Quantum Phenomena
The word quantum simply means the smallest piece of something. The
quantum of our money system is the penny, the quantum of negative charge is
the electron (at least as far as we know right now). There are several
quantities in physics which are quantized, that is, that occur in multiples of
some smallest value. Light is one of these quantities.
Photons & the Photoelectric Effect
In earlier chapters we treated light as a wave, but there are circumstances
when light behaves more like it is made up of individual bundles of energy.
These bundles are separate from each other but share a wavelength,
frequency, and speed. The quantum of light is called the photon.
In the late 19th century an effect was discovered by Heinrich Hertz which could
not be explained by the wave model of light. He shown ultraviolet light on a
piece of zinc metal, and the metal became positively charged. Although he
didn’t know it at the time, the light was causing the metal to emit electrons.
Using light to cause electrons to be emitted from a metal is called the
photoelectric effect. According to the theory of light at the time, light was
considered a wave, and should not be able to “knock” electrons off of a metal
surface.
At the turn of the 20th century, Max Planck showed that light could be treated
as tiny bundles of energy called photons, and the energy of a photon was
proportional to its frequency. Thus, a graph of photon energy E vs. frequency f
looks like this:
The slope of this line is a constant that occurs many times in the study of
quantum phenomena called Planck’s constant. Its symbol is h and its value is
6.63 x 10-34 J*s (or J/Hz). The equation for the energy of a photon is:
E = hf
And since f = c/λ
E=hc/λ
The energy of a photon is proportional to its frequency, but inversely
proportional to its wavelength. This means that a violet photon has a higher
frequency and energy than a red photon.
Often, when dealing with small amounts of energy like those of photons or
electrons, we may prefer to use a very small unit of energy called the electronvolt (ev). The conversion between joules and electron-volts is
1 ev = 1.6 x 10-19 J
Planck’s constant can be expressed in terms of electron-volts as
h = 4.14 x 10-15 eV*s
In 1905, Albert Einstein used Planck’s idea of the photon to explain the
photoelectric effect: one photon is absorbed by one electron in the metal
surface, giving the electron enough energy to be released from the metal. But
not just any photon will knock an electron off of a metal surface. The photon
must first have enough energy to “dig” the electron out of the metal, and then
have some energy left over to give the electron some kinetic energy to escape
completely.
Each metal that can exhibit the photoelectric effect has a minimum energy and
frequency called the threshold frequency f0 that the incoming photon must
meet to dig the electron out of the metal, and must exceed if the electron is to
have kinetic energy to escape. For example, the metal sodium has a threshold
frequency that corresponds to yellow light. If yellow light is shone on a sodium
surface, the yellow photons will be absorbed by electrons in the metal, causing
them to be released, but there will be no energy left over for the electrons to
have any kinetic energy. If we shine green light on the sodium metal, the
electrons will be released, have some energy left over to use as kinetic energy,
and jump off the metal completely (green light has a higher frequency and
energy than yellow light). If a brighter (more photons) green light is shone on
the surface, more electrons will be emitted, since one photon can be absorbed
by one electron. If these electrons are funneled into a circuit, we can use them
as current in an electrical device. If orange light were shone on the sodium
metal, no emission of electrons would take place, no matter how bright the
orange light is, since orange light is below the frequency for sodium.
The graph of maximum kinetic energy of a photoelectron vs. frequency of
incident light looks like this:
Note that the electrons have no kinetic energy up to the threshold frequency
(color), and then their kinetic energy is proportional to the frequency of the
incoming light.
The minimum energy needed to eject an electron from the atom completely is
called the work function, φ. The work function is proportional to the threshold
frequency f0 by the equation:
𝝋 = 𝒉𝒇𝟎
Thus the kinetic energy of the ejected electron from a photo-emissive surface
is equal to the difference between the energy of the absorbed photon and the
work function:
Since a photon has energy, does it follow that it has momentum? Recall that
momentum can be defined as p=mv (mass x velocity). But a photon has no
mass. It turns out that in quantum physics photons do have momentum by
virtue of their wavelength. The equation for the momentum of a photon is:
Kemax = Ephoton – φ = hf-hf0
𝒉
𝒑=
𝝀
So the momentum of a photon is inversely proportional to its wavelength.
Photons can, and do impart momentum to sub-atomic particles in collisions
that follow the law of conservation of momentum. This phenomenon was
experimentally verified by Arthur Compton in 1922. Compton aimed X-rays of
a certain frequency at electrons, and when they collided and scattered, the Xrays were measured to have a lower frequency, indicating less energy an
momentum. The scattering of X-ray photons from an electron with a loss in
energy is called the Compton Effect.
