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Math Jeans
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As is well known, Planck's radiation law for the distribution function of blackbody radiation used a then new concept of energy quanta in order to describe experimental data.
The distribution functions formulated by Wilhelm Wien and Lord Rayleigh, describing the same phenomena, were formulated from classical physics, and were accurate when describing opposite ends of the spectrum.
Mainly, Wien's law was accurate for low wavelengths, while the Rayleigh-Jeans law was for long wavelengths.
Similarly, both equations are easily obtained through applying a high or low wavelength limit to Planck's law, but here is the confusing part:
Wien's law was obviously derived through the use of a continuous energy spectrum (no quantization), and it follows that removal of such quantization would therefore yield Wien's law from Planck's law. In other words, if the value of Planck's constant approaches 0, the equation approximates Wien's law just as in the case of a short wavelength limit.
It is also the case that the Rayleigh-Jeans law failed in its use of the equipartition theorem in order to describe energy levels of each mode in the cavity. Upon applying energy quantization to his model, the final equation becomes Planck's radiation law (I have the math for this if it is necessary for discussion).
Here is my question: although use of energy quanta as opposed to equipartition in Lord Rayleigh's derivation will result in a correct final answer, and that Rayleigh himself admits to equipartition's failure at short wavelengths, is there any physical significance to the fact that if no wavelength limit is taken to Planck's formula, the Rayleigh-Jeans law follows from an increase in the magnitude of energy quantization (high limit of Planck's constant)?
I found this quite counter-intuitive.
The distribution functions formulated by Wilhelm Wien and Lord Rayleigh, describing the same phenomena, were formulated from classical physics, and were accurate when describing opposite ends of the spectrum.
Mainly, Wien's law was accurate for low wavelengths, while the Rayleigh-Jeans law was for long wavelengths.
Similarly, both equations are easily obtained through applying a high or low wavelength limit to Planck's law, but here is the confusing part:
Wien's law was obviously derived through the use of a continuous energy spectrum (no quantization), and it follows that removal of such quantization would therefore yield Wien's law from Planck's law. In other words, if the value of Planck's constant approaches 0, the equation approximates Wien's law just as in the case of a short wavelength limit.
It is also the case that the Rayleigh-Jeans law failed in its use of the equipartition theorem in order to describe energy levels of each mode in the cavity. Upon applying energy quantization to his model, the final equation becomes Planck's radiation law (I have the math for this if it is necessary for discussion).
Here is my question: although use of energy quanta as opposed to equipartition in Lord Rayleigh's derivation will result in a correct final answer, and that Rayleigh himself admits to equipartition's failure at short wavelengths, is there any physical significance to the fact that if no wavelength limit is taken to Planck's formula, the Rayleigh-Jeans law follows from an increase in the magnitude of energy quantization (high limit of Planck's constant)?
I found this quite counter-intuitive.