Two unusual systems
Quantum mechanics surprises with the statement that the microscopic world works very differently from the macroscopic world. Therefore, it took a while until quantum mechanics was formally established as the theory of the microworld. In particular, despite the fact that two of the natural systems on which theories of quantum mechanics could initially be tested were very simple, even from the point of view of the physicists of the time, one needed to introduce a number of novel concepts for their description. These two physical systems were the hydrogen atom and the spectrum of a thermal radiation source. The hydrogen atom was the lightest of all atoms with the most simply structured spectrum. It exhibited many regularities involving rational numbers relating its discrete energy levels. It could only be ionised once implying that it had only a single electron and from these reasons it was the obvious test case for any theory of mechanics in the quantum regime. Werner Heisenberg was the first to be successful in solving this quantum mechanical analogue of the Kepler-problem, i.e. the equation of motion of a charge moving in a Coulomb-potential, paving the way for a systematic understanding of atomic spectra, their fine structure, the theory of chemical bonds, interactions of atoms with fields and ultimately quantum electrodynamics.
The Planck-spectrum was equally puzzling: It is the distribution of photon energies emitted from a body at thermal equilibrium and does not, in particular, require any further specification of the body apart that it should be black, meaning ideally emitting and absorbing radiation irrespective of wave length: From this point of view it is really the simplest macroscopic body one could imagine because its internal structure does not matter. In contrast to the hydrogen atom it is described with a continuous spectrum. In fact, there are at least two beautiful examples of Planck-spectra in Nature: the thermal spectrum of the Sun and the cosmic microwave background. The solution to the Planck-spectrum involves quantum mechanics, quantum statistics and relativity, and unites three of the four the great constants of Nature: the Planck-quantum h, the Boltzmann-constant \(k_B\) and the speed of light c.
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| The spectrum (basically intensity against wavelength or frequency) of the light from the sun (in yellow) and a blackbody with the same temperature (grey). I'm actually surprised by how similar they are. |
Limits of the Planck-spectrum
Although criticised at the time by many physicists as phenomenological, the high energy part of the Planck-spectrum is relatively straightforward to understand, as had been realised by Wilhelm Wien: Starting with the result that photons as relativistic particles carry energies proportional to their frequency as well as momenta inversely proportional to their wave length (the constant of proportionality in both cases being the Planck-constant h), imposing isotropy of the photon momenta and assuming a thermal distribution of energies according to Boltzmann leads directly to Wien's result which is an excellent fit at high photon energies but shows discrepancies at low photon energies, implying that at low temperatures the system exhibits quantum behaviour of some type.
