Chapter 3
Wave Properties of Particles
1
The Wave Debate
Particle-ists
Wave-ists
Maxwell’s Equations
Interference
Diffraction
•
•
•
E = hf, p = h/ 
Photoelectric
Compton Effect
2
3
3.1 De Broglie waves
A moving body behaves in certain ways as though
it has a wave nature
4
Particle-like Behavior of Light
Planck’s explanation of blackbody radiation
Einstein’s explanation of photoelectric effect
5
A photon of light of frequency  has the momentum
hv h
p

c 
Why ?
6
Recall your last lecture…
E 2 = (pc)2 + (mc2)2
But isn’t the (rest) mass of a photon 0 ?
So for a photon, E = (pc)
E = hf
pc = hf
(photon momentum)
p = h/ 
7
Recall relativity
But isn’t the (rest) mass
of a photon = 0 ?
E 2 = (pc)2 + (mc2)2
So for a photon
E 2 = (pc)2
E = pc
But doesn’t
E = hf also..
pc = hf
(photon momentum)
p = h/ 
8
The wavelength of a photon is therefore specified by
its momentum according to the relation
Photon wavelength
h

mv
(3.1)
9
de Broglie: Suggested the converse
All matter, usually thought of as particles, should exhibit
wave-like behavior
Implies that electrons, neutrons, etc., are waves!
Prince Louis de Broglie
(1892-1987) 10
de Broglie Wavelength
Relates a particle-like property (p)
to a wave-like property ()
11
de Broglie
12
photon model
wave model
E = hf
p =
h

13
De Broglie suggested that Eq. (3.1) is a completely general
one that applies to material particles as well as to photons.
The momentum of a particle of mass m and velocity  is
p = mv, and its de Broglie wavelength is accordingly
De Broglie
wavelength
h

mv
(3.2)
The greater the particle’s momentum, the shorter its
wavelength.
In Eq. (3.2)  is the relativistic factor
 
1
1 v / c
2
2
(3.3)
14
Wave-Particle Duality
particle
wave function
15
A travelling wave
amplitude
c
16
Models of a free electron
electron
Wavelength  = h/p
amplitude
electron
Momentum p = mv
17
Example: de Broglie wavelength of an electron
Mass = 9.11 x 10-31 kg
Speed = 106 m / sec
6.63  1034 Joules  sec
10

.

7
28

10
m
31
6
(9.11  10 kg)(10 m/sec)
This wavelength is in the region of X-rays
18
Example: de Broglie wavelength of a ball
Mass = 1 kg
Speed = 1 m / sec
6.63  1034 Joules  sec

 6.63  1034 m
(1 kg)(1 m/sec)
This is extremely small ! Thus, it is very difficult to observe
the wave-like behavior of ordinary objects.
19
Light
Matter
Waves
Particles
So Light exhibits both WAVE and PARTICLE behaviour.
and matter PARTICLEs can exhibit WAVE behaviour.
20
3.2 Waves of what?
Waves of probability
21
Water waves : height of the water surface
Sound waves : pressure
Light waves : electric and magnetic field
Matter waves : Wave function
?
22
The wave function  itself, however, has no direct
physical significance.
Probability density
The probability of experimentally finding the body described by
the wave function
at the point x, y, z at the time t is

proportional to the value of 
2
there at t.
This interpretation was first made by Max Born in 1926.
23
Born
2

The probability of finding an electron at a given
location is proportional to the square of .
24
3.3 Describing a wave
A general formula for waves
25
Phase Velocity

How fast is the wave traveling?
The phase velocity is the
wavelength / period:
v = 
Since  = 1/:
x
The wave moves one wavelength,
, in one period, .
v = v
In terms of the k-vector, k = 2 and
the angular frequency,  = 2 this is:
It’s also helpful to define a phase delay, T,
that a wave experiences after propagating
a distance, d:
v =/k
T=d/v
26
How fast do de Broglie waves travel?
If we call the de Broglie wave velocity vp, we can apply the usua
l formula
v p  
to find vp. The wavelength l is simply the de Broglie wavelength
=h/mv.
27
To find the frequency, we equate the quantum expression E=hf
with the relativistic formula for total energy E=mc2 to obtain
hf  mc
f 
2
mc 2
h
The de Broglie wave velocity is therefore
De Broglie
phase Velocity
 mc 2
 p  f  
 h
 h  c 2

