Lecture 20 – Particle and Waves
Ch 9
pages 446-451; 455-463
Summary of lecture 19
 Planck’s theory of black body radiation proposed that energy is
emitted by oscillators in discrete packets E=hn. These packets,
called photons, are treated as energy particles
 Bohr extended Planck’s hypothesis to provide an explanation for
the stability of atoms and define energy levels for a hydrogenic
atom:
2 4
Z e m 1
En  2 2 2
8 0 h n
Combining the two, if an atom absorbs energy, its electron will be
promoted from the nth orbit to, say, the kth orbit, the frequency of
the energy particle or photon emitted by the atom is given by:
Z 2 e4 m  1
1 
E  Ek  En  hv  2 2  2  2 
8 0 h  n
k 
Wave-Particle Duality
Up until now we have dealt with the first fundamentally
different concept of quantum mechanics, quantization of
energy. The second even more radical departure from
classical principles (and every day’s intuition) is the dual
nature of matter
Several experimental observations suggested that light waves
can have particle-light properties
These observations can be summarized to say that energy can
have particle-like properties
Waves vs. Particles
• We began our discussion by defining light in
terms of wave-like properties.
• But Planck’s relationships suggest that light
can be thought of as a series of energy
“packets” or photons.
Wave-Particle Duality
Light incident on a metal surface can, under some
circumstances, eject electrons from the surface (photoelectric
effects). If the light is below a certain frequency, no electrons
are ejected. If the light is above a certain frequency, electrons
will be emitted regardless of how low the intensity of the light
is. The maximum kinetic energy of the ejected electrons is
independent of the intensity of the light and dependent on its
frequency
The scattering of X-rays from carbon and other materials is
explained by assuming the X-ray photons have particle-like
collisions with atoms and electrons (Compton effect); that is
to say, they have momentum though no mass. These
observations can be summarized to say that energy can have
particle-like properties.
The Photoelectric Effect
• Shine light on a metal and observe
electrons that are released.
metal
• Find that one needs a minimum
amount of photon energy to see
electrons (“no”).
• Also find that for n ≥ no,
number of electrons increases
linearly with light intensity .
The Photoelectric Effect
• Finally, notice that as frequency
of incident light is increased,
kinetic energy of emitted eincreases linearly.
0
n0
Frequency (n)
1
2
men  hn photon  
2
 = energy needed to release e-

• Light apparently
behaves as a particle.
The Photoelectric Effect
• For Na with  = 4.4 x 10-19 J,
what wavelength corresponds to no?
0
1
2
men  hn photon  
2
hn =  = 4.4 x 10-19 J
0
hc/l = 4.4 x 10-19 J
n0

Frequency (n)
6.626x1034 J.s3x10 8 m /s

hc
l

19
4.4 x10 J
4.4 x1019 J 
l = 4.52 x 10-7 m = 452 nm
Particles as waves
• Electrons shine through a crystal and look at pattern
of scattering.
• Diffraction can only be explained by treating electron
as a wave instead of a particle.
Wave-Particle Duality
At about the same time, Davisson and Germer made the
reciprocal observation; they diffracted electrons (which are of
course particles of with a certain mass) against single crystals
of nickel and observed a diffraction particle as would be
produced by X-ray diffraction. Incidentally, electron
diffraction is nowadays a major technique of structure
determination
Changing the electron speed they could change the
momentum and they measured the diffraction pattern as a
function of momentum; based on well-known classical wave
diffraction theory and the experimental results, they
calculated the wavelength associated with the electron and
they discovered the following relationship:
h
l
p
Wave-Particle Duality
l
h
p
This is called De Broglie’s wavelength and every
particle (e.g. neutron, protons, etc.) has been
experimentally demonstrated to have a
characteristic De Broglie’s wavelength which
depends on its momentum only.
In 1925, deBroglie proposed an explanation for why an
electron does not decay from a Bohr orbit. Recalling Planck’s
hypothesis that radiation is quantized in particles of energy
E=hn with momentum, deBroglie hypothesized that electrons
have wavelengths
Wave-Particle Duality
l
h
p
If the wavelength of an electron wave orbiting a nucleus is an
integral multiple of the length of the orbit, a standing electron
wave results (see diagram below). deBroglie’s equation states
that within a Bohr orbit:
nl  2R; n  1,2,3...
Using Bohr’s expression for the quantization of angular
momentum:
nh
nh
h
mVR 
2
 2R  nl 
mV
l 
mV

h
p
Particle and Waves
• What is a standing wave?
• A standing wave is a motion in
which translation of the wave
does not occur.
• In the guitar string analogy
(illustrated), note that standing
waves involve nodes in which no
motion of the string occurs.
• Note also that integer and halfinteger values of the wavelength
correspond to standing waves.
Particle and Waves
Louis de Broglie suggests that for the e- orbits envisioned
by Bohr, only certain orbits are allowed since they satisfy
the standing wave condition.
not allowed
De Broglie’s wavelength
DeBroglie’s particle wave hypothesis can be used to obtain
quantized energy expressions for very simple problems.
Example: For a particle in a box, the length of the box L must
equal an integral multiple of half DeBroglie’s wavelength (to
have constructive interference between waves, otherwise
there would be destructive interference and the waves would
cancel out):
h
l
l
n
p
n
l
2

