Summary of quantum numbers
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Start lecture ~ 8:55 am
This week’s honors lecture:
Prof. Ron Wakai, “Biomagnetic imaging”
Final Exam:
Monday May 12
12:25 - 2:25 pm
Ingraham B10
Wed. Apr 30, 2008
For hydrogen atom:
• n : describes energy of orbit
• ℓ describes the magnitude of orbital angular momentum
• m ℓ describes the angle of the orbital angular momentum
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Modified Bohr model
• Different orbit shapes
A
• Directions of ‘orbital bar magnet’ quantized.
• Orbital magnetic quantum number
Big angular
momentum
Small
angular momentum
These orbits have same energy, a) A,
but different angular momenta: b) C,
r r r
L=r"p
c) B,
Rank the angular momenta
d) B,
from largest to smallest:
(
)
Wed. Apr 30, 2008
!
B, A
C, A
A, C
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For example: ℓ=1
gives 3 states:
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Electron spin
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Include spin
N
New electron property:
Electron acts like a
bar magnet with N and S pole.
– m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values
– Determines z-component of L: Lz = m l h
– This is also angle of L
B, C
e) C, A, B
!
2
Orbital mag. quantum number mℓ
C
B
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• Quantum state specified by four quantum
numbers:( n, l, ml , ms )
S
– Three spatial quantum numbers (3-dimensional)
Magnetic moment fixed…
!– One spin quantum number
…but 2 possible orientations
of magnet: up and down
S
N
Described by
• Spin up
spin quantum number ms
ms = +1/2
• Spin down
ms = "1/2
z-component of spin angular momentum Sz = msh
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!
!
!
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Atomic Quantum number summary
Quantum Number Question
• Hydrogen atom states ( n, l, ml , ms )
How many different quantum states exist with n=2?
– n: principle quantum number
A. 1
B. 2
C. 4
D. 8
• Determines energy
• (n=1, 2, 3…)
!
– ℓ: orbital quantum
number
• Magnitude of orbital angular momentum L = h l(l + 1)
• ℓ=0, 1, 2, … n-1
ℓ = 0 : 2s2
m l = 0 : ms = 1/2 , -1/2
ℓ=1
– mℓ: orbital magnetic
quantum number
r
m l = +1: ms = 1/2 , -1/2
m l = 0: ms = 1/2 , -1/2
m l = -1: ms = 1/2 , -1/2
• Orientation of L ( Lz = m l h)
!
• mℓ = - ℓ, - ℓ + 1, … 0, … ℓ - 1, + ℓ
– ms: spin quantum
r number
• Orientation
! ! of S ( Sz = msh)
2 states
2 states
2 states
There are a total of 8 states with n=2
• m s=-1/2, +1/2
Wed. Apr 30, 2008
2 states
: 2p6
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!
Putting electrons on atom
Question
• Electrons obey Pauli exclusion principle
• Only one electron per quantum state (n, ℓ, mℓ, ms)
How many different quantum states are in a
5g (n=5, ℓ =4) sub-shell of an atom?
A. 22
B. 20
C. 18
D. 16
ℓ =4, so 2(2 ℓ +1)=18.
E. 14
In detail, m = -4, -3, -2, -1, 0, 1, 2, 3, 4
unoccupied
occupied
n=1 states
Hydrogen: 1 electron
one quantum state occupied
(n = 1,l = 0,ml = 0,ms = +1/2)
l
and ms=+1/2 or -1/2 for each.
18 available quantum states for electrons
!
Wed. Apr 30, 2008
Helium: 2 electrons
n=1 states
two quantum states occupied
(n = 1,l = 0,ml = 0,ms = +1/2)
(n = 1,l = 0,ml = 0,ms = "1/2)
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!
!
Other elements: Li has 3 electrons
" n=2 %
$
'
$ l=0 '
$ ml = 0 '
$
1'
$ ms = + '
#
2&
!
!
# n=2 &
%
(
% l=0 (
% ml = 0 (
%
1(
% ms = " (
$
2'
" n=2 %
$
'
$ l =1 '
$ ml = 0 '
$
1'
$ ms = + '
#
2&
# n=2 &
%
(
% l =1 (
% ml = 0 (
%
1(
% ms = " (
$
2'
" n=2 %
$
'
$ l =1 '
$ ml = 1 '
$
1'
$ ms = + '
#
2&
!
!
!
!
