Download Phys 102 – Lecture 26 Phys 102 Lecture 26

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