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Quantum Mechanics

Quantum Mechanics. Electrons behave as waves (interference etc) and also particles (fixed mass, charge, number). Lessons on http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon. or http://www.colorado.edu/physics/2000/quantumzone /. Science at the end of ~1900: Classical Mechanics.

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Quantum Mechanics

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  1. Quantum Mechanics

  2. Electrons behave as waves (interference etc)and also particles (fixed mass, charge, number) Lessons on http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon or http://www.colorado.edu/physics/2000/quantumzone/

  3. Science at the end of ~1900: Classical Mechanics

  4. Leading to Mech. Engg., Civil Engg., Chem. Engg.

  5. Science at the end of ~1900: Electromagnetics Rainbows Polaroids Lightning Northern Lights Telescope Laser Optics

  6. Science at the end of ~1900: Electromagnetics Electronic Gadgets

  7. Science at the end of ~1900: Electromagnetics Ion Channels Chemical Reactions Neural Impulses Biological Processes Chemistry and Biology

  8. But there were puzzles !!! Dalton (1808) What does an atom look like ???

  9. Solar system model of atom Continuous radiation from orbiting electron Spectrum of Helium mv2/r = Zq2/4pe0r2 Centripetal force Electrostatic force Pb1: Atom would be unstable! (expect nanoseconds observe billion years!) Pb2: Spectra of atoms are discrete! Transitions E0(1/n2 – 1/m2) (n,m: integers)

  10. Bohr’s suggestion From 2 equations, rn = (n2/Z) a0 a0 = h2e0/pq2m = 0.529 Å (Bohr radius) Only certain modes allowed (like a plucked string) nl = 2pr (fit waves on circle) Momentum ~ 1/wavelength (DeBroglie) p = mv = h/l (massive classical particles  vanishing l) This means angular momentum is quantized mvr = nh/2p = nħ

  11. Bohr’s suggestion E = mv2/2 – Zq2/4pe0r Using previous two equations En = (Z2/n2)E0 E0 = -mq4/8ħ2e0 = -13.6 eV = 1 Rydberg Transitions E0(1/n2 – 1/m2) (n,m: integers) Explains discrete atomic spectra So need a suitable Wave equation so that imposing boundary conditions will yield the correct quantized solutions

  12. What should our wave equation look like? ∂2y/∂t2 = v2(∂2y/∂x2) y w x k String Solution: y(x,t) = y0ei(kx-wt) w2 = v2k2 What is the dispersion (w-k) for a particle?

  13. What should our wave equation look like? ∂2y/∂t2 = v2(∂2y/∂x2) w k Quantum theory: E=hf = ħw(Planck’s Law) p = h/l = ħk (de Broglie Law) and E = p2/2m + U (energy of a particle) X X Thus, dispersion we are looking for is w k2 + U So we need one time-derivative and two spatial derivatives

  14. Wave equation (Schrodinger) iħ∂Y/∂t = (-ħ22/2m + U)Y w k Kinetic Potential Energy Energy Makes sense in context of waves Eg. free particle U=0 Solution Y = Aei(kx-wt) = Aei(px-Et)/ħ We then get E= p2/2m= ħ2k2/2m

  15. For all time-independent problems Oscillating solution in time iħ∂Y/∂t = (-ħ22/2m + U)Y = ĤY Separation of variables for static potentials Y(x,t) = f(x)e-iEt/ħ Ĥf =Ef, Ĥ = -ħ22/2m + U BCs : Ĥfn = Enfn (n = 1,2,3...) En : eigenvalues (usually fixed by BCs) fn(x): eigenvectors/stationary states

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