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3-5: The Discovery of Energy Quanta

As learned on the preceding page, Planck found Planck's formula which can well reproduce the energy distribution (spectrum) of the cavity radiation for all regions of frequency.
When we derived the Rayleigh-Jeans formula, we obtained the number of the normal oscillations in a vacuum (cavity), i.e. the number of the degrees of freedom of the oscillations of the electromagnetic field in a vacuum. The number of the normal oscillations of the frequency between and per unit volume is

If a constant energy kT is partitioned to all these degrees of freedom, then we obtain the Rayleigh-Jeans formula. Comparing Planck's formula and the Rayleigh-Jeans formula, we can easily see that, in realistic cases, the constant energy kT is not equally partitioned, but the energy partitioned actually is

Namely, the partitioned energy is reduced by a weight function

The form of this weight function must be

whose value is close to 1 when x is small, but becomes much smaller than 1 when x is large. The energy distribution would therefore be severely reduced for large x values. Thus, we should realize that, when is large, the law of equipartition of energy is not satisfied.

[Energy Quanta]
Thus, Planck discovered Planck's formula which is well fit to the experimental results. This discovery itself must be great, but Planck's greatness is in having not stayed there. Planck pursued the origin of Planck's formula, and finally he reached the epoch-making idea of energy quanta. Let us explain this below.
If we divide the matter into finer and finer portions, we ultimately reach molecules or atoms. Thus the matter is not continuous but the aggregate of a huge number of fundamental particles. We called this nature of the matter the "atomic nature". Moreover, we learned that there exists the fundamental unit of electric charge, i.e. the elementary charge. Namely, the electricity has also an atomic nature.
Planck thought that there would be fundamental units of energy. He called these energy quanta, which is just an atomic nature of energy. Standing on this viewpoint, he was successful in deriving Planck's formula (1900). This is a kind of revolution in the natural science, which can never be understood with the classical theory. This was the start of the new physics in the 20th century.

[Energy Quanta and Planck's Formula]
It was previously said that the cavity radiation is equivalent to a collection of a large number of harmonic oscillators. The energy of each oscillator is written in a form of

where q is the coordinate variable of the oscillator and p denotes the momentum variable.
As discussed in detail on the page: 3-2-A, we can get the mean value of the energy E using the Boltzmann distribution as

where the normalization constant is given by

If the energy E is a continuous quantity, the mean value of the energy would be given by This is the ordinary law of equipartition of energy. However, if the energy E is not a continuous quantity and cannot take any value freely, then p and q cannot be treated as continuous quantities.
Now, let the amount of the energy quantum be . Suppose that the energy of the oscillator must be an integral multiple of the energy quantum, . Then p and q should fulfill the relation

Considering this, the calculation of the mean value of E should be done not by the integration but by some other procedure. For this purpose, we need some mathematical expressions, so that we explain on the other page,
3-5-A: Derivation of Planck's formula.
The explanation is not so difficult but, if you feel somewhat tedious, you may skip it.

[The Discovery of Energy Quanta]
As explained on the page 3-5-A, if we consider that the energy of the harmonic oscillator, E, can take only an integral multiple of the energy quantum then the mean value would be given by the formula

This is nothing but Planck's formula which is very well fit to experiment.
This concludes that the energy of a cavity radiation with a frequency can take only an integral multiple of the energy quantum . People have usually thought that the energy is a continuous quantity, but, since the discovery of energy quanta, the energy would be discontinuous and has an elementary unit . Thus, an epoch-making idea that the atomic nature in the natural world exists not only in the matter and in the electricity but also in the energy was brought us by Planck at the beginning of the 20th century (1900). This was just the opening of the new science of the new century.

[Planck's Constant]
Precise measurements of Planck's constant have been done and its currently accepted value is

Planck's constant has a dimension of action; i.e., [energy] x [time] = [action], so that it is sometimes called an action quantum.

[Heat Capacities of Solids]
On the page Molecular Motions and Specific Heats we discussed the heat capacities of solids. There we learned that, if one mole of solid is regarded as a collection of 3NA harmonic oscillators, the theoretical value of the heat capacity obtained by using the law of equipartition of energy can well reproduce the experimental value at a temperature higher than the room temperature. However, if the temperature goes down, the law of equipartition of energy does not work well and theory is not fit to experiment.
As soon as the idea of energy quanta was proposed by Planck in 1900, Einstein and P. J. W. Debye (the Netherlands, 1884 - 1966) applied this idea to the cases of solids. They showed that, if we consider that the energy of an oscillation of frequency . can take only an integral multiple of , the theoretical heat capacities can amazingly well reproduce the corresponding experimental data all over the regions of temperature (1907, 1912).
Thus, it was clarified that the nature that the energy consists of "grains" -- we call this the atomic nature -- is held not only for the cavity radiation but also for solids.
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