Transcript Document

Last Time…
Bohr model of Hydrogen atom
Wave properties of matter
Energy levels from wave properties
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
Thu. Nov. 29 2007
Energy


Physics 208, Lecture 25
2
Quantum ‘Particle in a box’
Particle confined to a fixed region of space
e.g. ball in a tube- ball moves only along length L
L

Classically, ball bounces back and forth in tube.

This is a ‘classical state’ of the ball.

Identify each state by speed,
momentum=(mass)x(speed), or kinetic energy.

Classical: any momentum, energy is possible.
Quantum: momenta, energy are quantized
Thu. Nov. 29 2007
Physics 208, Lecture 25
3
Classical vs Quantum
Classical: particle bounces back and forth.


Sometimes velocity is to left, sometimes to right
L

Quantum mechanics:



Particle represented by wave: p = mv = h / 
Different motions: waves traveling left and right
Quantum wave function:

superposition of both at same time
Thu. Nov. 29 2007
Physics 208, Lecture 25
4
Quantum version

Quantum state is both velocities at the same time
  2L
One halfwavelength

L
momentum
h h
p 
 2L
Ground state is a standing wave, made equally of



Wave traveling right ( p = +h/ )

Wave traveling left ( p = - h/ )
Determined by standing wave condition L=n(/2) :
2 2 
 x  
sin  x 
L   
Thu. Nov. 29 2007
Quantum wave function:
superposition of both motions.
Physics 208, Lecture 25
5
Different quantum states

p = mv = h / 


Different speeds correspond to different 
subject to standing wave condition
integer number of half-wavelengths fit in the tube.
2 2 
sin  x 
Wavefunction:  x  
L   
  2L
One halfwavelength
L
Two halfwavelengths
Thu. Nov. 29 2007
momentum
h h
p 
 po
 2L
n=1

n=2
Physics 208, Lecture 25

momentum
h h
p    2po
 L
6
Particle in box question
A particle in a box has a mass m.
Its energy is all kinetic = p2/2m.
Just saw that momentum in state n is npo.
It’s energy levels
A. are equally spaced everywhere
B. get farther apart at higher energy
C. get closer together at higher energy.
Thu. Nov. 29 2007
Physics 208, Lecture 25
7
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
Thu. Nov. 29 2007
Physics 208, Lecture 25
8
Question
A particle is in a particular quantum state in a box of length L.
The box is now squeezed to a shorter length, L/2.
The particle remains in the same quantum state.
The energy of the particle is now
A. 2 times bigger
B. 2 times smaller
C. 4 times bigger
D. 4 times smaller
E. unchanged
Thu. Nov. 29 2007
Physics 208, Lecture 25
9
Quantum dot: particle in 3D box
CdSe quantum dots
dispersed in hexane
(Bawendi group, MIT)
Color from photon
absorption
Decreasing particle size


Determined by energylevel spacing
Energy level spacing increases
as particle size decreases.
i.e
2
E n 1  E n
Thu. Nov. 29 2007
n  1 h 2



8mL2
n 2h 2
8mL2
Physics 208, Lecture 25
10
Interpreting the wavefunction

Probabilistic interpretation
The square magnitude of the wavefunction ||2 gives the
probability of finding the particle at a particular spatial
location
Wavefunction
Thu. Nov. 29 2007
Probability = (Wavefunction)2
Physics 208, Lecture 25
11
Higher energy wave functions
L
n
p
n=3
h
3
2L
E
Probability
2
h
32
8mL2
n=2  2 h
2L
h2
2
8mL2
 h
2L
h2
8mL2
 n=1
Wavefunction
Thu. Nov. 29 2007
2
Physics 208, Lecture 25
12
Probability of finding electron


Classically, equally likely to find particle anywhere
QM - true on average for high n
Zeroes in the probability!
Purely quantum, interference effect
Thu. Nov. 29 2007
Physics 208, Lecture 25
13
Quantum Corral
D. Eigler (IBM)


48 Iron atoms assembled into a circular ring.
The ripples inside the ring reflect the electron quantum states of a
circular ring (interference effects).
Thu. Nov. 29 2007
Physics 208, Lecture 25
14
Scanning Tunneling Microscopy
Tip
Sample