2
L
nh
L
2p
p
2
p2
1  nh 
n2h2
En 

  
2m 2m  2 L 
8mL2
nh
2L
Wave-Particle Duality
l
h
p
Both electrons and light waves collide with surfaces
with finite momentum, although we normally associate
momentum with particles. Both electrons and X-rays
diffract off of surfaces, although we normally associate
diffraction and interference with radiation waves.
However, particles are localized in space, while waves
are not. How do we treat particles as waves and
viceversa?
Wave-Particle Duality
l
h
p
In classical physics, radiation waves are represented
by plane wave functions that are periodic in time and
space. An example of a plane wave traveling in the x
direction is
 ( x, t )  Ae i ( kx t )
Where A is the amplitude of the wave, k=2/l is the
propagation constant,  is the angular frequency   2n
e iz  cos z  i sin z, i   1
If the wave function is independent of time, we have a
stationary or standing wave
 ( x, t )  Ae i ( kx )
For simplicity we show only the sine component below
Wave-Particle Duality
X
181
166
151
136
121
106
91
76
61
46
31
1.5
1
0.5
0
-0.5
-1
-1.5
16
1 Wave
1
h
p
sin(kx)
l
A wave propagates through space at its wave velocity,
which is viewed as the time it takes for the wave peak
to shift by one wavelength. This diagram expresses the
fact that waves are continuous functions in space and
time, whereas particles are localized in space. How can
a wave function represent a particle?
Wave-Particle Duality
A matter wave packet can be localized to a single point in
space if we superimpose an infinite number of waves with
differing wavelengths l. For a stationary wave center at

x=x0:
sin  x0  x   0
1
  x 
0
2

ei ( x0  x ) k dk 
 0
 x0  x   0
In other words, if we superimpose a finite number of waves
2
with wavelengths varying between l   2 l   
0
0
0
0
The resulting wave packet has the form:
( x ) 
sin( x 0  x )k 0
( x 0  x )k 0
181
166
151
136
121
91
X
106
76
61
46
31
1
1.5
1
0.5
0
-0.5
-1
-1.5
16
181
163
145
127
91
X
109
73
55
37
sin(kx)
1 Wave
19
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
1
Sin(x)/x
Particle Wave
Wave-Particle Duality
As more waves are added the wave “packet” gets narrower.
In the limit of an infinite number of waves covering all
wavelength values, then we can localize a particle to a single
point in space
Using deBroglie’s expression, the width of the central lobe of
the packet is:
x 
1
0

l0
h

2 2 p0
Associating a packet of wave to a particle has an unexpected
consequence that leads to perhaps the most radical of all
ideas of quantum mechanics.
X
181
163
145
127
109
91
73
55
37
19
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
1
Sin(x)/x
Particle Wave
Heisenberg’s principle
x 
1
0

l0
h

2 2 p0
From the relationship given above, when we use packet of
waves to represent a particle localized within a certain space
we must superimposes waves covering a range of p such that
p=2p0. Thus
h
x ( 2 p 0 ) 
2
The range of momentum 2p0 represents the uncertainty with
which the momentum associated with the particle is known.
Thus, if we measure how well localized is a certain particle
and also its momentum, we cannot measure both with
infinite precision; in other words the precision with which we
can measure location and momentum is limited by the
relationship:
x 2 p 
h
2
Heisenberg’s principle
x 2 p 
h
2
Heisenberg’s observed that this is a best-case scenario. In
general:
1 h
x p 
2 2
This is the Heisenberg Uncertainty Principle, which limits
our ability to define the position of a particle at a particular
time. At best we can calculate the probability that a particle
is located at a particular position at the time of a
measurement
A similar expression can be found for energy and time:
E  t 
h
2
Heisenberg’s principle
x p 
1 h
2 2
E  t 
h
2
The Heisenberg Uncertainty Principle is perhaps that must
controversial theory of modern times. The basic principles of
classical mechanics can be summarized as follows:
 There is no limit to the accuracy with which dynamical
variables (e.g. position, momentum, time, and energy) can be
determined simultaneously, except the limit imposed by
instruments of measurement.
 There is no restriction on the number of dynamical
variables that can be measured simultaneously
 The velocity of a particle, and hence its kinetic energy, is a
continuous function. There are no restrictions on the values
that the energy may attain.
Heisenberg’s principle
x p 
1 h
2 2
E  t 
h
2
Heisenberg’s Uncertainty Principle imposes a limit on the
accuracy of measurements of the dynamical variables x and
px
Together with the quantization condition E=nhn,
Heisenberg’s Uncertainty Principle overturned the basic
principles of classical mechanics.