per unit volume of finding the electron there at that time. The stationary atomic states of Bohr are those whose probability density is independent of time; the electron in such a state is spread out in a wavelike manner, and does not follow an orbit. The real and imaginary parts of the one-particle wave function are separately visualizable, but not for n particles, as it exists in a 3 n-dimensional space.
Although the pictures of Heisenberg and Schrödinger are totally different, Schrödinger (and others) proved that their experimental consequences were identical. In 1926 the English physicist Paul Dirac showed that both pictures could be obtained from a more general version of quantum mechanics, called transformation theory, based on a generalization of classical mechanics. When he did this work, the basis of his doctoral thesis in 1926, he was the same age as Heisenberg. Dirac was born in Bristol, the son of an English mother and a Swiss father. The latter taught French language in the Merchant Venturers' School, where Dirac received his early education, going on to earn his Ph.D. at Cambridge. The influence of Dirac's treatise The Principles of Quantum Mechanics, published in 1930, has been compared by Helge Kragh to that of Newton's Principia Mathematica.
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