To make a form which gets the limiting cases correct as do Eqs. (9) and (11) and interpolates smoothly between them, Planck hypothesized that the allowed energies for each mode take the form of a series of levels at equal spacing proportional to the frequency. Thus,
The proportionality constant which controls the spacing, h, is known as Planck's constant.
Under the Planck Hypothesis, the partition function becomes
where we have used the familiar formula for summing an infinite geometric series to determine Z.
The average energy is then
The form Eq. (13) matches the experimental results for
perfectly (!) when h is assigned the value
