Summary Sheet - Atomic Physics, Quantum Mechanics, Relativity
A. Every substance has a characteristic absorption and emission spectrum. There are more
lines in an emission spectrum than in an absorption spectrum. The chemical properties of
the elements repeat in such a way that they can be arranged in a periodic table.
B. Thomson discovered cathode rays and determined that they are charged particles. When all
the data was in, it was concluded that cathode rays are electrons, pieces of atoms with
charge 1.60×10-19 C and mass 9.11×10-31 kg. Thomson formulated the “plum pudding”
atomic model.
C. Rutherford used the scattering of alpha particles to determine that the positive charge in an
atom is concentrated into a tiny nucleus. Rutherford concluded the atom is like a tiny solar
system with electrons orbiting around the nucleus like the planets orbit the sun.
D. All heated objects emit electromagnetic radiation. When Planck studied this radiation he
concluded that atoms are oscillators that have quantized energies. If an atom oscillates with
frequency f, then its energy is some integer multiple of h f, where h = 6.626×10-34 J·s.
E. In the photoelectric effect, light strikes a clean metal surface and electrons are ejected from
the metal. When scientists studied the effect, they realized that their theories could not
explain why the maximum electron energy depended upon the frequency and not the
intensity of the light. Einstein concluded that light behaved like particles, and each photon
had energy E = h f. This theory conflicted with the prevailing theories that light is a wave.
F. Bohr utilized the ideas of Planck and Einstein in order to improve Rutherford’s “solar
system” model of the atom. In Bohr’s model, the angular momentum L = m v r of the
nh
orbiting electrons is quantized, so that L = 2 π , where n is an integer. This meant that radius
of orbit is also quantized, rorbit = n2 r1, where r1 = 5.29×10-11 m. The energy of the electron
is therefore also quantized, and for a hydrogen atom E n = −13n.62 eV . Bohr also stated that the
electron did not radiate when it is in an allowed orbit. If it did, it would quickly lose energy
and crash into the nucleus.
Absorption and emission spectra are explained by the Bohr theory of the atom, which says
that a photon is absorbed or emitted only when an electron makes a jump from one allowed
orbit to another. The energy of the photon E = h f matches the energy difference between
the levels involved in the electron’s jump. Blue photons have higher energy and correspond
to larger energy jumps than red photons.
G. de Broglie hypothesized that particles behave like waves, much like light waves can behave
h
like particles. The de Broglie wavelength of any particle is λ =
. He updated the Bohr
mv
model of the atom by postulating that electrons form circular standing waves around the
nucleus instead of moving around in classical orbits. Electron diffraction is a key
experimental proof of de Broglie’s ideas.
H. Quantum mechanics describes all particles as waves and calls the matter-wave amplitude the
wave function Ψ(x) of the particle. The square of the wave function at x, Ψ 2(x), is the
probability of locating the particle at that particular position. The quantum mechanical atom
consists of “probability clouds” of electrons with quantized values for angular momentum,
energy, direction of angular momentum, and direction of electron spin. The quantum
mechanical atom explains the periodic table.
I.
The Heisenberg uncertainty principle says there is a fundamental limit of how accurately
both the position and the momentum of a particle can be simultaneously measured along the
same direction. The product of the uncertainties of the measurements must be greater than
Planck’s constant h: Δ y Δp y > h where Δy is the uncertainty of the position measurement,
Δpy is the uncertainty of the momentum measurement along the y direction, and h is
Planck’s constant.
J.
An inertial frame of reference (FOR) is one in which the law of inertia is valid. Any FOR
that has a constant velocity relative to an inertial FOR is also an inertial FOR. If a FOR is
accelerating, the law of inertia does not hold, the frame is not “inertial”, and the motion of
objects within the frame must be explained by introducing “inertial” or “fictitious” forces.
The earth is a noninertial frame of reference, but the inertial forces are so small compared to
those in our daily experience that we consider earth to be an inertial reference frame.
K. Observers in two inertial reference frames will disagree about how to describe a particular
object’s motion, but in both frames the object obeys the same laws of physics.
L. The two postulates of Einstein’s special theory of relativity are:
1.
The laws of physics are the same in all inertial reference systems.
2.
The speed of light in a vacuum is constant, regardless of the speed of the source or
the observer.
M. As a consequence of these two postulates, observers in different inertial reference frames
cannot agree on the simultaneity of events at different places. Neither can the two observers
synchronize clocks at different positions in both reference frames.
N. Time seems normal in one’s own inertial reference system, but time appears to slow down in
reference frames moving relative to one’s own frame. That is, moving clocks (relative to
your frame) run slow. If Mary is on earth and Ed is rapidly moving relative to earth, the
time elapsed on Mary’s clock is equal to the time elapsed on Ed’s clock multiplied by
adjustment factor γ:
Δ t Mary = γ Δ t Ed where γ =
1
1 − ( vc )
2
where v is the relative speed of the two
inertial reference frames and c is the
speed of light in a vacuum.
O. The Twin Paradox indicates that a person traveling at a high rate of speed relative to Earth
will age slower than people on Earth because of the time dilation effect described in point N
above. If time is dilated for moving systems, then length must be contracted. This means
that an attempt to measure length of an object that is moving relative to your inertial
reference frame will always result in a shorter length than if that object were measured in its
L Ed
own reference frame. If Ed and Mary both measure distance in Ed’s frame, L Mary =
γ
P. To preserve the conservation laws of momentum and energy we must introduce relativistic
momentum p = γmv and relativistic kinetic energy
KE = γmc2 - mc2, where the second term is the energy of the particle at rest. Thus the
Einstein rest-mass energy E0 = mc2 suggests that even at rest, particles have energy stored in
their mass. The fact that mass is another form of energy is utilized in nuclear power plants
and nuclear weapons.