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lecture_18 - 97 The Bohr Theory Matter Waves and Quantum...

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97 The Bohr Theory, Matter Waves, and Quantum Theory At the beginning of the 20 th century, classical physics was thought to be in “good shape”. There were only a few problems that could not be explained by Newton’s Laws. Matter was described by Newton’s Laws, and light was described as a wave, in accordance with Maxwell’s equations. Light waves are characterized by a speed, a wavelength, and a frequency: This relation is given by ν = c/ λ . As the frequency increases, the wavelength decreases. The electromagnetic spectrum shows that visible radiation is in the wavelength range from 400 to 700 nm. Some of the problems that could not be understood with the classical theory included: Blackbody radiation – the distribution of wavelengths of light emitted by an object at a given temperature The photoelectric effect The line spectrum of hydrogen Ultraviolet Catastrophe” Blackbody radiation can be understood as follows: put a fireplace poker in the fireplace. First, it glows dull red, then orange, then yellow. Analyze the light with a prism. The results are shown in the figure. Note that as T increases, the maximum in the wavelength shifts to shorter values. Theory said that oscillators associated with the atoms in the lattice emitted light. Classically, the average value of the energy of one of these oscillators is . kT = ε The result is the Ultraviolet Catastrophe, indicated by the red line.
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98 Planck said that the energy of an oscillator was proportional to a fundamental “quantum”, with energy nh ν . He used the Maxwell-Boltzmann distribution to calculate the average value: Let P( ε ) = the distribution function Then, for Planck’s model, () × = ε energies energy energy of . prob 1 e h nh ) ( P kT h energies ν = ν × ε = ε ν This last expression has the right behavior (goes to zero) at short wavelengths (high frequencies), and also at long wavelengths. By data fitting, Planck was able to fit the data in the plot. He found that a value of h = 6.6 × 10 -34 J sec fit the data. Planck won the Nobel Prize in 1918 for this work. Photoelectric effect:
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