MODULE IN MODERN PHYSICS
Chapter 3: Wave Properties of Particles
DE BROGLIE WAVES
In 1923, Louis de Broglie, a French physicist, proposed a hypothesis to explain the
theory of the atomic structure. By using a series of substitution de Broglie hypothesizes
particles to hold properties of waves. Within a few years, de Broglie's hypothesis was tested
by scientists shooting electrons and rays of lights through slits. What scientists discovered
was the electron stream acted the same was as light proving de Broglie correct.
Deriving the de Broglie Wavelength
De Broglie derived his equation using well established theories through the following series
of substitutions:
De Broglie first used Einstein's famous equation relating matter and energy:
E=mc2



E = energy,
m = mass,
c = speed of light
Using Planck's theory which states every quantum of a wave has a discrete amount of energy
given by Planck's equation:
E=hν



E = energy,
h = Plank's constant (6.626 x 10-34 J s),
ν= frequency
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Since de Broglie believed particles and wave have the same traits, he hypothesized that the
two energies would be equal:
mc2=hν
Because real particles do not travel at the speed of light, De Broglie submitted velocity (v) for
the speed of light (c).
mv2=hν
Through the equation λ, de Broglie substituted v/λ for ν and arrived at the final expression
that relates wavelength and particle with speed.
mv2 =
Hence De Broglie wavelength:
Although de Broglie was credited for his hypothesis, he had no actual experimental
evidence for his conjecture. However, he was able to show that it accounted in a natural way
for the energy quantization – the restriction to certain specific energy values – that Bohr had
to postulate in his 1913 model of the hydrogen atom (this will be discussed in the next
chapter). Within a few years, in 1927, Clinton J. Davisson and Lester H. Germer shot
electron particles onto a nickel crystal. What they saw was the diffraction of the electron
similar to waves diffraction against crystals (x-rays). In the same year, an English physicist,
George P. Thomson fired electrons towards thin metal foil providing him with the same
results as Davisson and Germer.
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WAVES OF PROBABILTY
In water waves, the quantity that varies periodically is the height of the water surface.
In sound waves, it is pressure. In light waves, electric and magnetic field varies. What is it
that varies in the case of matter waves?
Fundamental particles, such as electrons, may be described as particles or
waves. Electrons may be described using a wave function. The wave
function's symbol is the Greek letter psi, Ψ.
Wave function, in quantum mechanics, variable quantity that mathematically
describes the wave characteristics of a particle. The value of the wave function associated
with a moving body at a particular point x, y, z in space at the time t is related to the
likelihood of finding the body there at the time.
By analogy with waves such as those of sound, a wave function, designated by the
Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave
(or de Broglie wave), although for such waves amplitude has no physical significance. The
square of the wave function, Ψ2, however, does have physical significance: the probability of
finding the particle described by a specific wave function Ψ at a given point and time is
proportional to the value of Ψ2.
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ELECTRON MICROSCOPE
The wave nature of moving electrons is the basis of
the electron microscope, the first of which was built in
1932. The resolving power of any optical instrument,
which is limited by diffraction, is proportional to the
wavelength of whatever is used to illuminate the specimen.
In the case of a good microscope that uses visible light, the
maximum useful magnification is about 500x; higher
magnifications give larger image but do not reveal any
more detail. Fast electrons, however, have wavelengths
very much shorter than those of visible light and are easily
controlled by electric and magnetic fields because of their
charge. X-rays also have shorter wavelengths, but is not
yet possible to focus the, adequately.
In an electron microscope, current-carrying coils
produce magnetic fields that act as lenses to focus an
electron beam on a specimen and then produce an enlarged
image on a fluorescent screen or photographic plate
(Figure 3.5 on the right). To prevent the beam from being
scattered and thereby blurring the image, a thin specimen
is used and the entire system is evacuated.
The technology of magnetic ―lenses‖ does not
permit the full theoretical resolution of electron waves to
be realized in practice. For instance, 100 keV electrons
have wavelengths of 0.037 nm, but the actual resolution
they can provide in an electron microscope maybe only
about 0.01 nm. However, this is still a great improvement
on the ~200nm resolution of an optical microscope, and
magnifications over 1,000,000x have been achieved with
electron microscopes.
An electron microscope.
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Types of electron microscope
There are two main types of electron microscope – the transmission EM (TEM) and
the scanning EM (SEM).
TEM stands for Transmission
Electron Microscope (TEM). A TEM is a type of
electron microscope that uses a broad beam of
electrons to create an image of a sample’s internal
structure. A beam of electrons is transmitted
through a sample, creating an image that details a
sample’s morphology, composition, and crystal
structure.
The
transmission
electron
microscope is used to view thin specimens (tissue
sections, molecules, etc) through which electrons
can pass generating a projection image. The TEM
is analogous in many ways to the conventional
(compound) light microscope. TEM is used,
among other things, to image the interior of cells
(in thin sections), the structure of protein
molecules (contrasted by metal shadowing), the
organization of molecules in viruses and
cytoskeletal filaments (prepared by the negative
staining technique), and the arrangement of protein
molecules in cell membranes (by freeze-fracture).
