phys 102 – lecture 26 the quantum numbers and spin 1

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Phys 102 – Lecture 26The quantum numbers and spin

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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


“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


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


(n, ℓ, mℓ)

2

, , n mψ probability

Shell

Subshell

CheckPoint 2: orbitals


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


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?


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.


2He

eμ

m

2ee

e

m

μ L

0Hμ

Calculation: Zeeman effect


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


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


“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


–

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


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)


CheckPoint 3.2

How many electrons can there be in a 5g (n = 5, ℓ = 4) sub-shell of an atom?


ACT: Quantum numbers

How many total electron states exist with n = 2?


A. 2B. 4C. 8

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)

Summary of today’s lecture


• 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

phys 102 – lecture 26 the quantum numbers and spin 1

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