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SUCCESSESCorrect energy quantization & atomic spectra
FAILURESRadius & momentum quantization violatesHeisenberg Uncertainty Principle
Electron orbits cannot have zero L
Orbits can hold any number of electrons
2 4
2 2
1
2e
nm k e
En
1,2,3,...n
Recall: the Bohr modelOnly orbits that fit n e– wavelengths are allowed
2 22
02ne
nr n a
m ke
2rr p
nL n
e– wave
Phys. 102, Lecture 26, Slide 2
2 22 ( , , ) ( , , )
2 e
keψ r θ φ Eψ r θ φ
m r
, , n mψ
“Principal Quantum Number” n = 1, 2, 3, …
“Orbital Quantum Number” ℓ = 0, 1, 2, 3 …, n–1
“Magnetic Quantum Number” mℓ = –ℓ, …–1, 0, +1…, ℓ
2 4
2 2
1
2e
nm k e
En
Energy
Magnitude of angular momentum
Orientation of angular momentum
Quantum Mechanical Atom
Schrödinger’s equation determines e– “wavefunction”
( 1)L
zL m
3 quantum numbersdetermine e– state
Phys. 102, Lecture 26, Slide 3
“SHELL”
s, p, d, f “SUBSHELL”
ACT: CheckPoint 3.1 & more
How many values for mℓ are possible for the f subshell (ℓ = 3)?
A. n = 3B. n = 2C. n = 1
Phys. 102, Lecture 26, Slide 4
For which state is the angular momentum required to be 0?
A. 3B. 5C. 7
(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, 1, 0) (4, 1, 1) (4, 2, 0) (4, 2, 1) (4, 2, 2)
(4, 3, 0) (4, 3, 1) (4, 3, 2) (4, 3, 3) (15, 4, 0)(5, 0, 0)
(4, 0, 0)
Hydrogen electron orbitals
Phys. 102, Lecture 26, Slide 5
(n, ℓ, mℓ)
2
, , n mψ probability
Shell
Subshell
CheckPoint 2: orbitals
Phys. 102, Lecture 26, Slide 6
r
r
1s (ℓ = 0)
2p (ℓ = 1)
2s (ℓ = 0)
3s (ℓ = 0)
Orbitals represent probability of electron being at particular location
r
r
L
Angular momentum
( 1)L
L
zL m
0,1,2 1n
, 1,0,1,m Only one component of L quantized
Magnitude of angular momentum vector quantized
Other components Lx, Ly are not quantized
2L 6L
zL
0zL
zL
zL
0zL
zL
2zL
2zL
1 2
Phys. 102, Lecture 26, Slide 7
e–
r
–
+
Classical orbit picture
What do the quantum numbers ℓ and mℓ represent?
L
eμ
e–
r
–
+
Orbital magnetic dipole
Electron orbit is a current loop and a magnetic dipole
eμ IA2 e
eL
m
2ee
e
m
μ L
Recall Lect. 12
Dipole moment is quantized
What happens in a B field?
Phys. 102, Lecture 26, Slide 8
eμ
e– r– +
B
θzcoseU μ B θ
2 e
eBm
m
Recall Lect. 11
Orbitals with different L have different quantized energies in a B field
2Be
eμ
m
5 eV
5.8 10T
“Bohr magneton”
ACT: Hydrogen atom dipole
What is the magnetic dipole moment of hydrogen in its ground state due to the orbital motion of electrons?
2He
eμ
m
A.
B.
C.
Phys. 102, Lecture 26, Slide 9
2He
eμ
m
2ee
e
m
μ L
0Hμ
Calculation: Zeeman effect
Phys. 102, Lecture 26, Slide 10
eμ
e– –
+
B
z
Calculate the effect of a 1 T B field on the energy of the 2p (n = 2, ℓ = 1) level
2 costot n eE E μ B θ
2 2ne
eE Bm
m
0B 0
B
1m
0m
1m Energy level splits into 3, with energy splitting
2 e
e BE
m
For ℓ = 1, mℓ = –1, 0, +1
1m
ACT: Atomic dipoleThe H α spectral line is due to e– transition between the n = 3, ℓ = 2 and the n = 2, ℓ = 1 subshells.
