20160821105344atomic4.pdf - Atomic Nuclear Physics KYA 323...

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Atomic-4 1 KYA 323 Atomic & Nuclear Physics Dr Andrew Cole Rm 458 [email protected] KYA323 Atomic-4 2 Multielectron Atoms For each electron in a multi-electron atom, the Coulomb attraction of Z protons is partially screened by the repulsion due to to Z– 1 other electrons. Symmetry and spin-orbit interactions can dominate the structure of the atom and play a large role in determining the eigenfunctions and eigenvalues. The various interactions are typically of quite different strengths. To treat this system quantum mechanically we can tackle them one or two at a time in order of decreasing strength. Detailed quantitative results require massive computing power. KYA323
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Atomic-4 3 Identical Particles The finite extent of the electron wave functions means that wave functions may overlap. Where two or more identical particles overlap, they become indistinguishable. Quantum statistical physics is very different from classical statistical physics because no additional information can be added to the wave function describing an electron. E.g.: in classical statistical physics, I can paint one electron green and another one red, and tell them apart that way. KYA323 Atomic-4 4 Schrödinger Equation for non-interacting particles Retain the labels 1 and 2 just for notational purposes. We have a wave function ψ T that is some function of ψ 1 and ψ 2 . ψ 1 is evaluated at position 1, and ψ 2 at position 2. ψ T and V are functions of six variables ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ). 2 is the Laplacian operator applied first at ( x 1 , y 1 , z 1 ) and then at ( x 2 , y 2 , z 2 ). For identical particles m 1 = m 2 By separation of variables we find that for independent electrons KYA323 - ~ 2 m 1 r 2 1 + ~ 2 m 2 r 2 2 - V T = E T T ( x 1 , . . . , z 2 ) = 1 ( x 1 , y 1 , z 1 ) 2 ( x 2 , y 2 , z 2 )
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Atomic-4 5 Probability Densities for Identical Particles Adopt a shortened notation such that So if particle 1 is in state α and particle 2 is in state β then Particles 1 and 2 are indistinguishable, so there’s no reason not to put particle 2 in state α and particle 1 in state β : The probability densities shouldn’t change based on which notation we choose, i.e., we would like to observe: KYA323 nlm l m s ( x 1 , y 1 , z 1 ) (1) T ( x 1 , . . . , z 2 ) = (1) β (2) T ( x 1 , . . . , z 2 ) = β (1) (2) T T = (1) β (2) (1) β (2) = β (1) (2) β (1) (2) Atomic-4 6 However, the two expressions are not guaranteed to be equivalent if the two particles are moving independently. For example, So we have to conclude that the expressions for ψ T on the previous slide are not acceptable general eigenfunctions for a non-interacting, two identical-particle system. We can find eigenfunctions that work if we abandon the idea that the total eigenfunction ψ T must also be an eigenstate of the single-particle system. ψ T ψ α ψ !
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