 
 mv  
(3.3)
particle velocity
∵v < c
Phase velocity is what we have
been calling
wave velocity
∴vp >c
?????
28
Simple harmonic
If we choose t=0 when the displacement y
of the string at x=0 is a maximum, its
displacement at any future time t at the
time place is given by the formula
x=0
y  A cos 2ft
(3.4)
Figure 3.1
(a) The appearance of a wave in a stretched string at a certain time.
(b) How the displacement of
a point on the string varies with time.
29
y  A cos 2ft
(3.4)
The displacement y of the string
at x at any time t is exactly the
same as the value of y at x=0 at
the earlier time t-x/vp.
In eq. (3.4)
Wave formula
t → t - x/vp
x
y  A cos 2f (t  )
vp
y  A cos 2 ( ft 
(3.5)
fx
)
vp
x=0
Figure 3.2 Wave propagation.
x
y  A cos 2 ( ft  )

x=vpt
(3.6)
30
Angular frequency
 = 2
2
Wave number

k

 p
(3.7)
(3.8)
The unit of  is the radian per second and that k is the radian per meter.
An angular frequency gets its name from uniform circular motion, where
a particle that moves around a circle  times per second sweeps out
2 rad/s.
The wave number is equal to the number of radians corresponding to a
wave train 1 m long, since there are 2 rad in one complete wave.
31
Wave formula
y  A cos(t  kx)
(3.9)
32
3.4 Phase and Group velocities
A group of waves need not have the same velocity as the
waves themselves
33
If the velocity of the waves are the same, the velocity with
which the wave group travels is the common phase
velocity.
However, if the phase velocity varies with wavelength, the
different individual waves do not proceed together.
This situation is called
dispersion.
As a result the wave group has a velocity different from the
phase velocities of the waves that make it up. This is the
case with de Broglie waves.
Figure 3.3 A wave group
34
Figure 3.4
Beats are produced by the superposition of two waves with different
frequencies.
35
Phase and Group Velocities
phase   t  kx  const
This phase velocity:
 dt  k dx  0
vp 
dx 

dt k
The observable is the group velocity (the velocity of
propagation of a wave “packet” or wave “group”).
Let’s consider the superposition of two harmonic
waves with slightly different frequencies (>>,
k>>k):
y1  A cos t  kx 
y2  A cos     t   k  k  x 
" " 
2
k
1

1

cos   cos   2 cos       cos      
2

2

1

1

y  y1  y2  2 A cos   2    t   2k  k  x cos     t  k  x  
2

2

k 
 
The velocity of propagation
 2 A cos  t  kx  cos 
t
x
2 
 2
of the wave packet:
d
fast oscillations
vg 
“envelope”=
-the group velocity
within the wave
dk
36
wave group
group
Phase velocity
Group velocity
vp


k
vg



k
vg
d

dk
Angular frequency of de Broglie waves
2 m 0 c 2
2 mc 2
  2 

h
h 1 2
Wave number of de Broglie waves
2 m 0 v
2  2 m v


k

h
h 1 2
37
De Broglie group velocity
vg  v
the velocity of the body
The de Broglie wave group associated with a moving body travels with
the same velocity as the body.
De Broglie phase velocity

c2

vp 
k
v
∵v
<c
∴ vp > c
Vp has no physical significance because the motion of the wave group,
not the motion of the individual waves that make up the group,
corresponds to the motion of the body, and vg < c as it should be.
The fact that vp>c for de Broglie waves therefore does not violates
38
special relativity.
Material dispersion
Waveguide dispersion
Material dispersion comes from a frequency-dependent response of a
material to waves. For example, material dispersion leads to undesired
chromatic aberration in a lens or the separation of colors in a prism.
Waveguide dispersion occurs when the speed of a wave in a waveguide,
such as a coaxial cable or optical fiber depends on its frequency. This
type of dispersion leads to signal degradation in telecommunications
because the varying delay in arrival time between different components
39
of a signal "smears out" the signal in time.
Refraction & Dispersion
raction
f
e
r
Short wavelengths are “bent”
more than long wavelengths
dispe
rsion
Light is “bent” and the resultant colors separate (dispersion).
Red is least refracted, violet most refracted.
40
Fiber Dispersion
1
2
3
4
5
Fiber
Dispersion
t
t
41
Light Dispersion
Conceptual waves
42
The name "dispersion relation"
originally comes from optics. It is possible to make the effective
speed of light dependent on wavelength by making light pass
through a material which has a non-constant index of refraction,
or by using light in a non-uniform medium such as a waveguide.
In this case, the waveform will spread over time, such that a
narrow pulse will become an extended pulse, i.e. be dispersed.
In these materials,
is known as the group velocity and
correspond to the speed at which the peak propagates, a value
different from the phase velocity.
43
The phase velocity
is the rate at which the phase of the wave propagates in space.
This is the velocity at which the phase of any one frequency
component of the wave will propagate.
You could pick one particular phase of the wave and it would
appear to travel at the phase velocity. The phase velocity is
given in terms of the wave's angular frequency ω and wave
vector k by
vp 

