The quantum numbers and spin
1

Recall: the Bohr model
Only orbits that fit n e – wavelengths are allowed e –
+ wave
SUCCESSES
Correct energy quantization & atomic spectra
E n
  e
2
2

1 n
2 n

1, 2,3,...
FAILURES
Radius & momentum quantization violates
Heisenberg Uncertainty Principle r n
 n
2 2 m ke
2
 2 n a
0
    r
2
Electron orbits cannot have zero L
L n
 n
Orbits can hold any number of electrons
Phys. 102, Lecture 26, Slide 2

Quantum Mechanical Atom
Schrödinger’s equation determines e – “wavefunction”



2
 
2 m e ke
2 r


( , , )

( , , )

“SHELL”
“Principal Quantum Number” n = 1, 2, 3, …
ψ n , , m
3 quantum numbers determine e – state
E n
  e
1
Energy
2
2 n
2
s, p, d, f “SUBSHELL”
“Orbital Quantum Number” ℓ = 0, 1, 2, 3 …, n –1
L

(

1) Magnitude of angular momentum
“Magnetic Quantum Number” m ℓ
L z
 m
= – ℓ , …–1, 0, +1…, ℓ
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 m ℓ are possible for the f subshell (ℓ = 3)?
A. 3
B. 5
C. 7
Phys. 102, Lecture 26, Slide 4

Hydrogen electron orbitals
(2, 0, 0) (2, 1, 0) (2, 1, 1)
ψ n , , m
2  probability
Shell
Subshell
(n, ℓ, m ℓ
)
(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 being at particular location
1s (ℓ = 0) 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 m ℓ represent?
r
Magnitude of angular momentum vector quantized
L (

1)

0,1, 2 n

1
+
Only one component of L quantized
L z
 m m
 
,

1, 0,1,
Classical orbit picture
– e –

1
L
L z
 
L z

0
L z
 
L

2
Other components L x
, L y are not quantized
L z
 
2
L z
 
L

6

2
L z

0
L z
 
L z
 
2 Phys. 102, Lecture 26, Slide 7

Orbital magnetic dipole z
Electron orbit is a current loop and a magnetic dipole
μ e r
μ e
 IA  
2 e m e
L Recall Lect. 12
+
–
μ e
  e
2 m e
L
Dipole moment is quantized e –
L
What happens in a B field?
B e –
–
+
θ
μ e r
U
  μ B cos θ  e
Bm Recall Lect. 11
2 m e
Orbitals with different L have different quantized energies in a B field
μ
B

2 e m e

5.8 10

5 eV
T
“Bohr magneton”
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?
μ e
  e
2 m e
L
A.
B.
C.
μ
H
  e
2 m e
μ
H

0
μ
H
  e
2 m e
Phys. 102, Lecture 26, Slide 9

Calculation: Zeeman effect z
B
+ e – –
Calculate the effect of a 1 T B field on the energy of the 2p (n = 2, ℓ = 1) level
μ e m
 
1
E tot

E n

2
 μ B cos θ

E n

2
 e
2 m e
Bm
For ℓ = 1, m ℓ
= –1, 0, +1
B

0 B

0 m
 
1 m

0 m
 
1
Energy level splits into 3, with energy splitting e B
2 m e
Phys. 102, Lecture 26, Slide 10

ACT: Atomic dipole
The H α spectral line is due to e – transition between the n = 3, ℓ = 2 and the n = 2, ℓ = 1 subshells.
How many levels should the n = 3, ℓ = 2 state split into in a B field? E
A. 1
B. 3
C. 5
n = 3
ℓ = 2
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
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” s
Atom with ℓ = 0
“Stern-Gerlach experiment”
Phys. 102, Lecture 26, Slide 12

Spin angular momentum
Electrons have an intrinsic angular momentum called “spin”
S
 
(

1) with s = ½
S z
 
2
S

2
3
S z
 m s m s
  
2
, 1
2
S z
 
2
Spin also generates magnetic dipole moment
S
B
μ s
  e
2 m e g
S with g ≈ 2
S
Spin DOWN (– ½)
B

0
Spin UP (+½) m s
 
1
2 m s
 
1
2
U
  μ B cos θ  ge
2 m e
Bm s
Phys. 102, Lecture 26, Slide 13

Magnetic resonance e – in B field absorbs photon with energy equal to splitting of energy levels
Absorption m s
 
1
2 m s
 
1
2 hf
 
E
“Electron spin resonance”
Typically microwave EM wave
Protons & neutrons also have spin ½
μ prot
  e
2 m p g p
S  μ s since m p
 m e
“Nuclear magnetic resonance”
Phys. 102, Lecture 26, Slide 14

Quantum number summary
“Principal Quantum Number”, n = 1, 2, 3, …
E n
  e
2
2
1 n
2
Energy
“Orbital Quantum Number”, ℓ = 0, 1, 2, …, n –1
L

(

1)
Magnitude of angular momentum
“Magnetic Quantum Number”, m l
= –ℓ, … –1, 0, +1 …, ℓ
L z
 m Orientation of angular momentum
“Spin Quantum Number”, m s
= –½, +½
S z
 m s
Orientation of spin

Electronic states
Pauli Exclusion Principle: no two e – can have the same set of quantum numbers
E
s (ℓ = 0) p (ℓ = 1)
n = 2
+½ –½ +½ –½ +½ –½ +½ –½
– – – – – – – – m ℓ
= –1 m ℓ
= 0 m ℓ
= +1 m
S
= +½ m
S
= –½
n = 1 – –
Phys. 102, Lecture 26, Slide 16

The Periodic Table
Pauli exclusion & energies determine sequence
s (ℓ = 0)
p (ℓ = 1)
n = 1
2
3
d (ℓ = 2)
6
7
4
5
Also s
Phys. 102, Lecture 26, Slide 17 f (ℓ = 3)

CheckPoint 3.2
How many electrons can there be in a 5g (n = 5, ℓ = 4) subshell 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
C. 8
Phys. 102, Lecture 26, Slide 19

ACT: Magnetic elements
Where would you expect 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
Principal quantum number E n
 
Z
2 n
2 
13.6eV
Orbital quantum number
L

(

1) ,
Magnetic quantum number L z
 m z
, m
• Spin angular momentum e – has intrinsic angular momentum
• Magnetic properties

0,1, n

1
 
, , 0,
S z
 m s m s
 
1 , 1
2 2
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