Einstein and vibrational energy of the atom and its way to QM

Question

As suggested by one of the commentators on my last question, I am going through Bohr's Nobel prize lecture in order to understand how quantum mechanics was developed.

The lecture describes Planck's observations on radiations. I'd like to know how it was concluded from these observations that energy can only change in discrete amounts and also the derivation of Planck's formula.

The lecture then goes on to Einstein's idea about the restricting condition for vibrational energies of atoms and the specific heats of crystals. Can anyone describe exactly what Einstein's idea was about the specific heats of crystal, how that accentuated the concept of discrete energy variation and how it led to further development of QM?

Answer 1

Planck's work was in resolving the ultraviolet catastrophe. In a classical system every degree of freedom should contain the same amount of energy, but the number of degrees of freedom increases as the frequency increases and there should be an infinite amount of energy in short wavelength radiation. Planck proposed that energy can only be transferred in discrete units (of $h\nu$) and this prevents the short wavelength modes being excited and keeps the energy finite.

Einstein pointed out that similar reasoning applies to the excitation of vibrations in a crystal lattice (or indeed any system in which it is possible to excite vibrational modes).

Answer 2

Planck's work was not in resolving the ultraviolet catastrophe, it was in alignment with Wien's law (a thermodynamic approach).

Douglas Stone goes into this issue quite in depth in his book on the history of quantum mechanics. Planck's role is seminal but also overstated (and he disagreed vehemently with Einstein's arguments in his 1905 and 1906 papers on the quantization of the radiation field and on the specific heat of solids).

Answer 3

Planck's motivation to consider discrete energies came from the fact that the classic theory with continuous energies did not match the observed radiation out put from (or inside of vacant) "black bodies." Only by assuming energies are discrete, and inserting that into the mathematics, could he come up with a black body radiation equation that matched the experiments. A good description of the mathematical development is in Eisbergs's "Fundamentals of Modern Physics."