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II. Planck Moment - Born Style

In building quantum statistics for the blackbody radiation, Planck had at his disposal the two formulas for the spectral density of radiation, corresponding to the limit cases of high and low temperatures. Both these cases satisfy the Wien’s displacement law, as the only theoretical criterion for the choice of radiation laws. According to Born (Born, 1955), Planck’s first reasoning can be linked to the properties of the Gaussian probability distribution. It is worth going into a little detail along this path, inasmuch as it gives us the overall idea of this important gnoseological process. Born’s starting point is, like Planck’s, the energy density as a function of temperature, with its extreme cases

(1)

where β ≡ (kT)^-1, with k the Boltzmann constant, T the absolute temperature and ε₀ an energy that has to be proportional with the frequency of light in order to satisfy the requirement of Wien’s displacement law. Then Born proceeds to the first of Planck’s steps, which was to study the entropy of such a system, which he assimilated with a system of oscillators. The rationale should have been like this: the entropy is classically related to heat exchanged, at equilibrium, between two thermodynamic systems, or between a system and the Universe. Here we have the heat in the form of thermal radiation! Nothing more natural then, than considering the heat as determined by the energy of this radiation. According to Born, Planck’s intention has been facilitated by the important discovery that the coefficient u’(β) (prime denoting the derivative with respect to variable) has a simple statistical meaning. This can be obtained starting from Thermodynamics. Indeed, as the radiation just represents the heat exchanged in equilibrium, in the formula defining the entropy:

(2)

we only have to identify the amount of heat with (du) the differential of the density of radiation energy. In so doing, equation (2) can be rewritten as

(3)

At this point one has to recall the Einstein’s procedure whereby one identifies the thermodynamic entropy with the statistical entropy as related to the probability by the Boltzmann relation:

(4)

Here two arguments have to be used: the energy as carried by radiation is actually characterized by fluctuations and, as these fluctuations take place at equilibrium, the entropy has to be maximum according to classical precepts. Then the entropy, as a function of energy density can be expanded as

(5)

and the Boltzmann formula reveals (although approximately) a Gaussian describing the fluctuations represented by the thermal radiation. We write it in the normalized form as:

(6)

where X ≡ Δu and σ² is the variance of this process which, with equation (3), can be written in the form

(7)

provided u is a continuous function of temperature. In other words, the first derivative of the energy density with respect to the inverse temperature is actually the variance of a normal distribution characterizing the fluctuations of energy of the field representing the thermal radiation.

This was the statistical meaning apparently revealed by Planck, and the reason he insisted upon a close consideration of the equation (1) which, in view of this, can be rewritten as (Born, 1955)

(8)

Actually Planck himself worked directly with the entropy (Planck, 1900). As mentioned above, considering the energy density is the mark of an Einstein-Born style approach of the problem, but it closely parallels Planck’s own way, having besides this a hint towards the newest theoretical discoveries in the problem of radiation. The reason we undertake this approach will be shortly apparent. Now, keeping in mind equation (7), which shows that we are actually looking at the variance of a Gaussian process, the equation (8) can be interpreted as representing two copies of this process, for the cases of high and low temperatures. For the instance where these two processes are statistically independent, the variance of the compound process is the sum of the two component variances, so one can infer that, in general, the law of radiation could be represented by the following differential equation:

(9)

This is what Planck really did. This equation has an immediate particular solution

(10)

and from this moment on everything is recorded history. Mention should be made that the general solution depends on one arbitrary constant leading to the Bose statistics, which was revealed later on.

Fact is that equation (10) has a physical interpretation unveiling the statistical ensemble that has it for a mean. The physical interpretation goes on describing an ensemble of harmonic oscillators, each one having the energy an integer multiple of ε₀, of which we know nothing but that it is an energy - in order to make the equation (9) physically meaningful - proportional with the frequency - in order to make the equation (10) physically meaningful - for the density of thermal spectrum has to satisfy Wien’s displacement law. The partition function of this ensemble can be written as

(11)

so u(β) from equation (10) appears indeed as the mean of this ensemble:

(12)

The presence of quantum is here suggested by the random variable characterizing this distribution, which is a natural integer. The photon comes into play later on. However, it seems that the modern theory relies mostly upon what has been left behind along the way towards Quantum Mechanics, that started from this beginning.

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