Transcript Last Time… - Welcome | Department of Physics

Last Time…
Decreasing particle size
Quantum dots (particle in box)
Quantum tunneling
3-dimensional
wave functions
This week’s honors lecture:
Prof. Brad Christian, “Positron Emission Tomography”
Tue. Dec. 4 2007
Physics 208, Lecture 26
1
Exam 3 results
Phy208 Exam 3

Exam average =
76%
Average is at B/BC
boundary
Course evaluations:
Rzchowski: Thu, Dec. 6
Montaruli: Tues, Dec. 11
Average
25
20
AB
Count

30
15
B
10
C
A
D
5
F
0
10
Tue. Dec. 4 2007
BC
20
30
Physics 208, Lecture 26
40
50 60 70
Range
80
90 100
2
3-D particle in box: summary

Three quantum numbers (nx,ny,nz) label each state

nx,y,z=1, 2, 3 … (integers starting at 1)

Each state has different motion in x, y, z

Quantum numbers determine px 

h
n
 nx
x
h
2L

Momentum in each direction: e.g.

2
2
p
p
p
 y  z  E o n x2  n y2  n z2 
Energy: E 
2m 2m 2m
2
x
Some quantum states have same energy

Tue. Dec. 4 2007
Physics 208, Lecture 26
3
Question
How many 3-D particle in box spatial quantum
states have energy E=18Eo?
A. 1
B. 2
C. 3
D. 5
E. 6
Tue. Dec. 4 2007
E  E o n x2  n y2  n z2 
n ,n ,n  4,1,1, 1,4,1, (1,1,4)
x

y
z
Physics 208, Lecture 26
4
3-D Hydrogen atom

Bohr model:

Restricted to circular orbits

Found 1 quantum number n


Energy
13.6
E n   2 eV , orbit radius
n
rn  n ao
2
From 3-D particle in box, expect that

H atom should have more quantum
numbers


Expect different types of motion w/ same energy
Tue. Dec. 4 2007
Physics 208, Lecture 26
5
Modified Bohr model

Different orbit shapes
A
B
C
Big angular
momentum
Small
angular momentum
These orbits have same energy, a) A, B,
but different angular momenta: b) C, B,
Lrp
c) B, C,
Rank the angular momenta
d) B, A,
from largest to smallest:


A
A
C
e) C, A, B

Tue. Dec. 4 2007
C
Physics 208, Lecture 26
6
Angular momentum is quantized
orbital quantum number ℓ
Angular momentum quantized L 
ℓ is the orbital quantum number

1,
For a particular n, ℓ has values 0, 1, 2, … n-1
ℓ=0, most elliptical
ℓ=n-1, most circular

For hydrogen atom, all have same energy
Tue. Dec. 4 2007
Physics 208, Lecture 26
7
Orbital mag. moment


Orbital charge motion
produces magnetic dipole

Orbital
magnetic
dipole 
electron
Current
Proportional to angular momentum 
e
 B L /
B 
 0.9271023 A m2
 

  B

1


2m
Each orbit n,  has
2
 Same energy: E n  13.6 /n
eV
 Different orbit shape
1

(angular momentum): L 

 Different magnetic moment:   B L /

 
Tue. Dec. 4 2007
Physics 208, Lecture 26
8
Orbital mag. quantum number mℓ


Directions of ‘orbital bar magnet’ quantized.
Orbital magnetic quantum number



m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values
Determines z-component of L: Lz  m
This is also angle of L

For example: ℓ=1
gives 3 states:
Tue. Dec. 4 2007
Physics 208, Lecture 26
9
Question

For a quantum state with ℓ=2, how many different
orientations of the orbital magnetic dipole
moment are there?
A. 1
B. 2
C. 3
D. 4
E. 5
Tue. Dec. 4 2007
Physics 208, Lecture 26
10
Summary of quantum numbers
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
Tue. Dec. 4 2007
Physics 208, Lecture 26
11
Hydrogen
wavefunctions





Radial probability
Angular not shown
For given n,
probability peaks at ~
same place
Idea of “atomic shell”
Notation:
 s: ℓ=0




p:
d:
f:
g:
ℓ=1
ℓ=2
ℓ=3
ℓ=4
Tue. Dec. 4 2007
Physics 208, Lecture 26
12
Full hydrogen wave functions:
Surface of constant probability