Over the last 20 yrs, technology developed to controllably
position tip and sample 1-2 nm apart.
Is a very useful microscope!
Thu. Nov. 29 2007
Physics 208, Lecture 25
15
Particle in a box, again
L
Particle contained entirely
within closed tube.
Wavefunction
Probability =
(Wavefunction)2
Open top: particle can escape if
we shake hard enough.
But at low energies, particle
stays entirely within box.
Like an electron in metal
(remember photoelectric effect)
Thu. Nov. 29 2007
Physics 208, Lecture 25
16
Quantum mechanics says
something different!
Low energy
Classical state
Low energy
Quantum state
Quantum Mechanics:
some probability of the
particle penetrating
walls of box!
Nonzero probability of being outside the box.
Thu. Nov. 29 2007
Physics 208, Lecture 25
17
Two neighboring boxes


When another box is brought nearby, the
electron may disappear from one well, and
appear in the other!
The reverse then happens, and the electron
oscillates back an forth, without ‘traversing’ the
intervening distance.
Thu. Nov. 29 2007
Physics 208, Lecture 25
18
Question
Suppose separation between boxes increases by a factor of two.
The tunneling probability
A. Increases by 2
B. Decreases by 2
C. Decreases by <2
‘high’ probability
D. Decreases by >2
E. Stays same
‘low’ probability
Thu. Nov. 29 2007
Physics 208, Lecture 25
19
Example:
Ammonia molecule
N
H
H
H





Thu. Nov. 29 2007
Ammonia molecule: NH3
Nitrogen (N) has two equivalent
‘stable’ positions.
Quantum-mechanically tunnels
2.4x1011 times per second (24 GHz)
Known as ‘inversion line’
Basis of first ‘atomic’ clock (1949)
Physics 208, Lecture 25
20
Atomic clock question
Suppose we changed the ammonia molecule so
that the distance between the two stable
positions of the nitrogen atom INCREASED.
The clock would
A. slow down.
B. speed up.
C. stay the same.
N
H
H
H
Thu. Nov. 29 2007
Physics 208, Lecture 25
21
Tunneling between conductors



Make one well deeper:
particle tunnels, then stays in other well.
Well made deeper by applying electric field.
This is the principle of scanning tunneling microscope.
Thu. Nov. 29 2007
Physics 208, Lecture 25
22
Scanning Tunneling Microscopy
Tip, sample are quantum
‘boxes’
Tip
Potential difference induces
tunneling
Tunneling extremely sensitive
to tip-sample spacing
Sample


Over the last 20 yrs, technology developed to controllably
position tip and sample 1-2 nm apart.
Is a very useful microscope!
Thu. Nov. 29 2007
Physics 208, Lecture 25
23
Surface steps on Si
Images courtesy
M. Lagally,
Univ. Wisconsin
Thu. Nov. 29 2007
Physics 208, Lecture 25
24
Manipulation of atoms

Take advantage of tip-atom interactions to physically
move atoms around on the surface

This shows the assembly
of a circular ‘corral’ by
moving individual Iron
atoms on the surface of
Copper (111).

The (111) orientation
supports an electron
surface state which can
be ‘trapped’ in the corral
Thu. Nov. 29 2007
Physics 208, Lecture 25
D. Eigler (IBM)
25
Quantum Corral
D. Eigler (IBM)


48 Iron atoms assembled into a circular ring.
The ripples inside the ring reflect the electron quantum states of a
circular ring (interference effects).
Thu. Nov. 29 2007
Physics 208, Lecture 25
26
The Stadium Corral
D. Eigler (IBM)
Again Iron on copper. This was assembled to investigate quantum chaos.

The electron wavefunction leaked out beyond the stadium too much to to observe
expected effects.
Thu. Nov. 29 2007
Physics 208, Lecture 25
27
Some fun!
Kanji for atom (lit. original child)
Iron on copper (111)
Thu. Nov. 29 2007
Carbon Monoxide man
Carbon Monoxide on Pt (111)
Physics 208, Lecture 25
D. Eigler (IBM)
28
Particle in box again: 2 dimensions
Motion in x direction
Motion in y direction
Same velocity (energy),
but details of motion are different.
Thu. Nov. 29 2007
Physics 208, Lecture 25
29
Quantum Wave Functions
Probability
(2D)
Wavefunction
Ground state: same wavelength
(longest) in both x and y
Need two quantum #’s,
one for x-motion
one for y-motion
Use a pair (nx, ny)
Ground state: (1,1)
Probability = (Wavefunction)2
One-dimensional (1D) case
Thu. Nov. 29 2007
Physics 208, Lecture 25
30
2D excited states
(nx, ny) = (2,1)
(nx, ny) = (1,2)
These have exactly the same energy, but the
probabilities look different.
The different states correspond to ball bouncing
in x or in y direction.
Thu. Nov. 29 2007
Physics 208, Lecture 25
31
Particle in a box
What quantum state could this be?
A. nx=2, ny=2
B. nx=3, ny=2
C. nx=1, ny=2
Thu. Nov. 29 2007
Physics 208, Lecture 25
32
Next higher energy state


The ball now has same bouncing motion in both x
and in y.
This is higher energy that having motion only in x
or only in y.
(nx, ny) = (2,2)
Thu. Nov. 29 2007
Physics 208, Lecture 25
33
Three dimensions

Object can have different velocity (hence
wavelength) in x, y, or z directions.





