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Phys 102 – Lecture 26 Phys 102 Lecture 26 The quantum numbers and spin 1 Recall: the Bohr model Only orbits that fit n e– wavelengths are allowed SUCCESSES Correct energy quantization & atomic spectra me k 2 e 4 1 En = − ⋅ 2 2 2 n + n = 1, 1 2,3,... 23 FAILURES Radius & momentum quantization violates d l Heisenberg Uncertainty Principle e– wave n2 2 2 rn = ≡ n a0 2 me ke Δr ⋅ Δpr ≥ 2 Electron orbits cannot have zero L Ln = n Orbits can hold any number of electrons Phys. 102, Lecture 26, Slide 2 Quantum Mechanical Atom Schrödinger’s equation determines e– “wavefunction” 2 ⎛ ke 2 ⎞ 2 ∇ − ⎜− ⎟ ψ (r , θ , φ) = Eψ (r , θ , φ) ⇒ r ⎠ ⎝ 2me “SHELL” SHELL ψn, ,m 3 quantum numbers determine e– state “Principal Quantum Number” n = 1, 2, 3, … me k 2 e4 1 En = − 2 2 n2 Energy L= Magnitude of angular momentum s, p, d, f “SUBSHELL” “Orbital Quantum Number” ℓ = 0, 1, 2, 3 …, n–1 ( + 1) “Magnetic Quantum Number” mℓ = –ℓ, …–1, 0, +1…, ℓ Lz = m Orientation of angular momentum Phys. 102, Lecture 26, Slide 3 ACT: CheckPoint 3.1 & more For which state is the angular momentum required to be 0? A. n = 3 B. n = 2 C. n = 1 How many values for mℓ are possible for the f subshell (ℓ = 3)? A. 3 B. 5 C. 7 Phys. 102, Lecture 26, Slide 4 Hydrogen electron orbitals ψn, 2 ,m Shell ∝ probability ( ℓ ℓ) (n, ℓ, m (2, 0, 0) (2, 1, 0) (2, 1, 1) (3, 0, 0) (3, 1, 0) (3, 1, 1) (3, 2, 0) (3, 2, 1) (3, 2, 2) (4 0 0) (4, 0, 0) (4 1 0) (4, 1, 0) (4 1 1) (4, 1, 1) (4 2 0) (4, 2, 0) (4 2 1) (4, 2, 1) (4 2 2) (4, 2, 2) (5, 0, 0) (4, 3, 0) (4, 3, 1) (4, 3, 2) (4, 3, 3) (15, 4, 0) Subshell Phys. 102, Lecture 26, Slide 5 CheckPoint 2: orbitals Orbitals represent probability of electron being at partic lar location being at particular location 1s (ℓ = 0) r 2p (ℓ = 1) 2p (ℓ = 1) r 2s (ℓ = 0) r 3s (ℓ = 0) r Phys. 102, Lecture 26, Slide 6 Angular momentum What do the quantum numbers ℓ and mℓ represent? Magnitude of angular momentum vector quantized + – Classical orbit picture ( + 1) = 0,1, 2 n −1 Only one component of L quantized Lz = m e– L Lz = + =1 L =L= r Lz = 0 Lz = − m =− , − 1, 0,1, Other components Lx, Ly are not quantized Lz = +2 L= 2 Lz = + =2 L= 6 Lz = 0 Lz = − Lz = −2 Phys. 102, Lecture 26, Slide 7 Orbital magnetic dipole Electron orbit is a current loop and a magnetic dipole μe μe = IA = − r + – L μe = − e– e L 2me e L 2me Recall Lect. 12 Dipole moment is quantized What happens in a B field? B θ z e– – + μe U = − μe B cos θ = e Bm 2me Recall Lect. 11 Recall Lect. 11 Orbitals with different L have different quantized energies in a B field r μB ≡ e eV “Bohr magneton” = 5.8 ×10−5 2me T Phys. 102, Lecture 26, Slide 8 ACT: Hydrogen atom dipole What is the magnetic dipole moment of hydrogen in its ground state due to the orbital motion of electrons? d t t d t th bit l ti f l t ? μe = − e L 2me e A. μH = − A 2me B. μH = 0 e C. μH = + 2me Phys. 102, Lecture 26, Slide 9 Calculation: Zeeman effect B Calculate the effect of a 1 T B field on the energy of the 2p (n = 2, ℓ the energy of the 2p (n = 2 ℓ = 1) level = 1) level μe m = −1 z + Etot = En = 2 − μe B cos θ e F ℓ = 1, 1 = En =2 + Bm For ℓ mℓ = –1, 0, +1 2me B=0 e– – Energy level splits into 3, with energy splitting B>0 m = +1 m =0 m = −1 e B ΔE ≡ 2me Phys. 102, Lecture 26, Slide 10 ACT: Atomic