k
44
The group velocity
of a wave is the velocity with which the variations in the shape of
the wave's amplitude (known as the modulation or envelope of the
wave) propagate through space. The group velocity is defined by
the equation

vg 
k
where:
vg is the group velocity;
ω is the wave's angular frequency;
k is the wave number.
The function ω(k), which gives ω as a function of k, is known
as the dispersion relation.
45
Scheme
46
Differences in speed cause spreading or dispersion of wave
packets
47
The group velocity is the speed of the wavepacket
The phase velocity is the speed of the individual waves
Phase velocity = Group Velocity
The entire waveform—the component
waves and their envelope—moves as
one. non-dispersive wave.
Phase velocity = - Group Velocity
The envelope moves in the opposite
direction of the component waves.
Phase velocity > Group Velocity
The component waves move more
quickly than the envelope.
Phase velocity < Group Velocity
The component waves move more
slowly than the envelope.
Group Velocity = 0
The envelope is stationary while the
component waves move through it.
Phase velocity = 0
Now only the envelope moves over
stationary component waves. 48
Dispersion relation
In
physics, the dispersion relation is the relation between
the energy of a system and its corresponding momentum.
For example, for massive particles in free space, the dispersi
on relation can easily be calculated from the definition of kin
etic energy:
1 2 p2
E  mv 
2
2m
i.e. the dispersion relation in this case is a quadratic function
49
Electron Microscope
50
An electron microscope
51
Diffraction patterns produced by a beam of x-rays and electrons passing
through Al foil :
X-rays
electrons
Application: Electron
microscopy
52
Experimental Evidence for Electron Matter Waves
C.J. Davisson and L.H. Germer; G.P. Thomson (1927) Nobel
Prize for Physics 1937
53
3.5 Particle Diffraction
An experiment that confirms the existence of
de Broglie waves
54
Davisson-Germer experiment
55
(1927)
Figure 3.6 The Davisson-Germer experiment.
56
Davisson-Germer apparatus
57
Scattering of electrons from a crystalline
Ni target leads to electron diffraction.
58
Davisson and Germer -- VERY clean nickel crystal.
Interference is electron scattering off Ni atoms.
e
e
e e
e
e
Ni
e e det. e
e
scatter off atoms
e
e
move detector around,
see what angle electrons coming off
59
See peak!!
so probability of angle where detect
electron determined by interference
of deBroglie waves!
# e’s
0
e
e
e e
500
scatt. angle 
e e det. e
e
e
Ni
Observe pattern of scattering
electrons off atoms
Looks like ….
Wave!
60
PhET Sim: Davisson Germer
Careful…
near field view:
D = m doesn’t work here.
For qualitative use only!
61
http://phet.colorado.edu/simulations/schrodinger/dg.jnlp
Typical
polar graphs of electron intensity after the accident are
shown in Fig. 3.7.
The method of plotting is such that the intensity at any angle is
proportional to the distance of the curve at that angle from the
point scattering.
62
The “first order” diffraction maximum (n=1) is usually most intense.
Figure 3.7
Results of the Davisson-Germer experiment, showing how the number of scattered
electrons varied with the angle between the incoming beam and the crystal surface.
The Bragg planes of atoms in the crystal were not parallel to the crystal
surface, so the angles of incidence and scattering relative to one family of these
63
O
planes were both
65
(see Fig. 3.8).
Figure 3.8
The diffraction of the de Broglie waves by the target is
responsible for the results of Davisson and Germer. 64
Reflection
Constructive
interference
65
66
Incident x-ray
Bragg plane
2d sin  n : constructive interference
n  1,2,3,   
Fig. 2-20 X-ray scattering from a cubic crystal.
67
(1) Bragg equation for maxima in the diffraction pattern
n  2d sin
when
?
d =0.091 nm, =65O, n=1
  2 d sin  
2 0 . 091
nm
sin
(2) Use de Broglie formula
Ignore relativistic consideration
KE 
mv 

