Einstein's explanation in 1907 of the observed deviation at low temperature of the specific heat of simple solids from their classical value of 3Nk (N = the number of molecules in a gram, k = "Boltzmann's constant," a second universal constant from the blackbody formula) corroborated the quantum hypothesis. In Einstein's model of a solid the potential that an atom experiences near its equilibrium position is the same for all the atoms of the solid. Hence for small vibrations near their equilibrium point all the atoms oscillate with the same frequency ν. Quantization implies that each oscillator can only have an energy equal to ε
In his doctoral thesis on the electron theory of metals (1911), Niels Bohr concluded that atoms constructed according to the principles of classical physics could not represent the magnetic properties of metals. Working in Ernest Rutherford's laboratory in Manchester just after Rutherford proposed the nuclear model of the atom, Bohr seized upon it because its radical mechanical instability made it a promising candidate for repair by a quantum hypothesis (see Atomic Structure). Bohr stabilized the Rutherford atom by supposing that it could exist in various "stationary states" constrained by certain quantum rules but otherwise governed by the laws of classical mechanics. However, the laws of mechanics do not hold for the transition of the system between two stationary states during which the atom radiates a quantum of energy hν equal to the difference in energy between the two states. On Bohr's theory, radiation is not emitted (or absorbed) in the continuous way assumed by Maxwell-Lorentz electrodynamics.
Bohr's first postulate, which limited the validity of classical mechanics in the atomic domain, restricted the angular momentum of each atomic electron to an integral multiple of h/2π. The second postulate, which denied the validity of classical electrodynamics for radiative processes in atoms and made the frequencies of atomic spectral lines different from the orbital frequencies of the electronic motions, required surrendering the classical connection between the frequency ν of the emitted radiation and the mechanical frequency of the electron in its orbit.
With the help of these quantum rules Bohr accounted for the phenomenological regularities that had been discerned in the hydrogen spectrum, in particular, the Balmer formula for transitions to the n = 2 level, and also, and more dramatically, for the spectrum of ionized helium (1913–14). During World War I, Arnold Sommerfeld generalized Bohr's postulates to elliptical electron orbits and then to motions in three dimensions. He recorded his success in calculating regularities in doublet and triplet spectra, in the Zeeman effect, and in x-ray spectra in a long book, Atombau und Spektrallinien (first edition 1919), with which all physicists interested in quantum and atomic physics during the early 1920s began their work.
In the early 1920s Bohr gave a phenomenological explanation of the periodic table based on the occupancy by electrons of Coulomb-like orbits in multi-electron atoms. Thereafter, many theorists tried to justify Bohr's explanation, but, except for Wolfgang Pauli's formulation of the exclusion principle early in 1925, none of their efforts provided a stable foundation for the dynamics of atoms. They were seminal, however, in that they made manifest the problems a more complete quantum mechanics would have to solve.
In 1917 Einstein took what in retrospect was an important step toward this mechanics. Still flirting with the corpuscular nature of radiation, Einstein introduced the concept of the probability for the spontaneous emission of a light quantum by a "molecule" in a transition from one state to another. The concept allowed an easy derivation of Planck's blackbody formula. In 1923, Arthur Holly Compton's experiment on the scattering of X rays by electrons indicated that the shift in the wave length of the scattered X ray and the recoil energy of the electron could be derived on the assumption that the X rays acted as particles with energy hν and momentum hν/c (c the velocity of light). The positive result of the Compton experiment led Einstein to declare that there are "two theories of light, both indispensable, and ... without any logical connection." The corpuscular viewpoint accounted for the optical properties of atoms, whereas macroscopic phenomena like diffraction and interference required the wave theory of light. The two theories coexisted without any resolution during the early 1920s.
Another important guide to a more powerful quantum physics was the correspondence principle Bohr refined between 1913 and 1918. It stated that the frequencies calculated by Bohr's second postulate (during "quantum jumps") in the limit where the stationary states have large quantum numbers that differ very little from one another will coincide with the frequencies calculated with the classical theory of radiation from the motion of the system in the stationary states. Bohr's assistant Hendrik Kramers cleverly applied the correspondence idea to compute the intensity and polarization of the light emitted from simple atoms. Kramers and Werner Heisenberg extended the same idea to the dispersion of light and worked out ways to translate classical quantities involving a single stationary state into quantum mechanical quantities involving two or more states. Max Born, Heisenberg's teacher at the University of Göttingen, called for a "quantum mechanics" for calculating with the quantum mechanical quantities directly. That was in 1924. In less than a year Heisenberg provided him with one. Its guiding principles were satisfaction of Bohr's correspondence principle (in the appropriate limit the theory should yield the classical results); recognition that the troubles of the "old quantum theory" arose primarily from breakdown of the kinematics underlying classical dynamics; and restriction of the theory to relations between observable quantities.
