X 2 x

Quantised energy levels

N0, Nj, N2 = the number of oscillators at each frequency. The higher the energy, the greater the spacing between levels.

If only certain energy levels are possible, the integration is replaced by a series of discrete terms. The expression for average energy takes on a completely different look!

To evaluate the two infinite series, we use the binomial theorem:

and note that

(1 - x )-1 = 1 + x + x 2 + x 3 = S and (1 - x )-2 = 1 + 2x + 3x 2 + 4x 3 = S-

One can imagine the thrill experienced by Planck when he saw the expression appear in the denominator, which bore such a close resemblance to his empirical formula. The difference of course was that this time the expression emerged from a physical argument and not from a mathematical manipulation. The quantum hypothesis produced a fit to experimental data!

Quantum discrimination. The average energy of oscillators in thermal equilibrium is not = kT but a smaller value, depending not only on the temperature T but also on the frequency f. As the frequency increases, the number of oscillators decreases because they can only accept large quanta.

For low frequencies the average approaches the classical value.

Increasing f ^ quantum discrimination begins to bite.

(E) fi <<kT (decreasing rapidly as higher powers take over)