1
mv2
2
KE
65


0 . 165 nm
  h mv

 54 eV   m 0 c 2  0 . 51 MeV
2mKE
2 9 .1  10  31 kg 54 eV 1 .6  10 19 J
eV


 4 .0  10  24 kg  m s
 
h
 0 .166 nm
mv
Excellent agreement
!
68
Now we use de Broglie’s formula =h/mv to find the expected wavelength
of the electrons.
The electron kinetic energy of 54 eV is small compared with its rest ene
rgy m0c2 of 0.51 MeV, so we can let =1.
1 2
 2 mv  qV

h
 
mv

h

2qV
m
1.225

nm
V
m

h2

2mqV
When V=54 (V)
v
2qV
m
(6.63  10 34 ) 2
150
10
10


m
31
19
V
2  (9.1  10 )  (1.6  10 )
For electron

1.225
54

1.225
 0.166 (nm)
7.3484
69
O
25
The strong diffracted beam at =50O and V=54 V
arises from wavelike scattering from the family of
atomic planes shown, which have a separation
distance d=0.91 A.
70
The experiment consisted of firing an electron beam from an
electron gun on a nickel crystal at normal incidence (i.e.
perpendicular to the surface of the crystal). The electron gun
consisted of a heated filament that released thermally excited
electrons, which were then accelerated through a potential difference
of 54 V, giving them a kinetic energy of 54 eV. An electron detector
was placed at an angle θ = 50° to obtain a maximum reading, and
measured the number of electrons that were scattered at that
particular angle.[1]
According to the de Broglie relation, a beam of 54 eV had a
wavelength of 0.165 nm. The experimental outcome was 0.167 nm,
which closely matched the predictions of Bragg's law
for n = 1, θ = 50°, and for the spacing of the crystalline planes of
nickel (d = 0.091 nm) obtained from previous X-ray scattering
71
experiments on crystalline nickel.[1]
Neutron diffraction by a quartz crystal.
The peaks present directions in which constructive interference
occurred.
72
3.6 Particle in a box
Why the energy of a trapped particle is quantized
73
Figure 3.9
A particle confined to a box of width L.
The particle is assumed to move back and forth along a straight line
between the walls of the box.
74
De Broglie wavelength of trapped
particle
2L
 n
n
n=1, 2, 3,…
Figure 3.10 Wave functions of a particle trapped in a box L
wide.
75
Standing wave
n 
2L
n
n
 1, 2 , 3  
P2
h2

KE 
2m
2 m 2
n 2h 2
n  1, 2 ,3  
En 
2
8 mL
(1)
Energy in quantized by n
E 1 , E 2 , E 3   energy level
n
 quantum number
(2) since no counterpart in classical
physics
E 0
If
Why ?
mv  0 
then