Born, Heisenberg, and a fellow student of Heisenberg's, Pascual Jordan, soon developed the new mechanics into an elaborate mathematical formalism. They built a closed theory that displayed strikingly close analogies with classical mechanics, but at the same time preserved the characteristic features of quantum phenomena. Their work laid the foundations of a consistent quantum theory but at the price of relinquishing the possibility of giving a physical, visualizable picture of the processes it could calculate. Hence the relief felt by Planck, Einstein, and Lorentz when Erwin Schrödinger, who followed a route entirely different from Heisenberg's, began to publish his wave mechanics in 1926. It seemed to avoid the unconventional features of Heisenberg's formulation and rested on more traditional foundations and easier calculations: variational principles, differential equations, and the properties of waves.
Schrödinger had followed up insights and suggestions by Louis de Broglie and Einstein. In 1923 de Broglie published an idea that was the obverse of Einstein's attribution of particle properties to wave radiation—to endow discrete matter with wave properties.
By following sometimes fanciful analogies and the principle of relativity, de Broglie associated a wave of frequency ν and wavelength λ with a particle of momentum p and energy E according to ν = E/h, 1/λ = p/h. He thus extended the particle-wave duality of radiation to matter. Knowing the wavelength, Schrödinger soon found an appropriate differential equation for a wave of amplitude Ψ. He interpreted the Ψ function as describing a real material wave and considered the electron not a particle but a charge distribution whose density is given by the square of the wave function. In a short paper dated June 1926, Born rejected Schrödinger's viewpoint and proposed a probabilistic interpretation for the Ψ function. He stipulated that the wave function Ψ(x,t) determines the probability of finding the electron at the position x at time t. In 1927 two different sets of experimentalists—George P. Thomson (the son of Joseph John Thomson) in Britain and Clinton Davisson and Lester Germer in the United States—detected diffraction patterns from an electron beam.
Several physicists proved in 1926 that wave mechanics gave the same numerical answers as the "matrix mechanics" of Born, Heisenberg, and Jordan. Together they are known as quantum mechanics. In contrast to classical physics, which contained no scale and was assumed to apply both in the micro and macro domain, quantum mechanics asserted that the physical world presented itself hierarchically. Certain constants of nature layered the world. As P. A. M. Dirac emphasized in the first edition of his Principles of Quantum Mechanics, Planck's constant allows the parsing of the world into microscopic and macroscopic realms.
The conquest of the microrealm during the first years after the invention of quantum mechanics stemmed from the confluence of two factors: the apperception of an approximately stable ontology of electrons and nuclei, and the formulation of the dynamical laws governing the motion of electrons and other microscopic particles moving with velocities small compared to the velocity of light. Approximately stable meant that electrons and (non-radioactive) nuclei, the building blocks of atoms, molecules, simple solids, could be treated as ahistoric objects, with physical characteristics seemingly independent of their mode of production and lifetimes effectively infinite. These electrons and nuclei behaved as if they were "elementary," almost point-like objects specified only by their mass, their intrinsic spin, electric charge, and magnetic moment. In addition, the members of each species were indistinguishable: all electrons are identical, as are all protons, and all (stable) nuclei of a given charge and mass when in their ground state. Their indistinguishability implied that an assembly of them obeyed characteristic statistics depending on whether they had integral or half odd integral spin (measured in multiples of h/2π). Bosons (particles with zero or integral spins) can assemble in any number in a given quantum state. Fermions (particles with half odd integral spins) do not share a quantum state. A one-particle quantum state can be characterized either by the position and the spin state of the particle or by its momentum and its spin state. Thus no two identical Fermions can be at the same position if they have the same spin. More generally, the wave function describing a system of identical bosons remains unchanged under the interchange of any two particles, whereas that describing fermions changes sign under such a transposition.
The quantum mechanical explanation of chemical bonding resulted in a unification of physics and chemistry. In 1929, following the enormous success of nonrelativistic quantum mechanics in explaining atomic and molecular structure and interactions, Dirac, a main contributor to these developments, declared that "the general theory of quantum mechanics is now almost complete." Whatever imperfections still remained were connected with the synthesis of the theory with the special theory of relativity. But these were of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions. "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble." Dirac's assertion may still have the validity it had, but, as emphasized by Phillip Anderson, "the reductionist hypothesis does not by any means imply a "constructionist" one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe. In fact, the more the elementary particle physicists tell us about the nature of the fundamental laws, the less relevance they seem to have to the very real problems of the rest of science, much less to those of society. The constructionist hypothesis breaks down when confronted with the twin difficulties of "scale and complexity." Still, physics can be regarded as more foundational (not fundamental) than chemistry because the laws of physics encompass in principle the phenomena and the laws of chemistry.
J. L. Heilbron
Thomas S. Kuhn, Blackbody Theory and the Quantum Discontinuity, 1894–1912 (1978).
Abraham Pais, Inward Bound (1982). Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory
Olivier Darrigol, From C-Numbers to Q-Numbers (1992).
Mara Beller, Quantum Dialogue: The Making of a Scientific Revolution (1999).
Helge Kragh, Quantum Generations (1999).
From The Oxford Companion to the History of Modern Science
