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Quantum Concepts and Mechanics What everyone should know Patrik Callis Department of Chemistry and Biochemistry Montana State University Presented to MSU Philosophy Group April 27, 2009 and Chmy373 January 13, 2010 Outline Snippets of wisdom and insights insights, gathered during 48 years of studying quantum mechanics and quantum chemsitry. 1. The attraction of nuclei and electrons is ENORMOUS 2. Why doesn’t the electron fall into the nucleus? 3. A simple way to look at the Schrodinger Equation, and how it predicts ZERO POINT energy (thus predicting the electron won’t won t fall) and TUNNELING. TUNNELING 4. “Understanding” quantum mechanics 5 An amazing g thing g about quantum interference in the double slit experiment. 6. The secret life of wavefunctions (they oscillate in time with a real and imaginary part part.)) A Working Theory of Chemistry, Biology, and Geology Matter: Nuclei (+) and electrons (-) Potential Energy: Energ Electromagnetic --> > Coulomb’s Co lomb’s Law La = constant * charge1 g * charge2 g /distance Mechanics: Quantum Atoms: electrons seeking a more positive environment Chemical Energy: electrons seeking a more positive environment by pulling nucleli together _ + + _ + The Coulomb Force is Enormously Strong an electron 0.1 Angstrom g from a proton p would “weigh” 2.3 milligrams i e a MACROSCOPIC FORCE!! i.e., The forces (and therefore the energies) are the SAME in quantum mechanics and classical mechanics Birth of Quantum Theory 1905 Planck Quantization of energy ΔE = hν in matter to emit light of freq. ν 1905 Einstein: Quantization of energy ΔE = hν for LIGHT to absorb or emit light of freq. ν and Particle nature of light: momentum = h/wavelength ‘ 1912 Rutherford’s scattering of alpha particles revealed the nature of atoms: dense nucleus surrounded by empty space with electrons. This revealed a new concept: Zero Point Energy, i.e. the electron for some reason will not fall into the nucleus. Other radical new quantum concepts: ~1920 DeBroglie: Wave nature of particles wavelength = h/momentum ~1925 Heisenberg: Uncertainty Principle Δmomentum x Δposition ~ h The lowest energy gy state of the Hydrogen atom. + proton electron 1 Angstrom The electron will not fall into the nucleus! Figure 3. Density slice through the 1s orbital The blue line is the square of the wavefunction. wavefunction 1925 Schrodinger’s Equation: A simple equation that was discovered (not derived) Classical Mechanics Kinetic Energy + Potential Energy = Total Energy Quantum Mechanics (Schrodinger’s Equation without time) translated into English: -h2/8pi2mass x Curvature of Wavefunction + Potential Energy x Wavefunction = Energy x Wavefunction curvature operation (2nd derivative h/2pi mass Kinetic energy wavefunction potential energy Total energy Time independent Schrodinger Equation : ⎛ ∂2 ∂2 ∂ 2 ⎟⎞ ⎜ - 2× + 2 Ψ + potential E × Ψ = total E × Ψ 2 2 ⎜ 8π ∂z j ⎟⎠ all particles j ⎝ ∂x j ∂y j or : kinetic energy operator × Ψ + classical potential energy × Ψ h2 ∑ = total energy × Ψ HΨ = EΨ , where H = Hamiltonian = total energy operator Ψ * Ψ = probablility density for finding particle locations Ψ * is i the h complex l conjugate j . i.e., i change h all ll i - - > - i i = -1 Potential energy EXACTLY same as in Classical mechanics Three things are different from Classical mechanics: 1) The wavefunction (Schrödinger did not know what its physical meaning was at the time he published). Later the consensus was reached that the absolute square of the wavefunction gives the probability density for finding the particle.) 2) Kinetic energy is represented by the CURVATURE of the Wavefunction. In calculus, that is the 2nd derivative (i.e., the slope of the slope of the function) 3) h, Planck Planck's s constant, which was empirically adjusted so that the Schrödinger Equation gives agreement with experiment. This simple equation embodies the 5 seemingly distinct new "quantum concepts" focus on zero-point energy: which hi h is i so powerful f l that th t it kkeeps th the electrons l t ((-)) ffrom falling f lli iinto t th the nucleus(+) of an atom despite the enormous attraction. (Classically, the electron would sit in the nucleus at ordinary temperatures). When a force confines a particle, this means that its wavefunction is localized, which in turn means its CURVATURE is large. B 0.5 Wavefunction Wavefunction W 10 1.0 0.0 1 . 0 0 . 5 0 . 0 -0.5 - 0 . 5 -1.0 - 1 . 0 0 0 10 20 30 40 50 1 0 2 0 3 X X Large curvature = HIGH kinetic E Small curvature = LOW kinetic E Note that when wavefunction positive, the curvature is negative and vice versa, so the kinetic E is always positive 0 4 0 This, coupled with small mass, means kinetic energy is very high for confined electrons. Chemical energy is virtually all the change of zeropoint energy of electrons during chemical reactions. The dark energy of the universe that is responsible for the accelerating expansion of the universe is now thought by many to be quantum zero point energy. Understanding Quantum Mechanics? Richard Ri h d Feynman F lecturing l t i to t a lay l audience di att Cornell, circa. 1965: “There There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time... After they read the paper, quite a lot l t off people l understood d t d the th th theory off relativity... l ti it On O the th other hand, I think it is safe to say that no one p saying y g to “understands” qquantum mechanics... Do not keep your self “But how can it be like that?”, because you will get “down the drain” into a blind alley from whihc nobody has yet escaped escaped. NOBODY KNOWS HOW IT CAN BE LIKE THAT. “ --Richard P. Feynman Chapter 6, The Character of Physical Law 23rd Printing, 1998 Feynman, from Lectures on Physics III : Quantum Mechanics exactly describes the g behavior electrons and light. “Electrons and light do not behave like anything we have ever seen seen.” “There is one lucky break, however—electrons behave just like light” TIME DEPENDENT Schrodinger Equation : HΨ = EΨ , where H = Hamiltonian = total energy operator ∂Ψ ∂t = −i h HΨ 2π Ψ is proportional to e −i 2πEt h All wavefunctions oscillate with frequency such that E = hυ Time dependent Schrodinger Eq. Wavefunction = Space part x e-i2π(E/h)t Square of wavefunction has NO time dependence for energy eigen states Actual experimental results of a double-slit-experiment performed by Dr. A. Tonomura showing the buildup of an interference pattern of single electrons. Numbers of electrons are 10 (a), 200 (b), 6000 (c) 40000 (d) (c), (d), 140000 (e) (e). (Provided with kind permission of Dr. AkiraTonomura.) http://www.youtube.com/watch?v=oxknfn97vFE Click on the link below to see calculations and movies of an electron exhibiting interference when passing through two slits. http://rugth30.phys.rug.nl/quantummechanics/diffint.htm