76
Standing waves
First Harmonic
Standing Wave Pattern
Second Harmonic
Standing Wave Pattern
Third Harmonic
Standing Wave Pattern
77
Formation of Standing Waves
A standing wave pattern is a vibrational pattern created
within a medium when the vibrational frequency of the
source causes reflected waves from one end of the
medium to interfere with incident waves from the source.
78
If an upward displaced pulse is introduced at the left end, it will travel
rightward across the snakey until it reaches the fixed end on the right
side.
Upon reaching the fixed end, the single pulse will reflect and undergo
inversion. That is, the upward displaced pulse will become a downward
79
displaced pulse.
Note that there is a point on the diagram in the exact
middle of the medium that never experiences any
displacement from the equilibrium position.
80
The animation below depicts two waves moving through a medium in opposite
directions.
The blue wave is moving to the right and the green wave is moving to the left.
As is the case in any situation in which two waves meet while moving along
the same medium, interference occurs. The blue wave and the green wave
interfere to form a new wave pattern known as the resultant. The resultant in
the animation below is shown in black.
The resultant is merely the result of the two individual waves - the blue wave
and the green wave - added together in accordance with the principle of
superposition.
81
The result of the interference of the two waves above is a new
wave pattern known as a standing wave pattern. Standing waves
are produced whenever two waves of identical frequency interfere
with one another while traveling opposite directions along the
same medium. Standing wave patterns are characterized by
certain fixed points along the medium which undergo no
displacement. These points of no displacement are called nodes
(nodes can be remembered as points of no desplacement). The
nodal positions are labeled by an N in the animation above. The
nodes are always located at the same location along the medium,
giving the entire pattern an appearance of standing still (thus the
name "standing waves"). A careful inspection of the above
animation will reveal that the nodes are the result of the
destructive interference of the two interfering waves. At all times
and at all nodal points, the blue wave and the green wave
82
interfere to completely destroy each other, thus producing a node.
Midway between every consecutive nodal point are points which
undergo maximum displacement. These points are called
antinodes; the anti-nodal nodal positions are labeled by an AN.
Antinodes are points along the medium which oscillate back and
forth between a large positive displacement and a large negative
displacement. A careful inspection of the above animation will
reveal that the antinodes are the result of the constructive
interference of the two interfering waves.
In conclusion, standing wave patterns are produced as the result
of the repeated interference of two waves of identical frequency
while moving in opposite directions along the same medium. All
standing wave patterns consist of nodes and antinodes. The
nodes are points of no displacement caused by the destructive
interference of the two waves. The antinodes result from the
constructive interference of the two waves and thus undergo
maximum displacement from the rest position.
83
Standing waves – mathematically
Take two identical waves traveling in opposite directions
y1 = ym sin (kx - wt)
y2 = ym sin (kx + wt)
yT = y1 + y2 = 2ym cos wt sin kx
This uses the identity
sin a + sin b = 2cos½(a-b)sin½ (a+b)
Positions for which kx = np will ALWAYS have zero field.
If kx = np/2 (n odd), field strength will be maximum for
particular time
84
Standing waves - interpretation
y = 2ym cos wt sin kx
Positions which always have zero field (kx = np) are called
nodes.
Positions which always have maximum (or minimum) field
(kx = = np/2 (n odd)) are called antinodes.
The location of nodes and antinodes don’t travel in time,
but the amplitude at the antinodes changes with time.
85
Standing waves - if ends are fixed
If the amplitude must be zero at the ends of the medium
through which it travels, then standing waves will only be
created if nodes occur at the endpoints.
One example is a string with fixed ends, like a violin
string
Then the wavelength will be some fraction of 2L, where L
is the length of the string/antenna/etc.
86
Standing waves - if ends are fixed
If the amplitude must be zero at the ends of the medium through which
it travels, then standing waves will only be created if nodes occur at
the endpoints.
– One example is a string with fixed ends, like a violin string
Then the wavelength will be some fraction of 2L, where L is the length
of the string/antenna/etc.
L=n/2
L=n/2
87
Figure 3.11 Energy levels of an
electron confined to a box 0.1
nm wide.
88
3.7 Uncertainty Principle 1
We cannot know the future because
we cannot know the present
89
The Uncertainty Principle
Classical physics
– Measurement uncertainty is due to limitations of the measure
ment apparatus
– There is no limit in principle to how accurate a measurement
can be made
Quantum Mechanics
– There is a fundamental limit to the accuracy of a measureme
nt determined by the Heisenburg uncertainty principle
– If a measurement of position is made with precision x and
a simultaneous measurement of linear momentum is made
with precision p, then the product of the two uncertainties
can never be less than h/4
90
(a) A narrow de Broglie wave group.
The position of the particle can be precisely
determined, but the wavelength (and hence
the particle’s momentum) cannot be
established because these are not enough
waves to measure accurately.
x : small
 : not well defined
(b) A wide wave group. Now the wavelength
can be precisely determined but not the
position of the particle.
 : well defined
x : large
91
Figure 3.12
Uncertainty principle
It is impossible to know both the exact position and exact momentum
of an object at the same time.
This principle, which was discovered by Werner Heisenberg in 1927,
is one of the most significant of physical laws.
92
Wave train formed by Fourier transformation
Figure 3.13
An isolated wave group is the result of superposing an infinite number of
waves with different wavelengths.
The narrower the wave group, the greater the range of wavelengths involved.
A narrower de Broglie wave group thus means a well-defined position
(x smaller) but a poorly defined wavelength and a large uncertainty
p in
the momentum of the particle the group represents.
A wide wave group means a more precise momentum but, a less precise
93
position.
The Fourier Transform and its Inverse
The Fourier Transform and its Inverse:

