WAVES AND PARTICLES

1. We think of particles as matter highly concentrated in some volume of space, and of waves
as being highly spread out. Think of a cricket ball, and of waves in the ocean. The two are
completely different! And yet today we are convinced that matter takes the form of waves
in some situations and behaves as particles in other situations. This is called wave-particle
duality. But do not be afraid - there is no logical contradiction here! In this lecture we shall
first consider the evidence that shows the particle nature of light.
2. The photoelectric effect, noted nearly 100 years ago, was
crucial for understanding the nature of light. In the diagram
shown, when light falls upon a metal plate connected to
the cathode of a battery, electrons are knocked out of the
plate. They reach a collecting plate that is connected to
the battery's anode, and a current is observed. A vacuum
is created in the apparatus so that the electrons can travel
without hindrance. According to classical (meaning old!)
physics we expect the following:
a)As intensity of light increases, the kinetic energy of the ejected electrons should increase.
b)Electrons should be emitted for any frequency of light ν , so long as the intensity of the
light is sufficiently large.
But the actual observation was completely different and showed the following:
a)The maximum kinetic energy of the emitted ele
0
ctrons was completely independent of
the light intensity, but it did depend on .
b)For < (i.e. for frequencies below a cut-off frequency) no electrons are emitted no
matter how
ν
ν ν
large the light intensity was.
3. In 1905, Einstein realized that the photoelectric effect could be explained if light actually
comes in little packets (or quanta) of energy with the energy of each
34
quantum .
Here is a universal constant of nature with value 6.63 10 Joule-seconds, and is
known as the Planck Constant. If an electron absorbs a single photon, it would be able to
E h
h h
ν
−
=
= ×
leave the material if the energy of that photon is larger than a certain amount . is
called the work function and differs from material to material, with a value varying from
2-5 electron v
W W
max olts. The maximum KE of an emitted electron is then . We
visualize the photon as a particle for the purposes of this experiment. Note that this is
completely different from our earlier
K =hν −W
understanding that light is a wave!
Incident light Oscillating electron Emitted light
θ
e p
ν ′ Before After p
Electron
Incoming photon
ν p
scattered
scattered
4. That light is made of photons was confirmed by yet another experiment, carried out by
Arthur Compton in 1922. Suppose an electron is placed in the path of a light beam. What
will happen? Because light is electromagnetic waves, we expect the electron to oscillate
with the same frequency as the frequency of the incident light . But because a charged
particle radiates em waves, we
ν
expect that the electron will also radiate light at frequency
ν. So the scattered and incident light have the sameν. But this is not what is observed!
To explain the fact that the scattered light has a different frequency (or wavelength),
Compton said that the scattering is a collision between particles of light and electrons.
But we kno
( )2 22 2 4 1/ 2
w that momentum and energy is conserved in scattering between particles.
Specifically, from conservation of energy . The last
term is the energy of the scattered electr
e e e hν +mc =hν′+ pc +mc
on with mass . Next, use the conservation e m
of momentum. The initial momentum of the photon is entirely along the zˆ direction,
zˆ . By resolving the components and doing a bit of algebra, you can get
the change in waveleng
e
h
ν λ ν′ p= =p+p
( ) ( )
12
th 1 cos 1 cos , where the Compton
wavelength 2.4 10 m. Note that is always positive because cos
has magnitude less than 1. In other words, the frequency of the sca
c
e
c
e
h
m c
h
m c
λ λ θ λ θ
λ − λ λ θ
′− = − = −
= = × ′−
ttered photon is
always less than the frequency of the incoming one. We can understand this result
because the incoming photon gives a kick to the (stationary) electron and so it loses
energy. Since , it follows that the outgoing frequency is decreased. As remarked
earlier, it is impossible to understand this from a classical point of view. We shall now
see how the Compt
E=hν
on effect is actually observed experimentally.