# n=2 &
%
(
% l =1 (
% ml = 1 (
%
1(
% ms = " (
$
2'
# n=2 &
%
(
% l =1 (
% ml = "1 (
%
1(
% ms = + (
$
2'
# n=2 &
%
(
% l =1 (
% ml = "1 (
%
1(
% ms = " (
$
2'
n=2 states,
8 total, 1 occupied
!
# n =1 &
%
(
% l=0 (
% ml = 0 (
%
(
$ ms = "1/2'
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!
1s1
He
1s2
Li
1s22s1
Be
1s22s2
B
1s22s22p1
Ne
one spin up, one spin down
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Configuration
H
1s shell filled
(n=1 shell filled noble gas)
!
n=1 states,
2 total, 2 occupied
" n =1 %
$
'
$ l=0 '
$ ml = 0 '
$
'
# ms = +1/2&
Electron Configurations
Atom
11
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etc
2s shell filled
1s22s22p6 2p shell filled
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(n=2 shell filled noble gas)
12
!
2
The periodic table
Atoms with more than one electron
• Atoms in same column
have ‘similar’ chemical properties.
• Quantum mechanical explanation:
similar ‘outer’ electron configurations.
Be
2s2
Mg
3s2
Ca
4s2
Sc
3d1
Y
3d2
8 more
transition
metals
Wed. Apr 30, 2008
B
2p1
Al
3p1
Ga
4p1
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C
2p2
Si
3p2
Ge
4p2
N
2p3
P
3p3
As
4p3
O
2p4
S
3p4
Se
4p4
F
2p5
Cl
3p5
Br
4p5
He
1s2
Ne
2p6
Ar
3p6
Kr
4p6
13
Hydrogen
wavefunctions
p:
d:
f:
g:
14
– Ne core = 1s 2 2s2 2p6
(closed shell)
– 1 electron outside
closed shell
Na = [Ne]3s 1
• Outside (11th) electron
easily excited to other
states.
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Wed. Apr 30, 2008
• How does electron in excited
state decide to make a
transition?
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Na
16
Another possibility:
Stimulated emission
How do atomic transitions occur?
• Atom in excited state.
• Photon of energy hf=ΔE ‘stimulates’ electron to drop.
Additional photon is emitted,
Same frequency,
in-phase with stimulating photon
• One possibility: spontaneous
emission
• Electron ‘spontaneously’
drops from excited state
One photon in,
two photons out:
– Photon is emitted
‘lifetime’ characterizes average
time for emitting photon.
Wed. Apr 30, 2008
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• 11 electrons
ℓ=1
ℓ=2
ℓ=3
ℓ=4
Wed. Apr 30, 2008
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Na Optical spectrum
• Radial probability
• Angular not shown
• For given n, probability
peaks at ~ same place
• Idea of “atomic shell”
• Notation:
– s: ℓ=0
–
–
–
–
States fill in order of energy:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d
589 nm, 3p -> 3s
H
1s1
Li
2s1
Na
3s1
K
4s1
• Electrons interact with
nucleus (like hydrogen)
• Also with other electrons
• Causes energy to depend
on ℓ in addition to n
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hf=ΔE
light has been amplified
ΔE
Before
After
If excited state is ‘metastable’ (long lifetime for spontaneous
emission) stimulated emission dominates
Wed. Apr 30, 2008
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Ruby Laser
LASER :
Light Amplification by
Stimulated Emission of Radiation
Atoms ‘prepared’ in metastable excited states
…waiting for stimulated emission
Called ‘population inversion’
(atoms normally in ground state)
Excited states stimulated to emit photon from a spontaneous
emission.
• Ruby crystal has the atoms which will emit photons
Two photons out, these stimulate other atoms to emit.
• Flashtube provides energy to put atoms in excited state.
• Spontaneous emission creates photon of correct frequency,
amplified by stimulated emission of excited atoms.
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Relaxation to
metastable state
(no photon emission)
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• Hydrogen atom: single electron orbiting
around single positively-charged proton
• Hydrogen atom can be in different quantum
states, corresponding classically to different
orbits.
2 eV
1 eV
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Good description of atom
Ruby laser operation
3 eV
Wed. Apr 30, 2008
• Can also have more than one electron
orbiting around the nucleus.
Metastable state
• # of electrons determines the chemical
properties, and hence the element.
PUMP
Transition by stimulated
emission of photon
Ground state
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