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Standing Waves

Something more about…. Standing Waves. Wave Function. Differential Wave Equation. Standing Waves Boundary Conditions:. Separation of variables:. X=L. X=0. Wave Function:. Space: f(x). TIme: f(t). Equivalent to two ordinary (not partial) differential equations:. Space: X(x).

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Standing Waves

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  1. Something more about…. Standing Waves Wave Function Differential Wave Equation Standing Waves Boundary Conditions: Separation of variables: X=L X=0 Wave Function:

  2. Space: f(x) TIme: f(t) Equivalent to two ordinary (not partial) differential equations: Space: X(x) Time: T(t)

  3. EigenvalueCondition: n=0, ±1, ±2, ±3…… Eigenfunctions: Since any linear Combination of the Eigenfunctions would also be a solution General solution: Principle of superposition Fourier Series

  4. Fourier Series Any arbitrary function f(x) of period L can be expressed as a Fourier Series REAL Fourier Series COMPLEX Fourier Series

  5. qi = qr Wave Phenomena Reflexion Refraction InterferenceDiffraction Diffraction is the bending of a wave around an obstacle or through an opening. qi Wavelenght dependence n1 n2 qt q n1 sin (qi) = n2 sin (qt) w p=w sin(q)= ml bright fringes The path difference must be a multiple of a wavelength to insure constructive interference. q d ml p=d sin(q)= bright fringes

  6. Intensity pattern that shows the combined effects of both diffraction and interference when light passes through multiple slits. Interference and Diffraction: Huygens construction m=2 m=1 m=0

  7. Black-Body Radiation A blackbody is a hypothetical object that absorbs all incident electromagnetic radiation while maintaining thermal equilibrium. 

  8. Black-Body Radiation: classical theory Radiation as Electromagnetic Waves 1D 3D Since there are many more permissible high frequencies than low frequencies, and since by Statistical Thermodynamics all frequencies have the same average Energy, it follows that the Intensity I of balck-body radiation should rise continuously with increasing frequency. Breakdown of classical mechanical principles when applied to radiation !!!Ultraviolet Catastrophe!!!

  9. The Quantum of Energy – The Planck Distribution Law Physics is a closed subject in which new discoveries of any importance could scarcely expected…. However… He changed the World of Physics… Nature does not make a Jump Matter Discrete Energy Continuous Classical Mechanics Max Planck Energy Continuous Planck: Quanta E = hn h= 6.6262 x 10-34 Joule.sec An oscillator could adquire Energy only in discreteunits calledQuanta !Nomenclature change!: n →f

  10. Metal Photoelectric Effect: Einstein The radiation itself is quantized Fluxe Fluxe 1 2 n>no I n no • Below a certain „cutoff“ frequency no of incident light, no photoelectrons are ejected, no matter how intense the light. 1 • Above the „cutoff“ frequency the number of photoelectrons is directly proportional to the intensity of the light. 2 • As the frequency of the incident light is increased, the maximum velocity of the photoelectrons increases. • In cases where the radiation intensity is extremely low (but n>no) photoelectrons are emited from the metal without any time lag.

  11. Photon Energy of light: E = hn Kinetic Energy = Energy of light – Energy needed to escape surface (Work Function): ½ mev2= hn - hno Fo : It depends on the Nature of the Metal • Increasing theintensity of the light would correspond to increasing the number of photons. • Increasing the frequency of the light would correspond to increasing theEnergy of photons and the maximal velocity of the electrons.

  12. Light as a stream of Photons? E = hn discrete Zero rest mass!! Light as Electromagnetic Waves? E = eo |Eelec|2 = (1/mo) |Bmag|2continuous The square of the electromagnetic wave at some point can be taken as the Probability Density for finding a Photon in the volume element around that point. Energy having a definite and smoothly varying distribution. (Classical) A smoothly varying Probability Density for finding an atomistic packet of Energy. (Quantical) Albert Einstein

  13. The Wave Nature of Matter All material particles are associated with Waves („Matter waves“ E = hn E = mc2 mc2 = hn = hc/l or: mc = h/l De Broglie A normal particle with nonzero rest mass m travelling at velocity v mv = p= h/l

  14. Electron Diffraction Electron Diffraction Amorphous Material Crystalline Material Source Experimental Source Expected Conclusion: Under certain circunstances an electron behaves also as a Wave!

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