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Unformatted text preview: P460  Helium 1 MultiElectron Atoms • Start with Helium: He + same as H but with Z=2 • He  2 electrons. No exact solution of S.E. but can use H wave functions and energy levels as starting point. We’ll use some aspects of perturbation theory but skip Ritz variational technique (which sets bounds on what the energy can be) • nucleus screened and so Z(effective) is < 2 • “screening” is ~same as ee repulsion (for He, we’ll look at ee repulsion. For higher Z, we’ll call it screening) • electrons are identical particles. Will therefor obey Pauli exclusion rule (can’t have the same quantum numbers). This turns out to be due to the symmetry of the total wave function P460  Helium 2 Schrod. Eq. For He • have kinetic energy term for both electrons (1+2) • V 12 is the ee interaction. Let it be 0 for the first approximation, that is for the base wavefunctions and then treat it as a (large) perturbation • for the unperturbed potential, the solutions are in the form of separate wavefunctions ) ( ) ( ) ( 2 1 12 2 1 2 2 2 2 1 2 2 2 r r V r V r V V E V T T T T T T m T m r r r r h h − + + = = + ∇ − ∇ − ψ ψ ψ ψ 2 1 2 1 2 1 ) ( ) ( ) , ( E E E r r r r T T + = = r r r r ψ ψ ψ P460  Helium 3 Apply symmetry to He • The total wave function must be antisymmetric • but have both space and spin components and so 2 choices: • have 2 spin 1/2 particles. The total S is 0 or 1 • S=1 is spinsymmetric S=0 is spinantisymmetric sym antisym OR antisym symmetric either spin space spin space spin space He = ⊕ = = ⊕ = ⇒ = ψ ψ ψ ψ ψ ψ ψ 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ) ( 1 1 1 ) ( 1 1 ± ↓ ↑ − ↑ ↓ − − − ↓ ↓ ± ↓ ↑ + ↑ ↓ ↑ ↑ m m s s s spin m m m S ψ P460  Helium 4 He spatial wave function • There are symmetric and antisymmetric spatial wavefunctions which go with the anti and sym spin functions. Note a,b are the spatial quantum numbers n,l,m but not spin • when the two electrons are close to each other, the antisymmetric state is suppressed (goes to 0 if exactly the same point). Likewise the symmetric state is enhanced • b “Exchange Force” S=1 spin state has the electrons (on average) further apart (as antisymmetric space). So smaller repulsive potential and so lower energy • note if a=b, same space state, must have S=1 (“prove” Pauli exclusion) )) 1 ( ) 2 ( ) 2 ( ) 1 ( ( ) ( )) 1 ( ) 2 ( ) 2 ( ) 1 ( ( ) ( 2 1 2 1 b a b a space b a b a space asym sym ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ − = + = P460  Helium 5 He Energy Levels • V terms in Schrod. Eq.: • Oth approximation. Ignore ee term. • 0.5th approximation: Guess ee term. Treat electrons as point objects with average radius (for both n=1) a /Z (Z=2)....
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 Spring '08
 Johnson,C
 Atom, Energy, Quantum Physics, Pauli exclusion principle

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