The black body radiation
The quantum mechanics started by the black body radiation problem. At the end of the XIX century, physicists tried to explain the spectrum of frequencies originated by the thermal energy of a body. The concept was idealized to the black body, a body which absorbs all the energy received and doesn't reflect it; this allow us to know that all the radiation emitted by the black body comes from the thermal energy (and not by any other surrounding body which radiation is reflected in the black body). A black body is an hypothetical object, but which can be simulated by a rigid box with a cavity and a small hole where the internal radiation can escape. The box must in equilibrium with its environment. The radiation will start into the box, and escape from the some hole. As the hole is small compared with the box, we can consider the escaped radiation to be insignificant with the rest, which made the radiation still constant inside the box.
The spectrum which offers the black body has a maximum intensity at some frequency which is proportional to the temperature. But as you increase the temperature, the intensity decrease. Lord Rayleigh and Sir James Jeans imagined the radiation originated by oscillators (as the electrons rotating around the atomic nucleus) in the internal walls of the black body. By applying the classical mechanics, those oscillators will increase their energy as the temperature increase (E = kT/2, where E is the thermal energy, k the Boltzmann constant and T the temperature). Applying this energy to classical oscillators (E = kA2/2 = kx2/2 + mv2/2) we already see that, as the amplitude of the oscillators won't change, the velocity must; this leads to a higher frequency. The relation of the temperature and the frequency is exponential. This leads to the ultraviolet catastrophe. This means that higher the temperature is, higher the intensity at higher frequency is, and this is not true as the sun radiation should radiate us with high intensive ultraviolet rays, but doesn't.
So, another explanation should be giving to the black body problem. And Max Planck comes with the clue.
So, another explanation should be giving to the black body problem. And Max Planck comes with the clue.
Hypothesis of quanta
Max Planck started from the Boltzmann work which defines the entropy. He consider the entropy as an absolute value, as when you divide a gas in smaller parts until you have a single atom, the minimum equilibrium state of this single atom will define the initial state of the entropy. Adding more and more atoms, you increase the entropy by discrete quantities (as you introduce a discrete number of atoms). This was called the hypothesis of quanta.
Applying this to the oscillators introduced by Rayleigh and Jeans, we can first obtain the state of a microscopic oscillator by its coordinates, which are two: the moment, and the rate at which it varies. The region of this state plane defined by those two coordinates, will be constant and equals to h (the Planck's constant) which has the dimension of erg sec.
How does this affect the oscillators? Knowing that the amplitude of the oscillators will remain constant, and the product of the moment and its derivative too, there will be a region in the plane phase were the oscillator will be moving, which draws an ellipsis (with the moment in a semi-axis and its derivative in the other semi-axis). This lead to a energy state as E = nhv, where h is the Planck's constant, v is the frequency of the oscillator, and n is a positive integer which defines the region in the plane phase of the oscillator.
Coming back to the entropy, we must now connect this energy with the entropy. The second principle of the thermodynamics give it to us as dS = dE/T, and leads to:
Applying this to the oscillators introduced by Rayleigh and Jeans, we can first obtain the state of a microscopic oscillator by its coordinates, which are two: the moment, and the rate at which it varies. The region of this state plane defined by those two coordinates, will be constant and equals to h (the Planck's constant) which has the dimension of erg sec.
How does this affect the oscillators? Knowing that the amplitude of the oscillators will remain constant, and the product of the moment and its derivative too, there will be a region in the plane phase were the oscillator will be moving, which draws an ellipsis (with the moment in a semi-axis and its derivative in the other semi-axis). This lead to a energy state as E = nhv, where h is the Planck's constant, v is the frequency of the oscillator, and n is a positive integer which defines the region in the plane phase of the oscillator.
Coming back to the entropy, we must now connect this energy with the entropy. The second principle of the thermodynamics give it to us as dS = dE/T, and leads to:
At this point, we clearly see that for a given temperature, the energy don't tends to infinite for higher frequencies, which solve the ultraviolet catastrophe.
Planck postulate
From the energy state obtained, Planck enunciates its postulate as each physical entity with a single mechanical degree of freedom where its coordinates origins a simple harmonic motion can only have discrete values of energy represented by E = nhv (where n is a positive integer).
This quickly comes to the definition of the atomic structure, and helps Bohr to enunciate its atomic model. The Planck's constant was the entry point to the quantum mechanics, which forbids the energy to be continuous but discrete. This also helps the understanding of the wave-particle duality of the light.
