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Unformatted text preview: 5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next-simplest system: the Helium atom. In this situation, we have tow electrons with coordinates R z r 1 and r 2 orbiting a nucleus with charge Z = 2 located at the point R . Now, for the hydrogen atom we were able to ignore the motion of the nucleus r 2 by transforming to the center of mass. We then obtained a Schrdinger equation for a single y effective particle with a reduced mass that was very close to the electron mass orbiting the x origin. It turns out to be fairly difficult to r 1 transform to the center of mass when dealing with three particles, as is the case for Helium. However, because the nucleus is much more massive than either of the two electrons (M Nuc 7000 m el ) it is a very good z approximation to assume that the nucleus sits at the center of mass of the atom. In this approximate set of COM coordinates, then, R =0 r 2 and the electron coordinates r 1 and r 2 measure y the between each electron and the nucleus. Further, we feel justified in separating the motion of the nucleus (which will roughly x r 1 correspond to rigidly translating the COM of the atom) from the relative d the electrons orbiting the nucleus within the COM frame. Thus, in what follows, we focus only on the motion of the electrons and ignore the motion of the nucleus. We will treat the quantum mechanics of multiple particles (1,2,3) in much the same way as we described multiple dimensions. We will invent operators r 1 , r 2 , r 3 , and associated momentum operators p 1 , p 2 , p 3 . The operators for a given particle (i) will be assumed to commute with all operators associated with any other particle (j): [ r 1 , p 2 ] = [ p 2 , r 3 ] = [ r 2 , r 3 ] = [ p 1 , p 3 ] = ... 0 5.61 Physical Chemistry 25 Helium Atom page 2 Meanwhile, operators belonging to the same particle will obey the normal commutation relations. Position and momentum along a given axis do not commute: x , p = i y , p = i z , p = i 1 x 1 1 y 1 1 z 1 while all components belonging to different axes commute: = p y p p = p . x y y x y = p p etc 1 1 1 1 z 1 1 1 z 1 z 1 x 1 x 1 z 1 As you can already see, one of the biggest challenges to treating multiple electrons is the explosion in the number of variables required! In terms of these operators, we can quickly write down the Hamiltonian for the Helium atom: Kinetic Energy Nucleus-Electron 1 Of Electron 1 Electron-Electron Attraction Repulsion p 2 p 2 Ze 2 1 Ze 2 1 e 2 1 H 1 + 2 + 2 m 2 m e 4 0 r 1 4 0 r 2 4 0 r r e 1 2 Kinetic Energy Nucleus-Electron 2 Of Electron 2 Attraction This Hamiltonian looks very intimidating, mainly because of all the constants ( e , m e , ,...
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This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.
- Spring '08