Welcome to Quantum Mechanics 4, atoms.
Until now we've been concerned exclusively
with light. We saw that Max Planck was able
to describe the variation of thermal radiation
intensity versus frequency. But only by proposing
that energy is transferred in discrete qu
anta, thus giving birth to quantum theory.
Soon it became clear that these chunks of
energy correspond to momentum carrying particles
of light called photons. The wave nature of
light places fundamental limits on the precision
with which we can simultaneously know related
quantities such as photon momentum and position.
This uncertainty principle is one aspect of
the so-called wave-particle duality that characterizes
photons. Starting with this video, we want
to turn our attention to matter, specifically
the atom.
The concept of atoms has been around since
antiquity. Simple put, matter is either infinitely
divisible or it is not. If it isn't there
must be a smallest, indivisible, unit of a
substance that we call the atom. Democritus
and Lucretius were two of the ancients who
argued philosophically for this idea.
The scientific basis for atomic theory can
arguably be said to begin with John Dalton.
He proposed that the fundamental substances,
the elements, are composed of identical atoms.
Specific numbers of atoms of different elements
can combine to form molecules characteristic
of a compound substance. For example, two
hydrogen atoms and one oxygen atom combine
to form one water molecule. Consequently any
quantity of water will always be exactly two
parts hydrogen and one part oxygen. Chemical
reactions between compounds are simply the
rearrangement of their constituent atoms.
How atoms are bound into molecules and why
different atoms have different chemical properties
remained a mystery.
In addition to displaying unique chemical
properties, different materials often display
distinctive colors when they burn. By the
late 19th century this idea had been refined
with great precision through the study of
atomic spectra. In place of a flame, an electric
current can be discharged through a pure gas
sealed in a glass tube. The emitted light
can be resolved into a spectrum of constituent
colors using a prism or for more precision
a so-called diffraction grating. Each element
is found to emit a unique spectrum. The reason
for this was another mystery of the atomic
realm.
When an incandescent light bulb is viewed
through a diffraction grating we observe a
continuous rainbow spectrum. A graph of light
intensity versus wavelength shows a continuous
spectrum of the type Planck's law describes.
But when electric current is passed through
a gas, Helium in this case, the spectrum displays
a distinctive pattern of colors. It's characterized
by discrete spectral lines at specific wavelengths
instead of a continuous spectrum. Why does
Helium only emit light at these wavelengths?
And why do different elements emit different
spectra? What physical process gives rise
to these emissions? These questions were topics
of intense research near the end of the 19th
century.
As the first element in the periodic table,
and the lightest gas, hydrogen was presumably
the simplest atom. So, it became a focal point
for atomic research. Hydrogen displays a series
of spectral lines that are named after the
different researchers who studied them. In
the ultraviolet, near 100 nanometers is the
Lyman series. In the visible region is the
Balmer series and so on. Empirically it was
found that these spectral lines follow a simple
pattern described by the so-called Rydberg
formula. The inverse of any of these wavelengths
equals the Rydberg constant R times one over
n squared minus one over k squared where n
and k are integers with k greater than n.
Each n value corresponds to a particular series
while the k values correspond to the lines
in that series. For example, n equals 2 is
the Balmer series in the visible region, k
equals 3 is a red line while k equals 4 is
blue and larger k values are in the violet
end of the spectrum. Any theory of the hydrogen
atom must be able to explain the Rydberg formula.
A key piece of the puzzle fell into place
with the discovery of the electron in 1897
by J. J. Thomson.
He used an early cathode ray tube, the device
that evolved into the TV picture tube. Inside
an evacuated glass tube are metal plates connected
to a voltage source. There's a negative plate
called the cathode and two positive anode
plates with slits cut in their centers. It's
found that for a high enough voltage something
is emitted from the cathode and moves towards
the anodes. Some of that emission passes through
the slits and forms a so-called cathode ray
which travels to the end of the tube, strikes
a phosphor screen and produces a visible dot.
Thomson put two more metal plates inside the
tube parallel to this beam. Applying a voltage
to those plates created an electric field
between the plates. By varying this electric
field Thomson found he could bend the beam
thereby moving the dot on the screen. The
direction of bending showed that this beam
carried a negative electrical charge.
Thomson also placed wire coils on either side
of the tube. Passing electric current through
the coils produced a magnetic field, and an
electric charge moving through this magnetic
field is subject to a force. By varying the
magnetic field Thomson could also bend the
beam and move the visible dot. The electric
and magnetic fields can be combined and adjusted
so their effects on the cathode ray precisely
cancel.
Thomson assumed the ray was composed of particles
of mass m and charge minus e moving with velocity
v all three of which were unknown. The magnitude
of electric force is charge times the known
electric field capital E. The magnetic force
magnitude is charge times velocity times the
known magnetic field B. These had been adjusted
to cancel out so Thomson knew that the force
magnitudes were equal (although they pointed
in opposite directions). This gave him an
equation from which the velocity v is the
ratio of the know electric field E to the
know magnetic field B. Thomson found that
the cathode ray was traveling very fast, more
than one-tenth the speed of light.
