At the turn of the century, it was discovered that atoms are made of a very small positively charged nucleus surrounded by negatively charged electrons.
Planck discovered that the law of blackbody radiation could be fitted to a theoretical distribution only if the energy of the electromagnetic field came in discrete energy packets.
Einstein was a strong beleiver that matter was composed of atoms, a far from popular viewpoint at the time.
In 1913 Bohr came up with the concept that electrons orbit the nucleus only at a discrete set of closed orbits, and that photons are emitted as the electron makes a 'quantum jump' from one quantized state to another. The energy of the electron was quantized. Although this explained the observational evidence of discrete spectral lines emitted by hot gases, it was not known why the electron behaved in such an bizzare way !
[IMAGE of Bohr model of the atom, with quantized circular orbits ]
If electron were considered to be discrete point-like particles in circular orbit around the nucleus, they would be subject to strong centripedal accelerations. However according to Maxwell's equations of classical electromagnetism, charged particles undergoing accelerations emit radiation, they should be continuously losing their orbital energy and spiral into the nucleus. Atoms would decay, contrary to the evidence that atoms in their ground state are stable.
If an electron at rest in one inertial reference frame doesn't emit radiation, then neither does an electron in constant velocity motion as we can always find another inertial reference frame in which it is also at rest. From the postulates of special relativity; the laws of physics are invariant upon the particular choice of inertial reference frame, therefore electrons with constant velocity don't radiate. If we could somehow formulate the laws of the atoms in which only velocites occur without accelerations, this may explain the lack of radiation from atoms in their ground state. However from Erhenfest's theorem it isn't possible to have zero accelerations in a collection of charges localized to a certain region of space such as electrons around a much heavier nucleus. They must suffer to a certain degree from accelerations, and hence must radiate. Bohr's model leaves us with a paradox.
[IMAGE of electron 'spiralling' down into the nucleus]
Another problem was the quantum jump itself. How was a photon emitted during the discontinuous transition from a discrete orbit to another.
At about the same time, Heisenberg reasoned that dynamical variables such as position and momentum are non-commuting operators, and that the laws of mechanics obeyed equations in which pairs of these operators obey certain commutations rules. Particles were still considered a discrete points but with limits placed on what can be known about them through observations, i.e the Heisenberg uncertainty relations: If the position of a particle was observed with precision dx, we could never oberve its momentum with more precision than dp=h/dx. Similar statements could be made for other pair of congugate operators such as energy and time, photon number and field strength ect...
This solved the problem of the electron spiralling into the nucleus : When the electron is in a stable quatum state, it can't be considered as a point particle anymore, but has a certain degree of positional uncertainty or fuzziness which is of the order of the dimensions of the atom. Its momentum is also somewhat uncertain.
Schroedinger later conceived of the electron as a wavefunction or electron probability amplitude 'clouds', quantum uncertainty could be regarded within a more intuitive framework.Pauli showed that both of these formulations of quantum mechanics were equivalent representations, this completed the proof that electrons and photons could act as particles or waves, depending on the way in which they were observed.
[ IMAGE of some hydrogenic wavefunctions ]
The wavefunction is only a mathematical construct, a calculational tool to predict the probability of measuring the particle within a certain region of space. We cannot directly see this electron cloud. To reconstruct a wavefunction based on experimental laboratory observations requires that a huge number of individual measurements be taken at different points throughout space and time on quantum systems prepared in an identical way. Then statistics can be used to reconstruct the probability amplitude (like in scanning tunnelling microscopy) and even after all that effort the phase will be uncertain. An easier way would be to use diffraction of photons with wavelength of the order of the desired resolution, or even better: Holography using coherently prepared photons from an xray laser.
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