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 mk24 e 1 E = −⋅e n =1231,2,3,... n 2 22n + FAILURES Radius & momentum quantization violates Heisenberg Uncertainty Principle n22 2 Δr ⋅Δp ≥ rnan = 2 ≡ 0 r mkee 2 – e wave Electron orbits cannot have zero L
Lnn = Orbits can hold any number of electrons Phys. 102, Lecture 26, Slide 2 Quantum Mechanical Atom
Schrödinger’s equation determines e– “wavefunction”
⎛⎞ 22ke 3 quantum numbers −∇−2 ψ(,r θφ , ) =Eψ (,r θφ , ) ⇒ ψ determine e– state ⎜⎟ nm,, ⎝⎠2mre “SHELL” “Principal Quantum Number” n = 1, 2, 3, … mk24 e 1 E =− e Energy n 2 22n s, p, d, f “SUBSHELL” “Orbital Quantum Number” ℓ = 0, 1, 2, 3 …, n–1
L =+ (1) Magnitude of angular momentum
“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
2 ψ nm,, ∝ probability
Shell (n, ℓ, mℓ) (2, 0, 0) (2, 1, 0) (2, 1, 1) Subshell
(3, 0, 0) (3, 1, 0) (3, 1, 1) (3, 2, 0) (3, 2, 1) (3, 2, 2)
(4, 0, 0) (4, 1, 0) (4, 1, 1) (4, 2, 0) (4, 2, 1) (4, 2, 2)
(5, 0, 0) (4, 3, 0) (4, 3, 1) (4, 3, 2) (4, 3, 3) (15, 4, 0)
Phys. 102, Lecture 26, Slide 5 CheckPoint 2: orbitals
Orbitals represent probability of electron 1s (ℓ = 0) being at particular location r
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 + r L ==L (1) + = 0,1,2n − 1 Only one component of L quantized – – e Lz = m m =− ,1,0,1, − Classical orbit picture Other components L , L are not quantized L x y Lz = +2 L =+ L = 2 L = 6 z Lz = + L = 0 =1 z = 2 Lz = 0 L =− z Lz = −
Phys. 102, Lecture 26, Slide 7 Lz = −2 Orbital magnetic dipole
Electron orbit is a current loop and a magnetic dipole μ e e μe = IA =− L Recall Lect. 12 2me + r e μe =− L Dipole moment is quantized – 2m e– e L What happens in a B field? e B μ U = −μeB cosθ = Bm Recall Lect. 11 e 2m z θ e Orbitals with different L have different quantized energies in a B field – + e e – r −5 eV μB ≡ =×5.8 10 “Bohr magneton” 2me T Phys. 102, Lecture 26, Slide 8 ACT: Hydrogen atom dipole
What is the magnetic dipole moment of hydrogen in its ground stttate due to the orbita l motion of eltlectrons ?