Need three quantum numbers to label state
(nx, ny , nz) labels each quantum state
(a triplet of integers)
Each point in three-dimensional space has a
probability associated with it.
Not enough dimensions to plot probability
But can plot a surface of constant probability.
Thu. Nov. 29 2007
Physics 208, Lecture 25
34
Particle in 3D box


Ground state
surface of constant
probability
(nx, ny, nz)=(1,1,1)
2D case
Thu. Nov. 29 2007
Physics 208, Lecture 25
35
(121)
(112)
(211)
All these states have the same
energy, but different probabilities
Thu. Nov. 29 2007
Physics 208, Lecture 25
36
(222)
(221)
Thu. Nov. 29 2007
Physics 208, Lecture 25
37
The ‘principal’ quantum number

In Bohr model of atom,
n is the principal quantum number.

Arise from considering circular orbits.

Total energy given by principal quantum number
13.6
E n   2 eV
n
• Orbital radius is

Thu. Nov. 29 2007
rn  n 2 ao
Physics 208, Lecture 25
38
Other quantum numbers?

Hydrogen atom is three-dimensional structure


Should have three quantum numbers
Special consideration:


Coulomb potential is spherically symmetric
x, y, z not as useful as r, , 
Angular momentum warning!
Thu. Nov. 29 2007
Physics 208, Lecture 25
39
Sommerfeld: modified Bohr model

Differently shaped orbits
Big angular
momentum
Small
angular momentum
All these orbits have same energy…
… but different angular momenta
Energy is same as Bohr atom,
but angular momentum quantization altered
Thu. Nov. 29 2007
Physics 208, Lecture 25
40
Angular momentum question

Which angular momentum is largest?
Thu. Nov. 29 2007
Physics 208, Lecture 25
41
The orbital quantum number ℓ
In quantum mechanics, the angular momentum can
only have discrete values
L
 ,1
ℓ is the orbital quantum number
For a particular n,ℓ has values 0, 1, 2, … n-1
ℓ=0, most elliptical

ℓ=n-1, most circular
These states all have the same energy
Thu. Nov. 29 2007
Physics 208, Lecture 25
42
Orbital mag. moment
Orbital
magnetic
moment
electron
Current

Since




Electron has an electric charge,
And is moving in an orbit around nucleus…
… it produces a loop of current,
and hence a magnetic dipole field,
very much like a bar magnet or a
compass needle.
Directly related to angular momentum
Thu. Nov. 29 2007
Physics 208, Lecture 25
43
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
Area = r
evr e


mvr/
2 2m

  B L / 
Thu. Nov. 29 2007

B 

In quantum mechanics, L 
  B
2


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 25
44
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
Example: For ℓ=1,
m ℓ = -1, 0, or -1,
corresponding to
three different directions
of orbital bar magnet.
ℓ=1 gives 3
states:
Thu. Nov. 29 2007
m ℓ = +1
S
N
mℓ = 0
Physics 208, Lecture 25
m ℓ = -1
45
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
Thu. Nov. 29 2007
Physics 208, Lecture 25
46
Example: For ℓ=2,
m ℓ = -2, -1, 0, +1, +2
corresponding to
three different directions
of orbital bar magnet.
Thu. Nov. 29 2007
Physics 208, Lecture 25
47
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
Thu. Nov. 29 2007

n
13.6
  2 eV  z B
n
13.6
  2 eV  m B B
n
Physics 208, Lecture 25
48
Thu. Nov. 29 2007
Physics 208, Lecture 25
49
Summary of quantum numbers



n describes the energy of the orbit
ℓ describes the magnitude of angular momentum
m ℓ describes the behavior in a magnetic field due to the
magnetic dipole moment produced by orbital motion (Zeeman
effect).
Thu. Nov. 29 2007
Physics 208, Lecture 25
50
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.
Thu. Nov. 29 2007
Physics 208, Lecture 25
51
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.
Thu. Nov. 29 2007
Physics 208, Lecture 25
52
Electron spin orientations

Spin up

Spin down
Only two possible orientations
Thu. Nov. 29 2007
Physics 208, Lecture 25
53
Thu. Nov. 29 2007
Physics 208, Lecture 25
54
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.
Thu. Nov. 29 2007
Physics 208, Lecture 25
55
Include spin

We labeled the states by their quantum numbers.
One quantum number for each spatial dimension.