Now he turned off the magnetic field and observed
the effect of the electric field alone. The
electric field exists only in a region of
length L between the charged plates. Initially
the ray is moving to the right with velocity
v. Between the plates the electric force will
accelerate it upward. When it exits the plates
it will be traveling at an angle determined
by the original rightward velocity v and the
added upward velocity. The latter is the electric
force divided by the electron mass, that's
the upward acceleration, times the time spent
between the plates, which is the distance
L over the velocity v. By measuring this angle
Thomson could determine the charge to mass
ratio e over m. This was about 1,000 times
as large as the estimated value for a positively
charged hydrogen ion, what we now call a proton,
as obtained by electrochemical methods. Since
a proton and an electron together form an
electrically neutral hydrogen atom, they must
have opposite charges. So the difference in
their charge-to-mass ratios must be due to
a difference in mass. Thomson concluded that
the electron is only about one one-thousandth
the mass of a proton.
Following the discovery of the electron, Thomson
proposed an atomic model in which atoms consist
of positively and negatively charged components.
For hydrogen the bulk of the atomic mass is
contained in a cloud of positive charge e.
In this floats a much smaller and lighter
electron of negative charge minus e. This
model was very appealing because it seemed
to unlock one of the mysteries of the atom.
The existence of discrete spectral lines.
If the positive charge is uniformly spread
out through a sphere of radius a, and the
electron is displaced from the sphere's center
a distance r, then the electron feels a net
force only from the positive charge inside
the radius r, the dark blue region. The forces
from charge in the light blue region pull
in different directions and cancel out. The
charge q in the dark-blue region is the total
charge e times the fraction of the volume
occupied by the dark-blue region, namely,
r over a cubed (because the volume of a sphere
varies as its radius cubed). The force pulling
the electron back towards the center is the
product of charges, e q divided by the distance
r squared. When you put in the expression
for q you end up with a force which is a constant
times the displacement r. A force that varies
proportional to displacement is characteristic
of a spring. So in the Thomson model the electron
is bound to the atom by a type of electric
spring. A mass on a spring oscillates with
a single definite frequency. Calculating this
for hydrogen gives a frequency and wavelength
of ultraviolet light, near the Lyman spectral
series.
The radiation produced by an oscillating electron
is precisely described by electromagnetic
theory, which in Thomson's day had already
been known for decades. Here we show two views
of the electric field of an electron. When
the electron is at rest the electric field
is unchanging. As the electron begins to oscillate
its electric field peels off in waves which
radiate away at the speed of light. These
waves carry away energy, so the oscillation
energy of the electron will decrease.
This gives us a picture of how passing electric
current through a gas produces light. Suppose
a hydrogen atom with its electron at rest
at its center suffers an intense collision
which transfers energy to the atom and displaces
the electron. The atom's electric spring then
causes the electron to oscillate. As it does
it produces radiation which carries the energy
away causing the electron to come back to
rest and the radiation to stop. Another collision
can restart the process.
Now, a major shortcoming of this simple model
is that it predicts only a single radiation
frequency when we know that real atoms emits
series of spectral lines. Still, we seem to
be headed in the right direction.
The discovery of the electron showed that
atoms were not fundamental particles, but
built from more elementary objects. However,
Thomson's experiment measured only the electron
charge-to-mass ratio and did not determine
charge or mass independently.
Because of this it did not definitely establish
the electron as a specific particle. For example,
in a given electric field one could conceivable
have a series of particles, with different
charges and masses, but all having the same
charge to mass ratio. Maybe they were different
sized pieces of the same stuff. Yet their
behavior in the electric field would be identical.
To rule this out one needed to establish the
charge of a single electron.
The fundamental charge was measured in 1909
by Robert Millikan. Millikan achieved this
in his so-called oil-drop experiment. Here's
the idea. A chamber was formed between two
parallel metal plates. The top plate had a
hole in the middle. The distance between the
plates was d. You could look through the chamber
(using a microscope actually) at a ruler.
An oil mist was sprayed above the top plate
and occasionally a tiny oil drop would fall
through the hole into the chamber. Millikan
wanted to know the mass of the oil drop. In
principle you could measure the radius of
the drop directly with the ruler, but it's
so small that precise measurement isn't practical.
Instead, Millikan timed the oil drop falling
with its terminal velocity – which is very
small – over a relatively long distance
which could be measured accurately and gave
him the value of the terminal velocity. When
something is at terminal velocity the downward
pull of gravity (m times g) is balanced by
the upward air drag f d. There's a formula,
called Stoke's law, that gives the drag in
terms of known properties of air, the radius
of the oil drop, and the terminal velocity.
Knowing the effective density of the oil you
can relate the radius to the volume and hence
the mass. The equation m g equals f d then
has a single unknown – the radius r – which
can be solved for. Thus Millikan determined
the size, mass and weight of the oil drop.
Knowing the oil drop's weight, he then applied
a large voltage between the plates. This created
an electric field. If the oil drop carried
any static electricity this produced an upward
force that could be adjusted to just balance
the weight causing the oil drop to stay suspended
in the chamber. Millikan knew the electric
field in terms of the voltage and plate separation
so he could solve for the static electrical
charge. He did this many times and always
found that the charge was an integer number
of a fundamental charge which he identified
as the charge of the electron. Thus Millikan
was able to see the effects of single electrons,
and to show that these were identical in all
cases. Together with Thomson's results this
established the electron as a unique particle
with known charge and mass.