Uncertainty always exists in measurements we take on the quantum level. This
limit to accuracy at this level was formulated by Werner Heisenberg in 1928
and is called the Heisenberg Uncertainty Principle. It can be stated like this:
There is a limit to the accuracy of the measurement of the speed (or
momentum) and position of any sub-atomic particle. The more precisely we
measure the speed of a particular particle, the less accurately we can
measure its position, and vice-versa.
In 1924, Louis de Broglie reasoned that if a wave such as light can behave as a
particle, having momentum, then why couldn’t particles behave like waves? If
the momentum of a photon can be found with p = h/λ, then the wavelength
can be found by λ = h/p. De Broglie suggested that for a particle with mass m
and speed v, we could write the equation as :
This allowed the wavelength of a moving particle to be calculated. His
hypothesis was initially met with considerable skepticism until it was
experimentally verified in 1927.
Nuclear and particle physicists must take into account the wave behavior of
subatomic particles in their experiments. We typically don’t notice the wave
properties of objects moving around us because the masses are large in
comparison to subatomic particles and the value of Planck’s constant h is
extremely small. But the wavelength of any moving mass is inversely
proportional to the momentum of the object.
𝒉
𝝀=
𝒎𝒗
Atomic Physics
The ancient Greeks were the first to document the concept of the atom. They
believed all matter was made up of tiny indivisible particles. In fact, the word
atom comes from the Greek word atomos, meaning “uncuttable.” But a
working model of the atom didn’t begin to take shape until J.J. Thomson’s
discovery of the electron in 1897. He found that electrons are tiny negatively
charged particles and that all atoms contain electrons. He also recognized that
atoms are naturally neutral, containing equal amounts of positive and negative
charge, although he was not correct in his theory of how the charge was
arranged.
You may remember studying Thomson’s “plum-pudding” model of the atom,
with electrons floating around in positive fluid. A significant improvement on
this model was made by Ernest Rutherford in 1911, when he decided to shoot
alpha particles (helium nuclei) at very thin gold foil to probe the inner structure
of the atom. He discovered that the atom has a dense, positively charged
nucleus with electrons orbiting around it.
In 1913, Niels Bohr made an important improvement to the Rutherford model
of the atom. He observed that excited hydrogen gas gave off a spectrum of
colors when viewed through a spectroscope. But the spectrum was not
continuous; the colors were bright, sharp lines that were separate from each
other. It had long been known that every low pressure, excited gas emitted its
own special spectrum in this way, but Bohr was the first to associate the brightline spectra of these gases, particularly hydrogen, with a model of the atom.
The two postulates of the Bohr model of the atom are summarized below:
1.
Electrons orbiting the nucleus of an atom can only orbit in certain
quantized orbits, and no others. These orbits from the nucleus
outward are designated n = 1,2,3 …, and the electron has the energy in
each of these orbits E1, E2, E3, and so on. The energies of electrons are
typically measured in electrons-volts (eV). The lowest energy (in the
orbit nearest the nucleus) is called the ground state energy.
2. Electrons can change orbits when they absorb or emit energy.
a. When an electron absorbs exactly enough energy to reach a
higher energy level, it jumps to that level. If the energy offered to
the electron is not exactly enough to raise it to a higher level, the
electron will ignore the energy and let it pass.
b. When an electron is in a higher energy level, it can jump down to a
lower energy level by releasing energy in the form of a photon of
light. The energy of the emitted photon is exactly equal to the
difference between the energy levels between which the electron
moves.
For example, consider the energy level diagram for a particular atom
shown to the right.
Let’s say an electron in the ground state absorbs a photon and makes an
upward transition to n=4. Since the energy levels are labeled with the energy
of the electron above ground state in each case, the electron would need 13.22
eV to jump from the ground state to n = 4. If the electron drops from n = 4 to n
= 2, a photon of energy 10.2 eV is emitted. The second transition results in a
higher energy photon than the first, and since energy is proportional to
frequency the second photon must also have a higher frequency.
Sample Problem
Monochromatic red light (λ = 700 nm) is shone onto a metal surface that ejects
electrons with a kinetic energy of 1.1 eV. What is the work function of the
metal?
f = c/λ = (3.0 x 108)/(7 x 10-7) = 4.29 x
1014 Hz
E =hf =(4.14 x 10-15)(4.29 x 1014)=1.78 eV
1.78 eV-1.1 eV = 0.68 eV
Each electron has 1.1 eV of energy.
If a typical fission reaction releases 170 MeV, how may photons of the red light
would be required to produce electrons with a combined energy equal to the
energy of a single fission reaction?
1.70 x 108 / 1.1 = 1.54 x 108 photons
To generate this much energy, there
would have to be 1.54 x 108 red light
photons delivered each second. This is
equivalent to 1.78 e V times 1.54 x 108
photons, which = 2.74 x 108 eV. This
equals 4.38 x 10-11 J/s or Watts.