1s-state
n  1,
Tue. Dec. 4 2007
Spherically symmetric.
Probability decreases
exponentially with radius.
Shown here is a surface
of constant probability
 0, m  0
Physics 208, Lecture 26
13
n=2: next highest energy
2s-state
2p-state
n  2,
 0, m  0
n  2,
 1, m  0
2p-state
n  2,
 1, m  1
Same energy, but different probabilities
Tue. Dec. 4 2007

26
Physics 208, Lecture
14
n=3: two s-states, six p-states and…
3p-state
3s-state
3p-state
n  3,
 0, m  0
Tue. Dec. 4 2007

n  3,
 1, m  0

Physics 208, Lecture 26
n  3,
 1, m  1
15
…ten d-states
3d-state
3d-state
n  3,
3d-state
 2, m  0

Tue. Dec. 4 2007
n  3,
 2, m  1
n  3,
 2, m  2

Physics 208, Lecture 26
16
Electron spin
New electron property:
Electron acts like a
bar magnet with N and S pole.
Magnetic moment fixed…
…but 2 possible orientations
of magnet: up and down
Described by
spin quantum number ms

Spin up
ms  1/2

Spin down
ms  1/2
z-component of spin angular momentum Sz  ms
Tue. Dec. 4 2007

Physics 208, Lecture 26
17
Include spin

Quantum state specified by four quantum numbers:
n, , m , ms 

Three spatial quantum numbers (3-dimensional)

One spin quantum number

Tue. Dec. 4 2007
Physics 208, Lecture 26
18
Quantum Number Question
How many different quantum states exist with n=2?
A. 1
B. 2
C. 4
D. 8
ℓ = 0 : 2s2
ml = 0 : ms = 1/2 , -1/2
2 states
ℓ = 1 : 2p6
ml = +1: ms = 1/2 , -1/2
ml = 0: ms = 1/2 , -1/2
ml = -1: ms = 1/2 , -1/2
2 states
2 states
2 states
There are a total of 8 states with n=2
Tue. Dec. 4 2007
Physics 208, Lecture 26
19
Question
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
l
and ms=+1/2 or -1/2 for each.
18 available quantum states for electrons
Tue. Dec. 4 2007
Physics 208, Lecture 26
20
Putting electrons on atom


Electrons obey Pauli exclusion principle
Only one electron per quantum state (n, ℓ, mℓ, ms)
unoccupied
occupied
n=1 states
Hydrogen: 1 electron
one quantum state occupied
n 1,
 0,m  0,ms  1/2
Helium: 2 electrons
n=1 states
two quantum states occupied
n 1,  0,m  0,ms  1/2
n 1,  0,m  0,ms  1/2
Tue. Dec. 4 2007
Physics 208, Lecture 26
21
Atoms with more than one electron



Electrons interact with
nucleus (like hydrogen)
Also with other electrons
Causes energy to
depend on ℓ
Tue. Dec. 4 2007
Physics 208, Lecture 26
22
Other elements: Li has 3 electrons
 n  2 



0


 m  0 

1 
ms   

2 

 n  2 



0


 m  0 

1 
ms   

2 
 n  2 



1


 m  0 

1 
ms   

2 
 n  2 



1


 m  0 

1 
ms   

2 
 n  2 



1


 m  1 

1 
ms   

2 




 n  2 



1


 m  1 

1 
ms   

2 
 n  2 



1


m  1 

1 
ms   

2 
 n  2 



1


m  1 

1 
ms   

2 
n=2 states, 
8 total, 1 occupied
n=1 states,
2 total, 2 occupied
 n  1   n  1 

 

  0    0 
 m  0   m  0 

 

ms  1/2 ms  1/2
Tue. Dec. 4 2007
one spin up, one spin down
Physics 208, Lecture 26
23
Electron Configurations
Atom
Configuration
H
1s1
He
1s2
Li
1s22s1
Be
1s22s2
B
1s22s22p1
Ne
Tue. Dec. 4 2007
etc
1s shell filled
1s22s22p6
(n=1 shell filled noble gas)
2s shell filled
2p shell filled
Physics 208, Lecture 26
(n=2 shell filled noble gas)
24
The periodic table


