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Material Model 1 Basics in Quantum Mechanics 1 Basics in Quantum Mechanics • Dimension scaling 1: an atom has a typical dimension in the order of 1Å=10-10m, which means an atom to a baseball (~5cm) is like a baseball to the global earth (~15,000km)! • Dimension scaling 2: a nucleus has a typical dimension in the order of 10-5Å, which means a nucleus to the atom is like a rice grain (~1mm) to a huge building (~100m)! In this scheme, the electron is like a dust moving around the a rice grain with a distance in the range of the huge building. • Do we still believe that the motion of the electron must follow Newton’s laws? 2 Basics in Quantum Mechanics • Experimental evidence shown inconsistency in Newton’s theorem – – – – – 1. Black body radiation 2. Photon-electron effect 3. Atomic line spectrum 4. Atomic stability 5. Specific heat of solids • Mending works – 1. Plank’s quantized photon energy E=hv (1900) – 2. Einstein’s quantized momentum p=h/λ from p=E/c for photons (1905) – 3. Bohr’s quantized electron angular momentum J=nh/(2π), n=1,2,3,… (1913) – 4. de Broglie’s mater-wave duality 2πr=nλ, n=1,2,3,… (?) – 5. Einstein’s quantized phonon energy and momentum (1907) 3 Basics in Quantum Mechanics • Experimental confirmations – Compton’s photon-electron scattering experiment supported Plank-Einstein’s theorem (1923), so photons and phonons are particle like, or “real” waves (with zero static mass, reflecting the interaction between matters) can be particle like – Davison and Germer’s electron-Nickel crystal scattering experiment supported Bohr-de-Broglie’s theorem (1927), so electrons are wave like, or matters (with nonzero static mass) can be wave like 4 Basics in Quantum Mechanics A free particle in vacuum is described by: p2 E 2m Following de Broglie’s matter-wave duality, we have : E p k This free particle can be mapped to a plane traveling wave in vacuum with dispersion relation and traveling speed given as: d k p k 2 2m vg dk m m v The parabolic dispersion relation suggests that the plane traveling wave satisfy the following wave equation: (r , t ) 2 2 2 ( / j ) 2 j (r , t ) (r , t ) t 2m 2m 5 Basics in Quantum Mechanics In vacuum, the particle only bears the kinetic energy. In arbitrary space, however, energy conservation requires: (r , t ) 2 ( / j ) 2 j (r , t ) V (r )(r , t ) H(r , t ) t 2m This is SchrÖdinger’s equation (1926). 6 Basics in Quantum Mechanics • Interpretation of matter-wave – Wave packet? No, as otherwise the dispersion would make the particle (e.g., an electron) change its diameter, which is certainly absurd. – Statistical behavior of a large number of particles? No, as a single particle exhibit the wave behavior as well. – Born’s interpretation (1926): the intensity of the wave function at certain location in space is the probability of finding the particle at this location. • Therefore, the classical mechanics (Newton’s theory) predicts the trajectory of an object, whereas the quantum mechanics predicts the probability of an object’s emergence in space. 7 Basics in Quantum Mechanics • A classical view of a matter: (1) has mass, charge, etc., known as the particle property; they are the inherent property that defines the matter; (2) stay at a position, or move along a trajectory; they are not the inherent property of a matter! It was just introduced along with Newton’s theorem. • A quantum mechanics view of a matter: (1) the same; (2) stay at a position at certain time with certain probability. 8 Basics in Quantum Mechanics • Matter-wave properties: – Presentation (value quantity and operator) – Superposition and self-interference – Uncertainty principle 9