F ( ) 
 f (t ) exp(it ) dt
Fourier Transform

f (t )

1
2

 F ( ) exp(i t ) d 
Inverse Fourier Transform

So we can transform to the frequency domain and back.
Interestingly, these transformations are very similar.
There are different definitions of these transforms. The 2π
can occur in several places, but the idea is generally the 94
same.
The Scale Theorem in action
F()
f(t)
Short
pulse
The shorter the pulse,
the broader the
spectrum !
Mediumlength
pulse
t

t

This is the essence of
the Uncertainty Principle!
Long
pulse
t
95

Fourier Transform with respect to space
If f(x) is a function of position,
F (k )  


f ( x) exp(ikx) dx
x
Y {f(x)} = F(k)
We refer to k as the spatial frequency.
k
Everything we’ve said about Fourier transforms between the t and
 domains also applies to the x and k domains.
96
Pulse
Wave group
Wave train
Gaussian
distribution
Wave function
Fourier transform
Figure 3.14
The wave function and Fourier transform for (a) a pulse, (b) a wave
group, © a wave train, and (d) a Gaussian distribution.
A brief disturbance needs a broader range of frequencies to describe it
than a disturbance of greater duration.
97
The Fourier transform a Gaussian function is also a Gaussian function.
 ( x ), g ( k )
 Gaussian distribution
 : uncertaint y of f ( x 0 )
 ( x )  x
 ( k )  k
minimum value of
1
x  k 
2
1
x  k 
2
position of uncertainty
Fig.3.14(d)
wave number of uncertainty
98
Gaussian function
f x  
1
  x  x 0 2
e
2 
Px1x2 
/ 2
2
x2
 f x dx
x1
Px 0   
x0 
 f x  dx  0 .683
x0 
Fig. 3.15 A Gaussian distribution.
The probability of finding a value of x is given by the Gaussian function
f(x).
The mean value of x is x0 and the total width of the curve at half its
maximum value is 2.35, where  is the standard deviation of the
distribution.
The total probability of finding a value of x within a standard deviation of
99
x0 is equal to the shaded area and is 68.3 percent.
xk 
1
2
- analogous to minimum bandwidth/minimum pulsewidth
De Broglie wavelength of a particle
k
2


2p
h

h
p
hk
p
2
 xp  x 
hk
p 
2
hk
h
 x  k 
2
2
1
 x  k 
2
1 h
h
h
xp  


2 2 4 4
Uncertainty principle
h
xp 
4
(3.21)
100
H-bar
h
34

 1.054 10 J  s
2
The uncertainty principle becomes

xp 
2
3.22
101
Figure 3.16
The wave packet that corresponds to a moving packet is a composite of many
individual waves, as in Fig. 3.13.
The phase velocities of the individual waves vary with their wave lengths.
As a results, as the particle moves, the wave packet spreads out in space.
The narrower the original wavepacket－that is, the more precisely we know its
position at that time －the more it spreads out because it is made up of a102
greater span of waves with different phase velocities.
3.8 Uncertainty Principle II
A particle approach gives the same result
103
Figure 3.17
An electron cannot be observed without changing its momentum.
104
The Uncertainty Principle
Measurement disturbes the system
105
h

??
 x 
p 
momentum change of electron

h

photon momentum
p
A reasonable estimate of the minimum uncertainty in the me
asurement might be one photon wavelength, so that
x  
 x p  h