Atoms in same column
have ‘similar’ chemical properties.
Quantum mechanical explanation:
similar ‘outer’ electron configurations.
H
1s1
Li
2s1
Na
3s1
K
4s1
Be
2s2
Mg
3s2
Ca
4s2
Tue. Dec. 4 2007
Sc
3d1
Y
3d2
8 more
transition
metals
B
2p1
Al
3p1
Ga
4p1
C
2p2
Si
3p2
Ge
4p2
Physics 208, Lecture 26
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
25
Excited states of Sodium

Na level structure

11 electrons



Ne core = 1s2 2s2 2p6
(closed shell)
1 electron outside
closed shell
Na = [Ne]3s1
Outside (11th) electron
easily excited to other
states.
Tue. Dec. 4 2007
Physics 208, Lecture 26
26
Emitting and absorbing light
Zero energy
n=4
n=3
13.6
E 3   2 eV
3
n=2
13.6
E 2   2 eV
2
Photon
emitted
hf=E2-E1
n=1


E3  
13.6
eV
32
n=2
E2  
13.6
eV
22
E1  
13.6
eV
12
Photon
absorbed
hf=E2-E1
13.6
E1   2 eV
1
Photon is emitted when electron
drops fromone quantum
state to another
Tue. Dec. 4 2007
n=4
n=3
n=1


Absorbing a photon of correct
energy makeselectron jump to
higher quantum state.
Physics 208, Lecture 26
27



Optical spectrum of sodium
Transitions from
high to low energy
states
Relatively simple

1 electron
outside closed shell
Tue. Dec. 4 2007
Physics 208, Lecture 26
589 nm, 3p -> 3s
Optical spectrum
Na
28
How do atomic transitions occur?

How does electron in excited
state decide to make a
transition?

One possibility: spontaneous
emission

Electron ‘spontaneously’
drops from excited state

Photon is emitted
‘lifetime’ characterizes average time
for emitting photon.
Tue. Dec. 4 2007
Physics 208, Lecture 26
29
Another possibility:
Stimulated emission


Atom in excited state.
Photon of energy hf=E ‘stimulates’ electron to drop.
Additional photon is emitted,
Same frequency,
in-phase with stimulating photon
One photon in,
two photons out:
light has been amplified
E
hf=E
Before
After
If excited state is ‘metastable’ (long lifetime for spontaneous
emission) stimulated emission dominates
Tue. Dec. 4 2007
Physics 208, Lecture 26
30
: Light Amplification by
Stimulated Emission of Radiation
LASER
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.
Two photons out, these stimulate other atoms to emit.
Tue. Dec. 4 2007
Physics 208, Lecture 26
31
Ruby Laser
• Ruby crystal has the atoms which will emit photons
• Flashtube provides energy to put atoms in excited state.
• Spontaneous emission creates photon of correct frequency,
amplified by stimulated emission of excited atoms.
Tue. Dec. 4 2007
Physics 208, Lecture 26
32
Ruby laser operation
Relaxation to
metastable state
(no photon emission)
3 eV
2 eV
1 eV
Metastable state
PUMP
Transition by stimulated
emission of photon
Ground state
Tue. Dec. 4 2007
Physics 208, Lecture 26
33
The wavefunction


Wavefunction = 
= |moving to right> + |moving to left>
The wavefunction is an equal ‘superposition’ of the two
states of precise momentum.

When we measure the momentum (speed), we find one
of these two possibilities.

Because they are equally weighted, we measure them
with equal probability.
Tue. Dec. 4 2007
Physics 208, Lecture 26
34
Silicon


Tue. Dec. 4 2007
Physics 208, Lecture 26
7x7 surface
reconstruction
These 10 nm scans
show the individual
atomic positions
35
Particle in box wavefunction
 x  dx  Prob. Of finding particle in region dx about x
2
 x  L  ?
Particle is
never here

x=0
x=L
Particle is
never here x  0  ?
Tue. Dec. 4 2007
Physics 208, Lecture 26
36
Making a measurement
Suppose you measure the speed (hence, momentum) of the
quantum particle in a tube.
How likely are you to measure the particle moving to the
left?
A. 0% (never)
B. 33% (1/3 of the time)
C. 50% (1/2 of the time)
Tue. Dec. 4 2007
Physics 208, Lecture 26
37
Interaction with applied B-field



