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Unformatted text preview: chem 260/261 F11 Monday Septemeber 19, 2011 Lecture 6 1 Lecture 6 September 19, 2011 Chem 260/261 Today in Chemistry 260/261 •Harmonic oscillator •Describing the hydrogen atom with wave functions • quantum numbers for atomic systems • Hydrogen atom orbitals and energies Next in Chemistry 260/261 • Reading: ¡'SJ¢ £¤¥¦£¤§¨ ¡.PO¢ £¤£ • Solving the Schrödinger equation for more than one electron • Hartree orbitals • Aufbau Principle w NBOZ¦FMFDUSPO BUPNT¨ QFSJPEJD QSPQFSUJFT This simple potential model is widely used (after all we do not have many systems that can be solved exactly). unfortunately) V ( R AB ) = 1 2 k R AB ! R e ( ) 2 F = ma = -k(R-R e ) R e m 1 m 2 μ m 1 m 2 m 1 m 2 + F = ! k R AB ! R e ( ) Stiffness of the spring: k (spring constant). 1635-1703 The parabolic potential: Hooke ` s Law ! ! 2 2 μ d 2 dx 2 + ˆ V ( x ) " # $ % & ' ! ( x ) = E ! ( x ) ) ( ) ( 2 1 2 2 2 2 2 x E x kx dx d ψ ψ μ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − x = R ! R e The energy levels will be quantized because the wave function is bounded… ∫ ∞ ∞ − = 1 ) ( 2 dx x ψ Reduced mass E = n 2 h 2 8 mL ! " # $ % & 1 4 9 16 25 n=1 n=2 n=3 n=4 n=5 E = h ! n + 1 2 ! " # $ % & n=1 n=2 n=3 n=4 n=0 Harmonic Oscillator Wave functions Particle in a Box ! n x ( ) = ¡ L sin N " x L ! " # $ % & Ψ n (x) is ! a more complex formula, but behavior is similar. Energy x L 0.5 1.5 3.5 5.5 7.5 Energy x Harmonic Oscillator In both cases the lowest energy state has kinetic energy >0. This is a quantum phenomenon. No kinetic energy would mean Δx=0 and Δp=0 ç This cannot be! n=1,2,3… n=0,1,2,… ¡ = 1 2 ¢ k μ 1.6 Not\initeunlessE=hv(n+½) TheboundaryconditiondeterminesE http://phet.colorado.edu/ For the Harmonic Oscillator Simulator: http://phet.colorado.edu/en/simulation/bound-states Phase oscillation – like the oscillation of the string, but, as for a standing wave, the nodes are stationary for the energy eigenstates. n=0 P=0.843 P=0.157 Another Example of Quantum Weirdness … Tunneling E = h ! n + 1 2 ! " # $ % & 0.5 1.5 3.5 5.5 7.5 n=1 n=2 n=3 n=4 n=0 Energy x ! ( x ) 2 The probability distribution goes beyond the classical limit. In this region potential energy > total energy! (if the oscillator is classical) 16% of the time the oscillator would be measured in the l forbidden zone z Tunneling is responsible for a number of strange phenomena – but it is also useful… chem 260/261 F11 Monday Septemeber 19, 2011 Lecture 6 2 Nobel Prize 1986 Gerd Binning Heinrich Rohrer Build your own – see www.geocities.com/spm_stm/Project.html Scanning Tunneling Microsope http://eels.kuicr.kyoto-u.ac.jp/stm.en.html Perylene on graphite IBM Research: Fe on Cu surface Molecules on surfaces 48 iron atoms in a circular ring "corral" some surface state electrons and force them into "quantum" states of the circular structure. The ripples in the ring of atoms are the density distribution of a particular set of quantum states of the corral. The artists were delightedquantum states of the corral....
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