X-ray source
Target
Crystal
(selects
wavelength)
Collimator
(selects angle)
θ
Detector
5. In the apparatus shown, X-rays are incident
upon a target which contains electrons. Those
X-rays which are scattered in a particular
angle are then selected by the collimator
and are
θ
incident upon a crystal. The crystal
diffracts the X-rays and, as you will recall
from the lecture on diffraction, 2 sin
is the necessary condition. It was thus determined
that the
d θD =nλ
( )
changed wavelength of the scattered
follows 1 cos . c λ′−λ =λ − θ
6. Here is a summary of photon facts:
a)The relation between the energy and frequency of a photon is .
b)The relativistic formula relating energy and momentum for photons is . Note
E h
E pc
= ν
=
that in general 2 22 24 for a massive particle. The photon has 0.
c)The relation between frequency, wavelength, and photon speed is .
d)From the abbove, the momentum-wavelength re
E pc mc m
λν c
= + =
=
lation for photons is .
e)An alternative way of expressing the above is: and . Here 2
and 2 , ( is pronounced h-bar).
2
f)Light is always detected as packet
p h h
c
E p k
k h
ν
λ
ω ω πν
π
λ π
= =
= = =
= ≡
􀀽 􀀽
􀀽 􀀽
s (photons); if we look, we never observe half a
photon. The number of photons is proportional to the energy density (i.e. to square of
the electromagnetic field strength).
7. So light behaves as if made of particles. But do all particles of matter behave as if they
are waves? In 1923 a Frenchman, Louis de Broglie, postulated that ordinary matter can
have wave-like properties with the wavelength related to the particle's momentum
in the same way as for light, . We shall call the de Broglie (pronounced as
Deebrolee!) wavelength.
8. Let us estimate some typical
p
h
p
λ
λ= λ
34
34
De Broglie wavelengths:
a) The wavelength of 0.5 kg cricket ball moving at 2 m/sec is:
6.63 10 6.63 10 m
0.5 2
h
p
λ
−
− ×
= = = ×
×
θi
θi
10
2 2
2
This is extremely small even in comparison to an atom, 10 m.
b) The wavelength of an electron with 50eV kinetic energy is calculated from:
2 2 e e
K p h h
m m
λ
λ
−
= = ⇒ = 1.7 10 10m
2
Now you see that we are close to atomic dimensions.
9. If De Broglie's hypothesis is correct, then we can expect that electron waves will undergo
interference just like li
e m K
= × −
ght waves. Indeed, the Davisson-Germer experiment (1927) showed
that this was true. At fixed angle, one find sharp peaks in intensity as a function of electron
energy. The electron waves hitting the atoms are re-emitted and reflected, and waves from
different atoms interfere with each other. One therefore sees the peaks and valleys that
are typical of interference (or diffraction) patterns in optical experiments.
10. Let us look at the interference in some detail. When electrons fall on a crystalline
surface, the electron scattering is dominated by surface layers. Note that the identical
scattering planes are oriented perpendicular to the surface. Looking at the diagram,
we can see that constructive interference happens when (cos cos ) . When
this condition is satified, there is
r i a θ − θ =nλ
a maximum intensity spot. This is actually how we
find a and determine the structure of crystals.
a
θi
θr
cos i a θ
Detecting
screen
d sinθ
D
θ
d
Incoming coherent
beam of electrons
y
11. Let's take a still simpler situation: electrons are incident upon a metal plate with two tiny
holes punched into it. The holes - spearated by distance - are very close together. A
scree
d
n behind, at distance , is made of material that flashes whenever it is hit by an
electron. A clear interference pattern with peaks and valleys is observed. Let us analyze:
there will be a
D
maximum when sin . If the screen is very far away, i.e. ,
then will be very small and sin . So we then have , and the angular
separation between two adjacent minima is
d n D d
n
d
θ λ
λ
θ θ θ θ
λ
θ
= >>
≈ ≈
Δ ≈ . The position on the screen is y, and
tan . So the separation between adjacent maxima is and hence
. This is the separation between two adjacent bright spots. You can see
d
y D D y D
y D
d
θ θ θ
λ
= ≈ Δ ≈ Δ
Δ = from the
experimental data that this is exactly what is observed.
