1. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Standing Waves Wave Function Differential Wave Equation Something more about…. X=0 X=L Standing Waves Boundary Conditions: Standing Waves Boundary Conditions: Separation of variables: Wave Function:

2. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Equivalent to two ordinary (not partial) differential equations: Space: f(x) TIme: f(t) Space: X(x) Time: T(t)

3. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Eigenvalue Condition: Eigenfunctions: n=0, ±1, ±2, ±3…… General solution: Principle of superposition Since any linear Combination of the Eigenfunctions would also be a solution Fourier Series

4. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Fourier Series Any arbitrary function f(x) of period  can be expressed as a Fourier Series REAL Fourier Series REAL COMPLEX COMPLEX

5. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Wave Phenomena  i  rReflexionReflexionRefractionRefraction ii  t n 1 n 2 n 1 sin (  i ) = n 2 sin (  t ) Interference Diffraction Diffraction is the bending of a wave around an obstacle or through an opening.  p=w sin  w  p=d sin  d bright fringes m m m m The path difference must be a multiple of a wavelength to insure constructive interference. Wavelenght dependence

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

7. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Black-Body Radiation A blackbody is a hypothetical object that absorbs all incident electromagnetic radiation while maintaining thermal equilibrium.

8. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves 1D1D 3D3D 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!!! Black-Body Radiation: classical theory Radiation as Electromagnetic Waves

9. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves 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… Classical Mechanics Matter Discrete Energy Continuous Nature does not make a Jump Planck: Quanta E = h Energy Continuous h  x 10 -34 Joule.sec An oscillator could adquire Energy only in discrete units called Quanta !Nomenclature change!:  →  f Max Planck

10. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Photoelectric Effect: Einstein Metal Below a certain „cutoff“ frequency  of incident light, no photoelectrons are ejected, no matter how intense the light. Above the „cutoff“ frequency the number of photoelectrons is directly proportional to the intensity of the light. 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    photoelectrons are emited from the metal without any time lag. The radiation itself is quantized Flux e  Flux e   12 1 2

11. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Increasing the intensity of the light would correspond to increasing the number of photons. Energy of light: E = h Kinetic Energy = Energy of light – Energy needed to escape surface (Work Function): ½ m e v 2 = h  h  Kinetic Energy = Energy of light – Energy needed to escape surface (Work Function): ½ m e v 2 = h  h  PhotonPhoton Increasing the frequency of the light would correspond to increasing the Energy of photons and the maximal velocity of the electrons.   : It depends on the Nature of the Metal   : It depends on the Nature of the Metal

12. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Light as a stream of Photons? E = h  discrete Light as a stream of Photons? E = h  discrete Light as Electromagnetic Waves? E =   |E elec | 2 = (1/    |B mag | 2 continuous Light as Electromagnetic Waves? E =   |E elec | 2 = (1/    |B mag | 2 continuous Probability Density 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 Zero rest mass!!

13. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves The Wave Nature of Matter De Broglie All material particles are associated with Waves („Matter waves“ E = h E = mc 2 E = h E = mc 2 mc 2 = h  = hc  or: mc = h/ mc 2 = h  = hc  or: mc = h/ A normal particle with nonzero rest mass m travelling at velocity v mv = p  = h 

14. PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Source Electron Diffraction Experimental Expected Electron Diffraction Amorphous Material Crystalline Material Conclusion: Under certain circunstances an electron behaves also as a Wave!