Like a compass needle, it interacts with an external
magnetic field depending on its direction.
Low energy when aligned with field, high energy
when anti-aligned
13.6
E


eV   B
Total energy is then
2
This means that
spectral lines will split
in a magnetic field
Tue. Dec. 4 2007

n
13.6
  2 eV  z B
n
13.6
  2 eV  m B B
n
Physics 208, Lecture 26
38
Tue. Dec. 4 2007
Physics 208, Lecture 26
39
Orbital magnetic dipole moment
Can calculate dipole moment for circular orbit
charge
e
ev


Current =
period 2r /v 2r
Dipole moment
µ=IA
2
Area =  r
evr e


mvr/
2 2m

  B L /

In quantum mechanics, L 
  B
Tue. Dec. 4 2007



B 

e
 0.9271023 A  m 2
2m
 5.79105 eV /Tesla
1

1 magnitude of
orb. mag. dipole moment
Physics 208, Lecture 26
40
Tue. Dec. 4 2007
Physics 208, Lecture 26
41
Electron magnetic moment

Why does it have a magnetic moment?

It is a property of the electron in the same way that
charge is a property.

But there are some differences



Magnetic moment has a size and a direction
It’s size is intrinsic to the electron,
but the direction is variable.
The ‘bar magnet’ can point in different directions.
Tue. Dec. 4 2007
Physics 208, Lecture 26
42
Additional electron properties

Free electron, by itself in space, not only
has a charge, but also acts like a bar
magnet with a N and S pole.

Since electron has charge, could explain
this if the electron is spinning.

Then resulting current loops would
produce magnetic field just like a bar
magnet.

But…


Electron in NOT spinning.
As far as we know,
electron is a point particle.
Tue. Dec. 4 2007
Physics 208, Lecture 26
43
Spin: another quantum number

There is a quantum # associated with this
property of the electron.

Even though the electron is not spinning, the
magnitude of this property is the spin.

The quantum numbers for the two states are
+1/2 for the up-spin state
-1/2 for the down-spin state


The proton is also a spin 1/2 particle.
The photon is a spin 1 particle.
Tue. Dec. 4 2007
Physics 208, Lecture 26
44
Orbital
magnetic
moment
Orbital mag. moment

Since


Make a question out of this
electron
Current
Electron has an electric charge,
And is moving in an orbit around nucleus…

produces a loop of current,
and a magnetic dipole moment ,

Proportional to angular momentum
 
 B L /

magnitude of

orb. mag. dipole moment

Tue. Dec. 4 2007
e
B 
 0.9271023 A  m 2
2m
 5.79105 eV /Tesla
  B
Physics 208, Lecture 26

1
45
Orbital mag. quantum number mℓ


Possible directions of the ‘orbital bar magnet’ are
quantized just like everything else!
Orbital magnetic quantum number


m ℓ ranges from - ℓ, to ℓ in integer steps
Number of different directions = 2ℓ+1
For example: ℓ=1
gives 3 states:
m ℓ = +1
Tue. Dec. 4 2007
S
N
mℓ = 0
Physics 208, Lecture 26
m ℓ = -1
46
Particle in box quantum states
L
n
p
n=3
h
3
2L
E
h2
2
8mL2
 h
2L
h2
8mL2
Tue. Dec. 4 2007
Probability
2
h
32
8mL2
n=2  2 h
2L
 n=1
Wavefunction
2
Physics 208, Lecture 26
47
Particle in box energy levels
Quantized momentum
h
h
p n
 npo

2L
 Energy = kinetic
2
2
npo 
p

E

 n2Eo

2m
2m


Or Quantized Energy
Energy

n=5
n=4
En  n2Eo
n=3
n=quantum number
n=2
n=1
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Physics 208, Lecture 26
48
Hydrogen atom energies


Quantized energy levels:
Each corresponds to
different
 Orbit radius
 Velocity
 Particle wavefunction
 Energy
Each described by a
quantum number n
Zero energy
n=4
n=3
E3  
13.6
eV
32
n=2
E2  
13.6
eV
22
E1  
13.6
eV
12


n=1
13.6
E n   2 eV
n
Tue. Dec. 4 2007
Energy


Physics 208, Lecture 26
49
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50
Quantum numbers

Two quantum numbers
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Pauli Exclusion Principle
Where do the electrons go?
In an atom with many electrons, only one electron is
allowed in each quantum state (n, ℓ, mℓ, ms).
Atoms with many electrons have many atomic orbitals
filled.
Chemical properties are determined by the
configuration of the ‘outer’ electrons.
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Atomic sub-shells

Each
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53