A. 1B. 3C. 5
Phys. 102, Lecture 26, Slide 11
How many levels should the n = 3, ℓ = 2 state split into in a B field? E
n = 2
n = 3ℓ = 2
ℓ = 1
Intrinsic angular momentum
A beam of H atoms in ground state passes through a B field
Phys. 102, Lecture 26, Slide 12
“Stern-Gerlach experiment”
Atom with ℓ = 0
Since we expect 2ℓ + 1 values for magnetic dipole moment, e– must have intrinsic angular momentum ℓ = ½.
n = 1, so ℓ = 0 and expect NO effect from B field
Instead, observe beam split in two!
–
–
“Spin” s
S
S
Spin angular momentum
Phys. 102, Lecture 26, Slide 13
–
Spin UP (+½)Spin DOWN (– ½)
Electrons have an intrinsic angular momentum called “spin”
( 1)S s s
S
z sS m 1 12 2,sm
–
3
2S
2zS
2zS
2se
eg
m
μ S
Spin also generates magnetic dipole moment
with s = ½
cossU μ B θ2 s
e
geBm
m
with g ≈ 2
0B
12sm
12sm
B
Magnetic resonance
Phys. 102, Lecture 26, Slide 14
e– in B field absorbs photon with energy equal to splitting of energy levels
12sm
12sm
Absorption hf E “Electron spin resonance”
Typically microwave EM wave
Protons & neutrons also have spin ½
“Nuclear magnetic resonance”
2prot pp
eg
m
μ S s
μ since p em m
–
–
“Principal Quantum Number”, n = 1, 2, 3, …
“Orbital Quantum Number”, ℓ = 0, 1, 2, …, n–1
“Magnetic Quantum Number”, ml = –ℓ, … –1, 0, +1 …, ℓ
“Spin Quantum Number”, ms = –½, +½
Quantum number summary
2 4
2 2
1
2e
nm k e
En
Energy
Magnitude of angular momentum
Orientation of angular momentum
( 1)L
zL m
z sS m Orientation of spin
Electronic states
E
n = 1
n = 2
s (ℓ = 0) p (ℓ = 1)
Phys. 102, Lecture 26, Slide 16–
mS = +½
–
mS = –½
–
–½
–
+½
–
–½
–
+½
–
–½
–
+½
–
–½
–
+½
mℓ = –1 mℓ = 0 mℓ = +1
Pauli Exclusion Principle: no two e– can have the same set of quantum numbers
Pauli exclusion & energies determine sequence
The Periodic Table
f (ℓ = 3)
d (ℓ = 2)
p (ℓ = 1)n = 1
234567
Also ss (ℓ = 0)
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?
Phys. 102, Lecture 26, Slide 18
ACT: Quantum numbers
How many total electron states exist with n = 2?
Phys. 102, Lecture 26, Slide 19
A. 2B. 4C. 8
ACT: Magnetic elements
Where would you expect the most magnetic elements to be in the periodic table?
Phys. 102, Lecture 26, Slide 20
A. Alkali metals (s, ℓ = 1)B. Noble gases (p, ℓ = 2)C. Rare earth metals (f, ℓ = 4)
Summary of today’s lecture
Phys. 102, Lecture 25, Slide 21
• Quantum numbersPrincipal quantum numberOrbital quantum number Magnetic quantum number
• Spin angular momentume– has intrinsic angular momentum
• Magnetic propertiesOrbital & spin angular momentum generate magnetic dipole moment
• Pauli Exclusion PrincipleNo two e– can have the same quantum numbers
( 1) , 0,1, 1L n
1 12 2,z s sS m m
, , ,0,z zL m m
2 2 13.6eVnE Z n