SWESolnHatomLects4&5ME501F2011

SWESolnHatomLects4&5ME501F2011 - Purdue University...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lectures 4 and 5: SWE Solution for the Hydrogen Atom Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana Lucht@purdue.edu , 765-494-5623 (Phone) August 29, 2011 and September 2, 2011
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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics Internal SWE for the hydrogen atom. Angular momentum quantization. Solution of the radial part of the SWE. Quantum states and energy levels of the hydrogen atom.
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Patterned Border Template 3 Purdue University School of Mechanical Engineering The Internal SWE for the Hydrogen Atom • The solution of the internal SWE for the hydrogen atom will give us the energy and angular momentum states of the system. The Coulombic potential energy for the electron-proton system is given by Vr e r i ()  2 0 4  • The internal SWE is solved in a center of mass coordinate system to account for electron-proton mass ratio.
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Patterned Border Template 4 Purdue University School of Mechanical Engineering The Internal SWE for the Hydrogen Atom • The internal SWE is given by where  222 2 0 2 iii ii i V xyz        12 prot elec elec prot elec mm m m m  • The relative coordinates are defined in terms of ( r, , ) as 21 sin cos sin sin cos xx x r yy yr zz z r rx y z   
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Patterned Border Template 5 Purdue University School of Mechanical Engineering The Internal SWE for the Hydrogen Atom • The SWE is transformed into spherical coordinates, • The SWE equation is again solved by separation of variables,  2 2 22 2 2 2 2 11 1 2 sin ( ) 0 sin sin ii i i rV r rr r r r           (, ) () ( ) ) i rR r SR r   • This expression is substituted into the SWE and the result is a set of three equations for R(r) , ( ) , and ( ). Without getting into the details of the solution, let us examine the solutions and their physical significance.
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Patterned Border Template 6 Purdue University School of Mechanical Engineering Solutions of the Angular Part of the Internal
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SWESolnHatomLects4&5ME501F2011 - Purdue University...

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