What happens to a mass on a spring when quantum mechanics gets hold of it? We borrow only the answer. The energies come in discrete steps, and what is more, the spacing is the same everywhere.
The ladder in the figure is that. From the bottom, 0, \varepsilon, 2\varepsilon, 3\varepsilon, going on at the same width for ever. \varepsilon = \hbar\omega, so a stiffer spring means a wider spacing.
This ladder is what the previous piece meant by freezing out. Where kT is smaller than \varepsilon, nothing gets off the bottom rung. Here we carry the calculation of what actually happens all the way through.
Write down the partition function. Z = 1 + e^{-\beta\varepsilon} + e^{-2\beta\varepsilon} + \cdots. The n-th term is e^{-\beta\varepsilon} multiplied by itself n times, so this is a geometric series.
The bars in the figure are those terms, each shorter than the last by the same ratio. That ratio is e^{-\beta\varepsilon}, and being less than 1 it leaves a finite total however far you add.
The sum is Z = \dfrac{1}{1 - e^{-\beta\varepsilon}}. There are infinitely many levels and yet the partition function fits into that short a formula. The rest is the rule from the previous chapter: take the logarithm and differentiate in \beta.
The working gives \langle E \rangle = \dfrac{\varepsilon}{e^{\beta\varepsilon} - 1}. Planck writing this form is where quantum theory began, and we have now got it out of a partition function.
The bright curve in the figure is that. The faint straight line is the classical answer put there for comparison, the kT that equipartition calls for. An oscillator has position and velocity both entering as squares, so two lots of \dfrac{1}{2}kT make kT.
At high temperature the two lie exactly on one another, because kT far exceeds the spacing and the steps cease to be visible. At low temperature, though, only the bright one falls away to 0.
The heat capacity makes that falling away plain. Being the slope of the energy against temperature, it lies flat at the classical k when hot and drops to 0 when cold.
What matters is that the drop is exponential. Once kT falls below \varepsilon, the very chance of reaching the next rung is crushed by e^{-\beta\varepsilon}. It does not dull away by degrees: it falls off a cliff at a certain temperature.
The vertical line in the figure is where kT = \varepsilon. To its right, classical physics holds; to its left, the thing freezes out. That accounts for one step of the staircase in the previous piece.
To finish, treat a solid as a collection of oscillators. N atoms sit on a lattice, each vibrating in three directions. We count 3N oscillators. This is Einstein's model.
At high temperature each carries k, so the whole carries 3Nk, which is 3R per mole. That is the law of Dulong and Petit, found by experiment long before. The upper horizontal line in the figure is that height.
Experiments had also shown the heat capacity falling at low temperature. Classical physics could not account for it, and Einstein brought this calculation to bear on it. The curve in the figure is his answer, and as far as the falling goes it agrees. The fall is steeper than the measurements, though, and Debye's model is what corrected that, but that is another story.