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Blackbody Radiation

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Quantum Physics: Blackbody Radiation

Abstract

While Einstein’s theory of Relativity left many of the worlds leading scientists in awe about the big objects that govern our universe; Quantum Physics disrupted many ideas that Classical theory postulated and also forced a new way of perceiving our physical reality. The introduction of Quantum Physics in the early 20th century refuted conventional claims of Classical theory and hence launched Physics into a new era. Phenomena such as Blackbody Radiation that could not be explained by conventional physics forced scientists to compensate for the discrepancy of experimental data and classical theory. This compensation or explanation that quantum physics provides will be discussed in further detail.

Introduction

If one is confused about the kind of impact Quantum Physics has had on the scientific community one can compare it to Copernicus defying the Church and voicing out that the Earth revolves around the sun and not vice versa. Just the manner in which Copernicus’ discovery changed the way in which people think of their reality; quantum physics makes us redefine what we thought we knew about our world. Take into consideration a phenomenon known as blackbody radiation. First, a blackbody is an object that has the ability to absorb any form of electromagnetic radiation that it encounters. This means that there is no reflection or transmission of the incoming radiation. Hence this implies that these objects radiate energy; and this radiation is in its thermal form. The foremost property of a blackbody is that the energy (wavelength of the electromagnetic radiation) it emits is proportional to its temperature. At low temperatures, such as Temperature < 400º K, blackbodies emit radiation at low levels of visible radiation hence looking dark in colour (implicating the name blackbody) and at high temperatures they emit a high level of visible radiation in the form of colours such as orange, yellow et cetera. Summing up, a blackbody emits maxima of electromagnetic radiation for a given temperature. This emission is known as Blackbody Radiation.

The Discrepancy

Consider a blackbody, in this case a metal as shown in Figure 1. The black area represents that the body is hot and has a temperature T and is emitting radiation in the visible spectrum. According to classical mechanics the radiation (measure of frequency, ν, of the visible light) and magnitude of the radiation being emitted by the metal can be predicted given the equation:

1

Figure 1

ρ(ν, T) dν = {8πν2 ÷ c3} Ēosc dν (1)

where ρ is the spectral density which is a function of frequency, ν, and temperature, T;c is the speed of light and Ēosc is the average energy of the oscillating dipole inside the metal or blackbody. It can further be noted that the factor dν is present on both sides of the equation because the energy density is observed within the frequency interval of width dν which is centered at the measured frequency ν. Moreover, classical mechanics predicts that the average energy of the oscillator, Ēosc, is related to the temperature by:

Ēosc = kBT (2)

where kB is the Boltzmann constant, which relates temperature to energy. Substituting the value for (2) into (1) we obtain:

ρ(ν, T) dν = {8πkBTν2 ÷ c3} dν for T > 0º K (3)

Now given the classical definition of the spectral density of a blackbody radiator consider the graph in Figure 2.

2

Figure 2

According to the graph in Figure 2 it can be seen that classical theory (3) predicts that as the temperature of the blackbody is increased it would emit an infinitesimal amount of electromagnetic radiation, i.e., the spectral density distribution would be infinite. The experimental and theoretical (3) curve behave quite identical for low frequencies but the theoretical curve (3) keeps increasing as temperature increases. This is the discrepancy between experimental data and classical theory where the observations of an experiment are inconsistent with theoretical predictions.

The Explanation

The gap between the experimental observations and theoretical predictions were bridged together by a German physicist named Max Planck. He analyzed the system of blackbodies closely and in first understanding the classical reasoning for an infinitesimal spectral density with increasing temperature he was able to devise the reason for the discrepancy. Further, Planck also noted the discrepancies were observed at high temperatures and not at low temperatures. Classical reasoning for blackbody radiation indicated that the vibration of electric diploes emit radiation at the frequency at which they oscillate. Planck further devised that high-frequency radiation was inert at low temperatures implying they were active only at high temperatures. The radical change comes when Planck dismisses the idea that the dipoles emit radiation at the frequency at which they oscillate. He implicates that the energy (radiation) emitted by the oscillating diploes is proportional to the frequency. Therefore, this suggests that the radiation emitted is independent of frequency. Planck further assumed that radiation emitted by the dipoles is given by:

E = nhν (4)

where n is a positive integer, h is Planck’s constant (which was initially an unknown proportionality constant) and ν is the frequency. The frequency in (4) is related to energy; this is the first time that energy has been related to frequency. The classical interpretation of energy being continuous was obliterated by this new relationship which implied energy comes in “discrete packets” known as “quanta”. This implies the energy emitted by a blackbody is not continuous in nature and can only take on an integer set of values for each frequency. Moreover, using the relationship in (4) Planck produced a new definition for the average energy of the oscillating dipole:

Ēosc = hν ÷ [ehν/kT – 1] (5)

Substituting the value for (5) into (1) we obtain:

ρ(ν, T) dν = {8πhν3} ÷ {c3[ehν/kT – 1]} dν (6)

This is the new spectral density definition, devised by Planck, which enables the co-existence of observational data and theoretical predictions. The constant, h, was still undetermined but nevertheless he used the adjustable parameter, h, to reproduce experimental data at any given temperature. Hence the idea that Planck proposed that energy was related to frequency and not continuous (energy came in packets called “quanta”) proved worthy as he was able to provide an explanation of the discrepancy that classical mechanics encountered with blackbody radiation.

Conclusion

In closing, it can be seen that classical mechanics was unable to provide a viable explanation for a natural occurring phenomena such as blackbody radiation. Quantum mechanics provides the bridge between experimental observation and theoretical prediction in the case of blackbody radiation. The idea that energy comes in lumps or packets was a new idea during the early 20th century and has definitely revolutionized our perception of how the world works on a small (quantum) level. In comparison to the big scale of things quantum physics forces one to think about the workings of the world from a new stand point (i.e. in this case non-continuous energy) just like the way Copernicus suggested that the Earth revolves around the sun which inevitably revolutionized the perception that the conventional scientist at the time had of the world.