How does a TEM work?
An electron source sends a beam of electrons through an ultrathin sample. When the
electrons penetrate the sample, they pass through lenses below. This data is used to create
images directly on a fluorescent screen or onto a computer screen using a charge-coupled
device (CCD) camera.
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SEM stands for Scanning Electron Microscope.
A SEM is a kind of electron microscope that uses a
fine beam of focused electrons to scan a sample’s
surface. The microscope records information about
the interaction between the electrons and the
sample, creating a magnified image. SEM has the
potential to magnify an image up to 2 million
times. Because of its great depth of focus, a
scanning electron microscope is the EM analog of a
stereo light microscope. It provides detailed images
of the surfaces of cells and whole organisms that
are not possible by TEM. It can also be used for
particle counting and size determination, and for
process control. It is termed a scanning electron
microscope because the image is formed by
scanning a focused electron beam onto the surface
of the specimen in a raster pattern. The interaction
of the primary electron beam with the atoms near
the surface causes the emission of particles at each
point in the raster (e.g., low energy secondary
electrons, high energy back scatter electrons, Xrays and even photons). These can be collected
with a variety of detectors, and their relative
number translated to brightness at each equivalent
point on a cathode ray tube. Because the size of the
raster at the specimen is much smaller than the viewing screen of the CRT, the final picture is
a magnified image of the specimen. Appropriately equipped SEMs (with secondary,
backscatter and X-ray detectors) can be used to study the topography and atomic composition
of specimens, and also, for example, the surface distribution of immuno-labels.
How does SEM work?
To obtain a high-resolution image, an electron source (also known as an electron gun)
emits a stream of high-energy electrons towards a sample. The electron beam is focused
using electromagnetic lenses. Once the focused stream reaches the sample, it scans its surface
in a rectangular raster.
The interaction between the electron beam and the sample creates secondary
electrons, backscattered electrons, and X-rays. These interactions are captured to create a
magnified image.
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SEM vs TEM advantages
Scanning Electron Microscopes and Transmission Electron Microscopes each contain unique
advantages when compared to the other.
In comparison to TEMs, SEMs:
 Cost less
 Take less time to create an image
 Require less sample preparation
 Accept thicker samples
 Can examine larger samples
In comparison to SEMs, TEMs:
 Create higher resolution images
 Provide crystallographic and atomic data
 Create 2-D images that are often easier to interpret than SEM 3-D images
 Allow users to examine more characteristics of a sample
SEM vs TEM similarities and differences
There are many similarities between SEMs and TEMs. The components of these two
high-resolution microscopes are very similar. Each has an electron source/gun that emits an
electron stream towards a sample in a vacuum, and each contains lenses and electron
apertures to control the electron beam and capture images.
But the differences in function between the two are vast. They differ in how they
work, the types of samples that they require, the resolution of images that they create, and
more.
The table summarizes the differences between Scanning Electron Microscopes and
Transmission Electron Microscopes.
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PARTICLE DIFFRACTION
The Davisson—Germer experiment (1927) was the first measurement of the
wavelengths of electrons. C. J. Davisson, who worked in the Bell Research Laboratories,
received the Nobel Prize in Physics for the year 1937 together with George P. Thomson from
the University of Aberdeen in Scotland, who independently also found experimental
indications of electron diffraction. According to the Copenhagen Interpretation of Quantum
Mechanics, wave-particle duality leads to particles also exhibiting wave-like properties like
extension in space and interference.
Clinton J. Davisson (1881–1958) and Lester H. Germer (1896–1971) investigated the
reflection of electron beams on the surface of nickel crystals. When the beam strikes the
crystal, the nickel atoms in the crystal scatter the electrons in all directions. Their detector
measured the intensity of the scattered electrons with respect to the incident electron beam.
Their normal polycrystalline samples exhibited a very smooth angular distribution of
scattered electrons. Classica physics predicts that the scattered electrons will emerge in all
directions with a moderate dependence of their intensity on scattering angle and even less on
the energy of the primary electrons.
In early 1925, one of their samples was inadvertently recrystallized in a laboratory
accident that changed its structure into nearly monocrystalline form. Now the results were
very different. Instead of a continuous variation of scattered electron intensity with angle,
distinct maxima and minima were observed whose positions depended upon the electron
energy! As a result, the angular distribution manifested sharp peaks at certain angles. As
Davisson and Germer soon found out, other monocrystalline samples also exhibited such
anomalous patterns, which differ with chemical constitution, angle of incidence and
orientation of the sample. Typical polar graphs of electron intensity after the accident are
shown in Figure 3.7. The method of plotting is such that the intensity at any angle is
proportional to the distance of the curve at the angle from the point of scattering. If the
intensity were the same at all scattering angles, the curves would be circles centered on the
point of scattering.