dipole The H α spectral line is due to e– transition between the n transition between the n = 3, ℓ = 3 ℓ = 2 =2 and the n = 2, ℓ = 1 subshells. How many spectral lines will appear in a B H t l li ill i B field? fi ld? E A. 1 B. 3 ℓ = 2 n = 3 C. 5 n = 2 ℓ = 1 Phys. 102, Lecture 26, Slide 11 Intrinsic angular momentum A beam of H atoms in ground state passes through a B field n = 1, so ℓ = 0 and expect NO effect from B field Atom with ℓ = 0 Instead, observe beam split in two! Since we expect 2ℓ + 1 values for magnetic dipole moment, e– must have intrinsic angular momentum ℓ = ½. “Spin” Stern Gerlach experiment “Stern‐Gerlach experiment” Phys. 102, Lecture 26, Slide 12 Spin angular momentum Electrons have an intrinsic angular momentum called “spin” S = S = s( s + 1) S z = ms with s = ½ ms = − 12 , + 12 Sz = + Sz = − 2 S= 3 2 2 Spin also generates magnetic dipole moment S e gS μs = − 2me B S Spin DOWN (– ½) B>0 Spin UP (+½) ms = + 12 ms = − 12 U = − μs B cos θ = with g ≈ 2 ge Bms 2me Phys. 102, Lecture 26, Slide 13 Magnetic resonance e– in B field absorbs photon with energy equal to splitting of energy levels of energy levels Absorption ms = + 12 ms = − hf = ΔE “Electron spin resonance” Typically microwave EM wave 1 2 Protons & neutrons also have spin ½ μ prot e g p S << μs =+ 2m p since m p >> me “Nuclear magnetic resonance” Phys. 102, Lecture 26, Slide 14 Quantum number summary “Principal Quantum Number”, n = 1, 2, 3, … me k 2 e 4 1 En = − 2 2 n2 Energy “Orbital Quantum Number”, ℓ Q , = 0, 1, 2, …, n–1 , , , , L= ( + 1) Magnitude of angular momentum “Magnetic Magnetic Quantum Number Quantum Number”, m ml = –ℓ, … –1, 0, +1 …, ℓ = ℓ 1 0 +1 ℓ Lz = m Orientation of angular momentum “Spin Quantum Number”, ms = –½, +½ S z = ms Orientation of spin Electronic states Pauli Exclusion Principle: no two e– can have the same set of quantum numbers quantum numbers E s (ℓ = 0) s (ℓ = 0) +½ –½ n = 2 – – p (ℓ = 1) p (ℓ = 1) +½ –½ – – mℓ = –1 +½ –½ – – mℓ = 0 +½ –½ – – mℓ = +1 = –½ ½ mS = +½ = +½ mS = n = 1 – – Phys. 102, Lecture 26, Slide 16 The Periodic Table Pauli exclusion & energies determine sequence s (ℓ ( = 0)) Also Also ss p (ℓ = 1) n = 1 2 d (ℓ = 2) 3 4 5 6 7 f (ℓ = 3) Phys. 102, Lecture 26, Slide 17 CheckPoint 3.2 How many electrons can there be in a 5g (n = 5, ℓ = 4) sub‐ shell of an atom? shell of an atom? Phys. 102, Lecture 26, Slide 18 ACT: Quantum numbers How many total electron states exist with n = 2? A. 2 B 4 B. C. 8 Phys. 102, Lecture 26, Slide 19 ACT: Magnetic elements Where would you expect the most magnetic the most magnetic elements to be in the periodic table? A. Alkali metals (s, ℓ = 1) B. Noble gases (p, ℓ = 2) C. Rare earth metals (f, ℓ = 4) Phys. 102, Lecture 26, Slide 20 Summary of today’s lecture • Quantum numbers E = −13.6 13 6 eV V n2 Principal quantum number P i i l t b Orbital quantum number L = ( + 1) , = 0,1, n − 1 0… Magnetic quantum number Lz = mz , m = − ,… , 0, Magnetic quantum number • Spin angular momentum e– has intrinsic angular momentum has intrinsic angular momentum S z = ms ms = − 12 , 12 • Magnetic properties Orbital & spin angular momentum generate magnetic Orbital & spin angular momentum generate magnetic dipole moment • Pauli Exclusion Principle No two e– can have the same quantum numbers Phys. 102, Lecture 25, Slide 21