This result is consistent with Eq. (3.22), xp  .
2
(3.24)
(3.25)
106
Details
107
The act of observation (Compton Scattering)
Observations of particle motion by means of scattered illumination.
When the incident wavelength is reduced to accommodate the size of
the particle, the momentum transferred by the photon becomes large
enough to disturb the observed motion.
108
Compton Scattering: Shining light to observe electron
109
Act of watching: A Through Experiment
110
Diffraction by a circular aperture (Lens)
111
Resolving Power of Light thru a Lens
Image of 2 separate point sources formed by a converging lens
of diameter d, ability to resolve them depends on  and d
because of the inherent diffraction in image formation.
112
Putting it all togethor: act of observing an electron
113
3.9
Applying the uncertainty principle
A useful tool, not just a negative statement
114
It is worth keeping in mind that the lower limit of
for xp is rarely attained.
More usually
xp  h
/2
xp   , or even (as we just saw)
115
Example 3.7
A typical atomic nucleus is about 5.0x10-15 m in radius.
Use the uncertainty principle to place a lower limit on the energy
an electron must have if it is to be part of a nucleus.
Solution
r  5  10  15 m
p 
r
x
r

 1.1  10  20 kg  m/s
2x
If this is the uncertainty in a nuclear electron’s momentum,
the momentum p itself must be at least comparable in
magnitude.
An electron with such a momentum has a kinetic energy
KE many times greater than its rest energy moc2 (=0.51 MeV).
KE  pc  3.3 1012 J  20 MeV
KE  20 MeV
20 MeV electron is never found !
Electron can not exit in a nuclear !!
116
Ex. 3.8
A hydrogen atom is 5.3 x10-11 m in radius.
Use the uncertainty principle to estimate the minimum energy
an electron can have in this atom.
Solution
5 .3  10 11 m   x
p 

 9 . 9  10
2x
 25
kg  m/sec
An electron whose momentum is of this order of magnitude behaves like
a classical particle, and its kinetic energy is
p2
KE 
 5.4  10 19 J
2m
which is 3.4 eV. The kinetic energy of an electron in the lowest energy
level of a hydrogen atom is actually 13.6 eV.
117
Energy and time
We might wish to measure the energy E emitted during the time in
terval t inan atomic process.
If the energy is in the form of em waves, the limited time available
restricts the accuracy with which we can determine the frequency
 of the waves.
Let us assume that the minimum uncertainty in the number of
waves we count in a wave group is one wave.
Since the frequency of the waves under study is equal to the
number of them we count divided by the time interval, the
uncertainty  in our frequency measurement is
1
 
t
In general
N
 
t
118
1
 
t
E  h 
h
E 
t
Uncertainty in
energy and time

E  t 
2
(3.26)
119
Example 3.9
An “excited” atom gives up its excess energy by emitting a photon of
characteristic frequency, as described in Chap. 4.
The average period that elapses between the excitation of an atom
and the time it radiates is 1.0x10-8 s.
Find the inherent uncertainty in the frequency of the photon.
Solution
 t  10  8 sec
h
The photon energy is uncertain by the amount

E 
2t
 E  5.3  10  27 J
The corresponding uncertainty in the frequency of light is
Natural linewidth
 
E
 8  10 6 Hz
h
120
Energy-time uncertainty relation
Et   / 2
Transitions between energy levels of atoms are not perfectly
sharp in frequency.
n=2
An electron in n = 3 will spontaneo
usly decay to a lower level after a
lifetime of order t~10-8 s.
n=1
There is a corresponding ‘spread’ in
the emitted frequency
Intensity
E  h 32
n=3
 32
121
 32 Frequency
This is the irreducible limit to the accuracy with which we can determine
the frequency of the radiation emitted by an atom.
As a result, the radiation from a group of excited atoms does not appear
with the precise frequency .
For a photon whose frequency is, say 5.0x1014 Hz, /=1.6x10-8.
In practice, other phenomena such as the
doppler effect contribute
more than this to the broadening of spectral lines.
122
Doppler broadening
In atomic physics, Doppler broadening is the broadening
of spectral lines due to the Doppler effect in which the
thermal movement of atoms or molecules shifts the
apparent frequency of each emitter.
The many different velocities of the emitting gas result in
many small shifts, the cumulative effect of which is to
broaden the line. The resulting line profile is known as a
Doppler profile.
The broadening is dependent only on the wavelength of
the line, the mass of the emitting particle and the
temperature, and can therefore be a very useful method
for measuring the temperature of an emitting gas.
123
Line Broadening
Molecular absorption takes place at distinct
wavelengths (frequencies, energy levels)
Actual spectra feature absorption “bands”
with broader features
1. Natural line width from
uncertainty
principle (very small)
(Lorentzian line shape)
2. Pressure (collisional) broadening
(Lorentzian line shape)
3. Doppler broadening (Gaussian line shape)
124