The Hydrogen Atom Michael Fowler 11/22/06 Factoring Out the Center of Mass Motion The hydrogen atom consists of two particles, the proton and the electron, interacting via the Coulomb potential , where as usual ()212/V rrer−=GG12rrr=−GG. Writing the masses of the two particles as Schrödinger’s equation for the atom is: 1,mm2()()2222212121212,,22er rEr rmmrψψ⎛⎞−∇ −∇ −=⎜⎟⎝⎠GG==.G GG GButare not the most natural position variables for describing this system: since the potential depends only on the relative position, a better choice is 12,rrGG,r RGGdefined by: 1 12 21212,m rm rrrrRmm+=−=+GGGGGGso RGis the center of mass of the system. It is convenient at the same time to denote the total mass by 12,Mmm=+and the reduced mass by 1212.m mmmm=+Transforming in straightforward fashion to the variables ,rRGGSchrödinger’s equation becomes ()()22222,,22Rre.R rER rMmrψψ⎛⎞−∇−∇ −=⎜⎟⎝⎠GGGG==GGWriting the wave function ()()(),R rRrψψ= ΨGGGGwe can split the equation into two: ()()()()()222222RRrrRERMVrrErmψψ⎛⎞−∇Ψ=Ψ⎜⎟⎝⎠⎛⎞−∇ +=⎜⎟⎝⎠GGG=G=GGG
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