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Unformatted text preview: MSE 803: N ANOMATERIALS AND N ANOTECHNOLOGY Professor Xudong Wang 213 MSE Building Spring 2010 Class 4 Quantum Mechanics: Review of Fundamentals Why Quantum? To understand the “ nano ” world: understanding the behavior of photons and electrons is essential ! Electrons and photons both exhibit “duality” can behave like particles and waves Quantum mechanics resolves the wave particle duality nature of “ nano ”scale materials h E Background to Nanomaterials Quantum Mechanics: describes the interactions of electrons with matter presence of electron only manifested by its “actions” TV, electrical current Electron microscope, STM, etc In each of these observations the electron behaves as a “particle” and as a “wave”; respectively Thus the “duality” of behavior: particlewave like Similarly light, normally described as a “wave” exhibits “particle” like behavior – photoelectric effect (Hertz 1887, Einstein 1905) Based on concept of a “photon”  (Planck – 1901) Wave is a “disturbance” that is periodic in position and time In contrast a vibration is periodic in position or time Wave Equation (WE): t z U t v t z U z , 1 , 2 2 2 2 2 Where o is the phase velocity of the wave. kz t i a z t U exp , Wave Properties Thus: u k z u iku z u 2 2 2 , u t u u i t u 2 2 2 , u u k 2 2 2 v Gives: k v phase velocity Differentiating wrt. z and t , and substituting back into WE: Thus o the phase velocity of the wave is: v 2 2 2 v k k Gives: 2 k Wave Properties Note for a wave: ω= 2πν where ν is the frequency of the wave Thus substituting and defining the wave number as: Thus, the phase velocity of a wave is number of waves passing a point in one second, multiplied by the wavelength of the wave Wave theory successfully described most of properties of light: Except when interacting with metalsphotoelectric effect. In 1905, Einstein based on Plank’s famous hypothesis (1901) that light was composed of photons explained the PE effect. In 1924 Louis de Broglie proposed that because photons have both wave and particle characteristics, all forms of matter have wave as well as particle properties Schrodinger (1926) gave the idea of Wave Properties of Matter mathematical rigor G.P Thomson (1928) discovered electron diffraction h p Wave Properties of Particles Wave Property of Particles Describe properties of electrons by a harmonic wave: by a Wave Function: Ψ = A sin(kxωt) = A expi(wt  kz) Contains time and space components, but has no physical significance The superposition of these two waves yields a new wave: Y Y 1 + Y 2 = A sin(kx t) + A sin[(k +...
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This note was uploaded on 03/23/2010 for the course MATERIAL S 803 taught by Professor Samuel during the Spring '10 term at Wisconsin.
 Spring '10
 Samuel

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