e L 2m e the Bohr magneton

Transcription

1 e L μl = L = μb 2m with : μ B e e 2m e the Bohr magneton

2 Classical interation of magnetic moment and B field: (Young and Freedman, Ch. 27) E = potential energy = μ i B = μbcosθ τ = torque = μ B, perpendicular to μ

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4 THE ELECTRON S FINAL PROPERTY: CHARGE = X C MASS = X kg + SPIN ANGULAR MOMENTUM: Quantum no. s = ½ s = s(s + 1) = 3/ 4 s =± / 2 μ SPIN MAGNETIC MOMENT: e s = s = 2 m with = = z μ B e = e 2m e μ B x10 J / T s the Bohr magneton e

5 Degeneracy in the hydrogenic atom States Degeneracy Degeneracy without spin with spin n = 1: 1s Not 0 2 n = 2: 2s 0, 2p +1,2p 0,2p -1 or 4 8 2s 0, 2p x,2p y,2p z n = 3: 3s 0, 3p x,3p y,3p z, 3d 3z2-r2,3d x2-y2,3d xy,3d yz,3d xz 9 18 n = general n 2 2n 2

6 A FINAL ATOMIC INTERACTION actually relativistic

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8 SPIN-ORBIT SPLITTING IN HYDROGEN 2p (SMALL!):

9 LINE SPECTRA OF DIFFERENT SOURCES: Atomic hydrogen Sodium Sodium- D-line emiss. Helium Neon Mercury Molecular hydrogen = H 2 Sodium- D-line absorp. The Sun: blackbody emission plus absorption WHY LINES?

10 Spin-orbit splitting in spectroscopy: the sodium doublet From Serway, Moses, and Moyer Modern Physics

11 All reading and lecture slides posted Solutions to problems in Chapters 7, 8, and 10 posted Final exam: Wed., 6/15, 4:00-6:00 PM, here Coverage: Entire course Approx. 60% on material since midterm, 40% on material through midterm Old final posted after finish of lectures Extra office hours and question/review sessions before final: times TBA

12 What properties do wave functions of overlapping (thus indistinguishable) particles have? electrons as example: ψ = ψ ( r 1,s 1;r 2,s 2 ), including spin of both electrons But labels can' t affect any measurable quantity. E.g. probability density : ψ(r,s ;r,s ) (r,s ;r,s ) = ψ Therefore ψ(r 1,s 1;r 2,s 2 ) =± 1 ψ(r 2,s 2;r 1,s1 ) P 12ψ (r 1,s 1;r 2,s 2 ) with P12 = permutation operator r 1,s 1;r 2,s2 and eigenvalues of ± 1 Finally, all particles in two classes : FERMIONS : ( incl. e ' s ) : ψ antisymmetric s =,,... P12ψ = 1ψ BOSONS : ( incl. photons ) : ψ symmetric s = 0,1,2,... P ψ =+ 1ψ 12 Probability of finding two electrons at the same point in space with the same spin is zero: the Fermi Hole e 2- e 1- e 3- the Exchange Interaction Hund s 1 st rule & magnetism

13 Antisymmetry and the Pauli Exclusion Principle: Try Helium, 2 electrons in ground state 1s wave functions, 1s 2" Simple normalized antisymmetric trial wave function is 1 ψ (r 1,s 1;r 2,s 2) = ϕ1s(r 1,s 1 = ) ϕ1s(r 2,s 2 = ) ϕ1s(r 1,s 1 = ) ϕ1s(r 2,s 2 = ) 2 int erchanging labels gives 1 ψ (r 2,s 2;r 1,s 1 ) = ϕ1s(r 2,s 2 = ) ϕ1s(r 1,s 1 = ) ϕ1s(r 2,s 2 = ) ϕ1s(r 1,s 1 = ) 2 = ψ(r,s ;r,s ), as required Can t tell which electron is spin up--indistinguishable Also, if we try to put both electrons in 1s with spin-up ( ), first term always cancels second term, and ψ = 0! Therefore, we have the Pauli Exclusion Principle

14 Thus, in a many-electron system: Anti-symmetry of total wave function implies: Pauli Exclusion Principle: No two electrons can have all the same quantum nos. n,, m, m s or, if spin-orbit split n,, j, m j Electronic structure determined by filling n, (or n,, j ) levels from lowest to highest energy (E n from radial Schroedinger Eqn. with Z eff ) Partially filled subshells n, (or n,, j) have their lowest energy when a maximum no. of electrons have parallel spins = highest total spin angular momentum = S (Hund s First Rule), and then they couple to yield highest total orbital angular momentum = L (Hund s Second Rule)

15 The Hydrogenic Atom Schroedinger Equation: Spherical Polar Coordinates Classically: e - r p Quantum mechanically: π φ sinθdθ 0 0 π 0 2π dφ = cos θ 2π = 4π +Ze L = r p is conserved V(r) = -Ze 2 /4πε 0 r ˆK Polar angle = θ = arc cos(z/r) Azimuthal angle = φ = arc tan(y/x) Converting to new coordinates Ĥ ˆK

16 Intraatomic electron screening in many-electron atoms--a simple model In many-electron atoms: For a given n, s feels nuclear charge more than p, more than d, more than f Lifts degeneracy on in hydrogenic atom k C 1/(4πε 0 ) [Z eff ]

17 Filling the Atomic Orbitals: 1s ± ± Maximum Occupation FillingTotal = Degeneracy degeneracy No e s 2p x s 2 2 3p x d x2 x for nf

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20 SOME PERIODIC PROPERTIES: (plus see

21 From with much more:

22 Hund s First Rule: highest total spin angular momentum

23 The quantum mechanics of covalent bonding in molecules: H 2+ with one electron ϕ - = ϕ antibonding ϕ 1sa - ϕ 1sb ϕ + = ϕ bonding ϕ 1sa + ϕ 1sb Total Energy =R

24 H a H b ε positive (unoccupied) ε a.u. = ev Anti-Bonding ϕ anti MO ϕ 1sa - ϕ 1sb ε negative (occupied) Bonding ϕ bonding MO ϕ 1sa + ϕ 1sb ε a.u. = ev (Compare for H atom 1s)

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