Now there is an extra quantum number: spin.

A quantum state is specified four quantum
numbers: n, , m , ms 

An atom with several electrons filling quantum
states starting with the lowest energy, filling
quantum
states until electrons are used.

Thu. Nov. 29 2007
Physics 208, Lecture 25
56
Quantum Number Question
How many different quantum states exist with n=2?
1
2
4
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
Thu. Nov. 29 2007
Physics 208, Lecture 25
57
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.
Thu. Nov. 29 2007
Physics 208, Lecture 25
58
Number of electrons
Which of the following is a possible number of
electrons in a 5g (n=5, l=4) sub-shell of an atom?
22
20
17
l=4, so 2(2l+1)=18.
In detail, ml = -4, -3, -2, -1, 0, 1, 2, 3, 4
and ms=+1/2 or -1/2 for each.
18 available quantum states for electrons
17 will fit.
And there is room left for 1 more !
Thu. Nov. 29 2007
Physics 208, Lecture 25
59
Putting electrons on atom


Electrons are obey exclusion principle
Only one electron per quantum state
unoccupied
occupied
n=1 states
Hydrogen: 1 electron
one quantum state occupied
Helium: 2 electrons
n=1 states
two quantum states occupied
Thu. Nov. 29 2007
Physics 208, Lecture 25
60
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 

 

 1/2
m s 2007
1/2
m sThu.
 29
Nov.
one spin up, one spin down
Physics 208, Lecture 25
61
Electron Configurations
Atom
Configuration
H
1s1
He
1s2
Li
1s22s1
Be
1s22s2
B
1s22s22p1
Ne
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etc
1s shell filled
1s22s22p6
(n=1 shell filled noble gas)
2s shell filled
2p shell filled
Physics 208, Lecture 25
(n=2 shell filled noble gas)
62
The periodic table



Elements are arranged in the periodic table so that
atoms in the same column
have ‘similar’ chemical properties.
Quantum mechanics explains this
by similar ‘outer’ electron configurations.
If not for Pauli exclusion principle,
all electrons would be in the 1s state!
H
1s1
Li
2s1
Na
3s1
Be
2s2
Mg
3s2
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B
2p1
Al
3p1
C
2p2
Si
3p2
Physics 208, Lecture 25
N
2p3
P
3p3
O
2p4
S
3p4
H
1s1
F
2p5
Cl
3p5
He
1s2
Ne
2p6
Ar
3p6
63
Wavefunctions and probability

Probability of finding an electron is given by the
square of the wavefunction.
Probability large here
Probability small here
Thu. Nov. 29 2007
Physics 208, Lecture 25
64
Hydrogen atom:
Lowest energy (ground) state



1s-state
n 1,
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Spherically symmetric.
Probability decreases
exponentially with radius.
Shown here is a surface
of constant probability
 0, m  0
Physics 208, Lecture 25
65
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
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
25
Physics 208, Lecture
66
n=3: two s-states, six p-states and…
3p-state
3s-state
3p-state
n  3,
 0, m  0
Thu. Nov. 29 2007

n  3,
1, m  0

Physics 208, Lecture 25
n  3,
1, m  1
67
…ten d-states
3d-state
3d-state
n  3,
3d-state
 2, m  0
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
n  3,
 2, m  1

Physics 208, Lecture 25
n  3,
 2, m  2
68
Electron wave around an atom
Wave representing
electron


Electron wave extends around
circumference of orbit.
Only integer number of
wavelengths around orbit
allowed.
Thu. Nov. 29 2007
Wave representing
electron
Physics 208, Lecture 25
69
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
E1  
13.6
eV
12
Photon is emitted when electron
drops fromone quantum
state to another
Thu. Nov. 29 2007
n=4
n=3
n=1


Absorbing a photon of correct
energy makeselectron jump to
higher quantum state.
Physics 208, Lecture 25
70
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.
Thu. Nov. 29 2007
Physics 208, Lecture 25
71
Silicon


Thu. Nov. 29 2007
Physics 208, Lecture 25
7x7 surface
reconstruction
These 10 nm scans
show the individual
atomic positions
72
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  ?
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Physics 208, Lecture 25
73
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)
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Physics 208, Lecture 25
74