2 electrons; the helium atom and its peculiarities More than one electron: ÆPauli principle Wolfgang Pauli Nobel 1945 Non-central forces due to electron repulsion MNW-L4 The Helium atom; classical approximation a Without repulsion: a/2 a/2 2 E = 2E1 = −2Z E0 = −8⋅()13.6 eV = −108.8eV With repulsion; electrons at fixed distance a0, Repulsive energy: e2 = 2E0 = 27.2eV 4πε0a0 Calculated: Etot = −81.6eV Experimental: Etot = −79eV MNW-L4 The Helium atom: quantum mechanics 2 2 2 2 h 2 2 2e 2e e H = − (∇1 + ∇2 )− − + 2me 4πε0r1 4πε0r2 4πε0r12 +2e Ekin attraction repulsion r r r r Two-particle Schrödinger equation: HΨ(r1,r2 ) = EΨ(r1,r2 ) (0) First approximation: unperturbed system (no repulsion) H = H1 + H2 2 2 Each: h 2 2e Hi = − ∇i − 2me 4πε0ri r r r r Product wave function: Ψ(r1,r2 ) =ψ nlm (r1)ψ nlm (r2 ) =ψ1ψ 2 (0) r r H Ψ(r1,r2 )(= H1 + H2 )ψ1ψ 2 = (H1ψ1)()ψ 2 +ψ1 H2ψ 2 = E1ψ1ψ 2 + E2ψ1ψ 2 = Eψ1ψ 2 MNW-L4 The Helium atom: quantum mechanics 2 2 2 2 h 2 2 2e 2e e H = − (∇1 + ∇2 )− − + 2me 4πε0r1 4πε0r2 4πε0r12 +2e Ekin attraction repulsion 2 2 2e2 Z 4 Each:H = − h ∇2 − has solution E = −R = −R i 2m i 4πε r i 2 2 e 0 i ni ni ⎛ 1 1 ⎞ Hence: E = −8R⎜ + ⎟ ⎜ 2 2 ⎟ ⎝ n1 n2 ⎠ MNW-L4 Intermezzo Æ electron-electron repulsion as a perturbation 2 ()1 e First order perturbation E = n1l1m1;n2l 2m2 n1l1m1;n2l 2m2 = J 4πε0r12 5 ⎛ Ze2 ⎞ This is a mathematical problem – with a solution: J = ⎜ ⎟ ⎜ ⎟ 4 ⎝ 4πε0a0 ⎠ So: ⎛ 5 ⎞ Etot = −2Eb + J = ⎜− 8 + ⎟R ⎝ 2 ⎠ Ecalc= -5.5 R ; Eexp = -5.8 R MNW-L4 The orbital angular momentum of an electron in orbit r e r Semiclassical approach Hence μ = − L 2me μr So a magnetic moment -e is associated with +Ze an angular momentum 2πν r L From classical electrodynamics: Angular velocity ω = 2πν What happens to a magnetic dipole in a B-field ? Magnetic moment It rotates, and alines in the field μ = IA = πr2eν Angular momentum Magnetic interaction: 2 2 L = mvr = mωr = 2πmvr r r VM = −μ ⋅ B = −μzBz Zeeman effect MNW-L4 Space quantisation From classical electrodynamics: Origin of the spin-concept -Stern-Gerlach experiment; -A magnetic dipole moment μ undergoes space quantization Otto Stern no force in a magnetic field B Nobel 1943 -A magnetic dipole undergoes a torque, is oriented in a B field -A magnetic dipole μ undergoes a force in an inhomogeneous B-field, i.e. a field gradient (Stern-Gerlach experiment) -Theory: the periodic system requires an additional two-valuedness MNW-L4 Electron spin No classical analogue for this phenomenon 1 s = 2 h 1 s = 2 h Spin is an angular momentum, so it should satisfy Pauli: 2 2 There is an additional “two-valuedness” S s,ms = h s()s +1 s,ms in the spectra of atoms, behaving like an angular momentum Sz s,ms = hms s,ms 1 1 s = ,m = ± Goudsmit and Uhlenbeck 2 s 2 This may be interpreted/represented as an angular momentum Note: the spin of the electron cannot be explained from a classically “spinning” electronic charge Problem to be solved MNW-L4 The Helium atom and the Pauli principle 1) 2 electrons cannot occupy the same orbital (quantum state) 2) The 2 electrons must have different quantum numbers: α and β: 3) He must have an anti-symmetric wave function (under interchange of the 2 electrons) 1 Ψ()2,1 = []ψα ()2 ψ β ()1 −ψ β ()2 ψα ()1 = −Ψ()1,2 2 If α = β then Ψ = 0 Two possibilities for having an anti-symmetric wave function: A S A Ψ ()1,2 =ψ (r1,r2 )χ ()1,2 spatial ψ and spin χ A A S Ψ ()1,2 =ψ (r1,r2 )χ ()1,2 MNW-L4 EXTRA Addition of spins in a 2-electron system r r r M = m + m S = s1 + s2 S s1 s2 ; S= 0, 1 MS = -1, 0, 1 S =1, M S = 1 = ↑,↑ 1 A triplet of symmetric S =1, M S = 0 = ( ↑,↓ + ↓,↑ ) 2 spin wave functions S =1, M S = −1 = ↓,↓ 1 A singlet of an anti-symmetric S = 0, M S = 0 = ( ↑,↓ − ↓,↑ ) 2 spin wave function Two distinct families of quantum states in Helium: A S Ortho-helium: triplet states ψ χ S A Para-helium: singlet states ψ χ MNW-L4 Energy levels and Spectral lines in Helium “Singly-excited” states in Helium, with one electron in (1s) He (1s)(nl) configurations Selection rules: similar as in H ψ S χ A ψ Aχ S Δn = free Δl = ±1 Δm = −1,0,+1 ΔS = 0 (1s)(2s) levels are metastable MNW-L4 EXTRA Ordering of singlet and triplet states: qualitative Symmetric and Anti-symmetric spatial wave functions r r 1 r r r r Ψ± ()r1,r2 = []ψ n l m ()r1 ψ n l m ()r2 −ψ n l m ()r1 ψ n l m ()r2 2 1 1 1 2 2 2 2 2 2 1 1 1 r r r r 2 If: r1 ≈ r2 density becomes small for Ψ− ()r1,r2 Fermi-hole r r 2 While density of Ψ+ ()r1,r2 increases Effect of “exchange”; a “force” related to the Pauli principle Æ quantum interference MNW-L4 EXTRA Ordering of singlet and triplet states: qualitative Fermi-hole in case of an anti-symmetric spatial wave function r r 2 Ψ− ()r1,r2 In this state the electrons tend to be distanced from each other Æ less repulsion Æ more binding For the triplet states (or symmetric spin states) Conversely, in case of symmetric spatial function r r 2 Ψ+ ()r1,r2 Electrons tend to be close to each other Æ more repulsion Æ less binding energy ÆTriplet states are lower in energy than corresponding singlets MNW-L4.