Quantum Mechanics: Blackbody Radiation

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1 Blackbody Radiation Quantum Mechanics Origin of Quantum Mechanics Raleigh-Jeans law (derivation)-ultraviolet catastrophe, Wien s Distribution Law & Wein s Displacement law, Planck s radiation law (calculation of average energy of oscillator). Wien s law and Stefan s law from Planck s radiation law, Rayleigh-Jenas & Wien s law-limiting case of Planck s law. Origin of Modern Physics Modern physics in the form of quantum mechanics originated in early 20th century from an apparent collapse of classical deterministic physics related with the phenomena both connected to light as electromagnetic waves described by Maxwell's equations: Ultra-violet catastrophe of blackbody radiation: infinite energy, Non-existence of an aether as a medium carrying electromagnetic waves The ultra-violet catastrophe in black body radiation gave birth to quantum mechanics and the non-existence of aether to relativity theory. Other important unsolved phenomena were photoelectric effect, quantisation of atomic energy level, existence of line spectra etc and classical physics were completely failure to establish the realistic explanation. Ultimately Max Planck, Einstein, Scrodinger and many more came to put the realistic contribution to open up the door of new era. Later Planck came up with a resolution related with the concept of light as a deterministic wave phenomenon described by Maxwell's equations, to describe the light as particles or quanta of energy named photons. Later with the combination of Newton's corpuscular theory of light and Maxwell's wave theory, Planck gave up deterministic continuum physics for statistics of particles and finally opened the door of modern physics with wave-particle duality viewed as a resolution of the inescapable contradiction between wave and particle. Einstein picked up Planck's quanta in 1905, and established an explanation of experimentally discovered phenomenon of photo-electricity in 1923 Einstein the Nobel Prize in Physics, ultimately the particle nature of radiation is established. Both Planck and Einstein introduced discrete quanta of energy in order to avoid the ultraviolet Catastrophe long before the quantum mechanics of atoms was formulated in the 1920s in the form of Schrodinger's wave equation. Wave nature of light carries massive evidence and it is well described by Maxwell's Equations. In various aspects, light and matter interaction is connected in emission and absorption of light which are viewed to be difficult to describe as wave mechanics, with blackbody radiation as the basic problem. What is Blackbody? A blackbody is a theoretical idealized object described as something absorbing all incident radiation" commonly pictured as a cavity or empty bottle/box in which waves/photons are bouncing back and forth between walls at a certain temperature defining the temperature of the cavity. A blackbody is supposed to capture an essential aspect of the radiation from a real body like the visible glow from a lump of iron at C, the Sun at C or the invisible infrared faint glow of a human body at 37 0 C. A lump of iron, the Sun or a human body thought of as an empty bottle with a peephole, because Planck used this image in his proof of Planck's Law of blackbody radiation based on statistics of energy quanta/photons in a box. Planck's mathematical proof required a certain set up and that set up came to define the idealized concept of a blackbody as an empty bottle with peephole. But actual construction of such a blackbody is impossible. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 1

2 The Blackbody spectrum is shown below Rayleigh Jeans Law According to Rayleigh the radiation waves in blackbody can be compared to stand standing waves in cubical cavity. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 2

3 Total number of modes in the range and +d will be given by Total energy density in the range and +d will be U d=n λ E av dλ = (8/ 4 )E av d Rayleigh and Jeans assumed that each oscillator in the wall absorbed and emitted radiation constantly, with each oscillator having its own characteristic frequency. For continuous operation of any given oscillator, standing waves must be set up in the enclosure. For the enclosure having reasonable size the differences between neighbouring frequencies are so small that the radiation appears to be continuous. Rayleigh and jeans assumed the concept of classical harmonic oscillator having kinetic energy of each oscillator per degree freedom is (½)kT based on the equipartition theorem. The total energy per oscillator is equal to the sum of kinetic energy (½)kT plus the potential energy (½)kT. Therefore the average energy of kt to each mode of vibration leads to an energy density U λdλ for waves with wavelength between λ and λ+dλ given by The above equation is known as Rayleigh-Jeans formula for black body radiation and that is valid only for longer wavelength region. Mostly non-realistic part of Rayleigh-Jeans distribution law is that the area under the curve of black-body spectrum is infinite. The area physically represents the total energy radiated by the black body. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 3

4 Quantum calculation of black-body spectral irradiance: The Planck radiation law The solution to the ultraviolet catastrophe suggests that the equipartition theorem breaks down for high frequency radiation. Max Planck explored the consequences of a novel hypothesis that the resonant modes could only store integer multiples of a fundamental energy quantum that was directly proportional to the frequency of the mode. That is, the energy of a mode could only take on values given by E nhνwith n 0, 1, 2... He realized that this would make it increasingly unlikely that a sufficiently high frequency mode would store any energy at all since the available amounts of random thermal energy, kt would be far less than that required to create even a single photon. To calculate the average energy in a mode, Planck used another result from classical thermodynamics that the probability of a system at thermal equilibrium storing an energy E was proportional to the so-called Boltsmann factor, e -E/kT Thus, the average energy stored in a resonant mode would be Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 4

5 Total energy density in the range and +d will be U d=n λ E av dλ = (8/ 4 )E av d - is the desired expression for the Planck s radiation law. - is the Stefan-Boltzman Law, where Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 5