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Lecture%2011 - EEE 352: Lecture 11 The Atom and its Energy...

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1 EEE 352: Lecture 11 The Atom and its Energy Levels * Describe atomic structure [web] [web] * The Schrödinger equation for the hydrogen atom Spherical coordinate representation * Quantum numbers * Real atomic structure and the periodic table * Molecules and bonding 25 nm gate length FET Describing Atomic Structure One of the triumphs of quantum mechanics is its ability to explain ATOMIC STRUCTURE * This CANNOT be accounted for using the principle of classical physics * HYDROGEN has the simplest atom and consists of a single ELECTRON That circulates around a central NUCLEUS r e r dr e dr r F V r V Nucleus the From r ce tan Dis a at Electron of Energy Potential r e r F oton Pr Electron the Between Force Coulomb o rr o o πε 4 4 0 ) ( ) ( ) ( : 4 ) ( : & 2 2 2 2 2 = + = = = ∫∫ ∞∞ + e - e r THE HYDROGEN ATOM an attractive force! Describing Atomic Structure An earlier treatment of the shell model of the atom was given by Bohr in 1912 (this has become known as the Bohr-Sommerfeld model). Bohr received the Nobel prize for his model. ¾ Electrons were located in shells, whose radius was determined by classical quantization (just like oscillations on a string) ¾ There was no radiation decay… ¾ Transitions between shells agreed with optical properties of atoms for some series of lines, but not for others ( a problem remained ). ¾ It remained for quantum mechanics, as described by both Heisenberg and Schrödinger, to fully explain the properties. Niels Hendrik David Bohr Nobel Prize in physics, 1922
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2 The Schrödinger Equation for the Hydrogen Atom We solve the Schrödinger equation for the ELECTRON in the presence of the electric field of the PROTON * To do this we solve the time INDEPENDENT Schrödinger equation using a spherical coordinate representation ψ πε E r e m E V m o = = + 4 2 2 2 2 2 2 2 h h TIME INDEPENDENT SCHRÖDINGER EQUATION FOR THE HYDROGEN ATOM y z x r θ φ POLAR COORDINATE SYSTEM cos sin sin cos sin r z r y r x = = = The Schrödinger Equation for the Hydrogen Atom ' In one dimension, we had one quantum number , n ' This quantum number related to the energy . ' Hence, in three dimensions, we expect three quantum numbers! ' In spherical coordinates, one of these quantum numbers will be associated with the energy n . This is the radial number. ' The other two will be associated with the angular variations: ' We connect θ with the quantum number l. ' We connect φ with the quantum number m z . We use m z to distinguish from the mass m The Schrödinger Equation for the Hydrogen Atom The radial quantum number n gives the energy E n : 2 2 2 2 2 0 2 4 2 2 1 32 1 B n ma n me n E h h = = ε π n = 1, 2, 3, … Bohr radius 2 0 2 4 h me a B = The polar angle ( θ ) solutions are Legendre polynomials P l (cos θ ) l = 0, 1, 2, …, n 1 The polar angle ( φ ) solutions are simple exponentials m z = 0, ± 1, ± 2, ± 3, ± l z im e Quantum States ± Now, we have shown the Schrödinger equation for the HYDROGEN ATOM . The solutions of this are characterized by QUANTUM NUMBERS PRINCIPAL
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This note was uploaded on 09/19/2009 for the course EEE 352 taught by Professor Ferry during the Spring '08 term at ASU.

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Lecture%2011 - EEE 352: Lecture 11 The Atom and its Energy...

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