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De Broglie’s hypothesis suggested that electron waves were being diffracted by the
target, much as x-rays are diffracted by planes of atoms in a crystal. This idea received
support when it was realized that heating a block of nickel at high temperature causes the
many small individual crystals of which it is normally composed to form into a single large
crystal, all of whose atoms are arranged in a regular lattice.
The electron wavelength is therefore
which agrees well with the observed wavelength of 0.165 nm. The Davisson-Germer
experiment thus directly verifies de Broglie’s hypothesis of the wave nature of moving
bodies.
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UNCERTAINTY PRINCIPLE 1
We cannot know the future because we cannot know the present
In everyday life, calculating the speed and position of a moving object is relatively
straightforward. We can measure a car traveling at 60 miles per hour or a tortoise crawling at
0.5 miles per hour and simultaneously pinpoint where the objects are located. But in the
quantum world of particles, making these calculations is not possible due to a fundamental
mathematical relationship called the uncertainty principle.
Formulated by the German physicist and Nobel laureate Werner Heisenberg in 1927,
the uncertainty principle states that for particles exhibiting both particle and wave nature, we
cannot know both the position and speed of a particle, such as a photon or electron, with
perfect accuracy; the more we nail down the particle's position, the less we know about its
speed and vice versa.
In other words, if we could shrink a tortoise down to the size of an electron, we would
only be able to precisely calculate its speed or its location, not both at the same time.
To understand the general idea behind the uncertainty principle, think of a ripple in a
pond. To measure its speed, we would monitor the passage of multiple peaks and troughs.
The more peaks and troughs that pass by, the more accurately we would know the speed of a
wave—but the less we would be able to say about its position. The location is spread out
among the peaks and troughs. Conversely, if we wanted to know the exact position of one
peak of a wave, we would have to monitor just one small section of the wave and would lose
information about its speed. In short: the uncertainty principle describes a trade-off between
two complementary properties, such as speed and position.
The rollercoaster above serves as an analogy for how the uncertainty principle works at scales much smaller than this. Left: When the rollercoaster car reaches
the peak of the hill, we could take a snapshot and know its location. But the snapshot alone would not give us enough information about its speed. Right: As the
rollercoaster car descends the hill, we can measure its speed over time but would be less certain about its position. The uncertainty principle is a trade-off
between two complementary variables, such as position and speed.
Credit: Lance Hayashida/Caltech
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This principle was discovered by Werner Heisenberg in 1927 and is one of the most
significant of physical laws. More specifically, if one knows the precise momentum of the
particle, it is impossible to know the precise position, and vice versa. This relationship
also applies to energy and time, in that one cannot measure the precise energy of a system in
a finite amount of time. Uncertainties in the products of ―conjugate pairs‖
(momentum/position) and (energy/time) were defined by Heisenberg as having a minimum
value corresponding to Planck’s constant divided by 4π. More clearly:
Where Δ refers to the uncertainty
in that variable and h is Planck's constant.
What does it mean?
It is hard to imagine not being able to know exactly where a particle is at a given
moment. It seems intuitive that if a particle exists in space, then we can point to where it is;
however, the Heisenberg Uncertainty Principle clearly shows otherwise. This is because of
the wave-like nature of a particle. A particle is spread out over space so that there simply is
not a precise location that it occupies, but instead occupies a range of positions. Similarly, the
momentum cannot be precisely known since a particle consists of a packet of waves, each of
which have their own momentum so that at best it can be said that a particle has a range of
momentum.
Probability Distribution
Matter and photons are waves, implying they are spread out over some distance. What
is the position of a particle, such as an electron? Is it at the center of the wave? The answer
lies in how you measure the position of an electron. Experiments show that you will find the
electron at some definite location, unlike a wave. But if you set up exactly the same situation
and measure it again, you will find the electron in a different location, often far outside any
experimental uncertainty in your measurement. Repeated measurements will display a
statistical distribution of locations that appears wavelike. (See Figure 1.)
After de Broglie proposed the wave nature of matter, many physicists, including Schrödinger
and Heisenberg, explored the consequences. The idea quickly emerged that, because of its
wave character, a particle’s trajectory and destination cannot be precisely predicted for each
particle individually. However, each particle goes to a definite place (as illustrated in Figure
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1). After compiling enough data, you get a distribution related to the particle’s wavelength
and diffraction pattern. There is a certain probability of finding the particle at a given
location, and the overall pattern is called a probability distribution. Those who developed
quantum mechanics devised equations that predicted the probability distribution in various
circumstances.
Sample Problem
The uncertainty in the momentum Δp of a football thrown by Tom Brady during the
superbowl traveling at 40m/s is 1×10−6 of its momentum. What is its uncertainty in
position Δx?
Mass= 0.40kg
Note: 1 J= 1 kg.m/s
p
=mv
=(0.40kg)(40m/s)
=16 kg.m/ s
Note: π = 3.1
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