12. The double-slit experiment is so important that we need to discuss it further. Note the
following:
a) It doesn't matter whether we use light, electrons, or atoms - they all behave as waves.
in this experiment. The wavelength of a matter wave is unconnected to any internal
size of particle. Instead it is determined by the momentum, .
b) If one slit is close
h
p
λ =
d, the interference disappears. So, in fact, each particle goes through
both slits at once.
c) The flux of particles arriving at the slits can be reduced so that only one particle arrives
at a time. Interference fringes are still observed !Wave-behaviour can be shown by a
single atom. In other words, a matter wave can interfere with itself.
d) If we try to find out which slit the particle goes through the interference pattern
vanishes! We cannot see the wave/particle nature at the same time.
All this is so mysterious and against all our expectations. But that's how Nature is!
13. In real life we are
perfectly familiar with seeing a cricket ball at rest - it
has both a fixed position and fixed (zero) momentum.
But in the mic
Heisenberg Uncertainty Principle.
roscopic world (atoms, nuclei, quarks,
and still smaller distance scales) this is impossible.
Heisenberg, the great German physicist, pointed out
that if we want to see an electron, then we have to
hit it with some other particle. So let's say that a
photon hits an electron and then enters a detector.
It will carry information to the detector of the position
and velocity of the electron, but in doing so it will have
changed the momentum of the electron. So if the
electron was initially at rest, it will no longer be so
afterwards. The act of measurement changes the state
of the system! This is true no matter how you do the
experiment. The statement of the principle is:
"If the position of a particle can be fixed with accuracy , then the maximum accuracy
with which the momentum can be fixed is , where / 2." We call the
position uncertainty
x
p xp x
Δ
Δ Δ Δ ≥ 􀀽 Δ
, and the momentum uncertainty. Note that their product is fixed.
Therefore, if we fix the position of the particle (make very small), then the particle
will move about randomly very
p
x
Δ
Δ
fast (and have Δp very large).
14. By using the De Broglie hypothesis in a simple gedanken experiment, we can see how
the uncertainty principle emerges. Electrons are incident upon a single slit and strike a
screen far away
y
. The first dark fringe will be when sin . Since is small, we can
use sin , and so . But, on the other hand, tan and . So,
. But is really the uncertainty in
y y
x x
W
p p
W p p
p
W
h/ W
θ λ θ
λ
θ θ θ θ θ
λ
λ
=
Δ Δ
≈ ≈ = ≈
Δ
= the y position and we should call it Δy.
Momentum uncertainty in
the y component
32 ν Frequency
Δν 32
I
nt
ens
it
y
y Thus we have found that . The important point here is that by localizing the
position of the electron to the width of the slit, we have forced the electron to acquire
a momentum
p y h
y
Δ Δ ≈
y in the direction whose uncertainty is .
15. In a proper course in quantum mechanics, one can give a definite mathematical meaning
to and etc, and derive the uncertainty relations:
x
y p
x p
Δ
Δ Δ
/ 2, / 2, / 2
Note the following:
a) There no uncertainty principle for the product . In other words, we can know
in principl
x y z
y
x p y p z p
x p
Δ Δ ≥ Δ Δ ≥ Δ Δ ≥
Δ Δ
􀀽 􀀽 􀀽
e the position in one direction precisely together with the momentum in
another direction.
b) The thought experiment I discussed seems to imply that, while prior to experiment
we have well defined values, it is the act of measurement which introduces the
uncertainty by disturbing the particle's position and momentum. Nowadays it is more
widely accepted that quantum uncertainty (lack of determinism) is intrinsic to the
theory and does not come about just because of the act of measurement.
16. There is also an Energy-Time Uncertainty Principle which states that / 2.
This says that the principle of energy conservation can be violated by amount ,
but only for a short time given by . The quantity is called the uncertain
E t
E
t E
Δ Δ ≥
Δ
Δ Δ
􀀽
ty in
the energy of a system.
17. One consequence of / 2 is that the level of an atom does not have an exact
value. So, transitions between energy levels of atoms are never perfectly
ΔEΔt ≥ 􀀽
8
sharp in
frequency. So, for example, as shown below an electron in the 3 state will decay
to a lower level after a lifetime of order 10 s. There is a corresponding "spread"
in
n
t −
=
≈
the emitted frequency.
n = 3
n = 2
n = 1
32 E=hν
The Microscopic World
•ATOMS--10-10 m
•NUCLEI--10-14 m
•NUCLEONS--10-15 m
•QUARKS--???