Quantum Physics
27.1 Blackbody Radiation & Planck
1900 Major Unsolved Problems re Light:
|
|
|
Why do hot gases emit spectral lines, with
different sets of |
|
Why does the spectrum of a blackbody (or other incandescent source) disagree with that predicted by electromagnetic theory? |
|
|
“ultraviolet catastrophe” |
|
|
Predicted |
Observed |
1900 Max Planck
Light emitted by resonators, each
capable of emitting only at discrete (quantized) energies:
n is a quantum number, and h is Planck’s Constant.
27.2 The Photoelectric Effect & Particle Theory of Light
|
Put negative V on E: No light More light
Put positive V on E: Some current if V not too
negative |
|
Even if
|
All of the “things” on the right would imply that the energy of light is not determined by the intensity, but by the frequency.
1905 Einstein
Light behaves like a particle whose
energy
- “photons”. When interacting with matter, the exchange in energy is an
“all-or-nothing” proposition.
|
The maximum KE of the photoelectron
is
Cutoff wavelength:
|
|
|
Contradicts the Wave Model of Light |
Explained by the particle Model of Light |
|
No electrons
are emitted if the incident light falls below some cutoff frequency |
The effect is
not observed below a certain cutoff frequency follows from the fact that the
photon energy must be greater than or equal to |
|
The maximum kinetic energy of the photoelectrons is independent of the light intensity. According to wave theory, light of higher intensity should carry more energy into the metal per unit time and therefore eject photoelectrons having higher kinetic energy. |
That |
|
The maximum kinetic energy of the photoelectron increases with increasing light frequency. The wave theory predicts no relationship between photoelectron energy and incident light frequency. |
That |
|
Electrons are emitted from the surface almost instantaneously (less than 10-9 sec after the surface is illuminated), even at low light intensities. Classically, we expect the photoelectrons to require some time to absorb the incident radiation before they acquire enough kinetic energy to escape from the metal. |
That the electrons are emitted almost instantaneously is consistent with the particle theory of light, in which the incident energy arrives at the surface in small spatial packets and there is a one-to-one interaction between photons and photoelectrons. In this interaction, the photons energy is imparted to an electron that then has enough energy to leave the metal. This is in contrast to the wave theory, in which the incident energy is distributed uniformly over a large area of the metal surface. |
Example: Problem #12
Electrons
are ejected from a metallic surface with speeds ranging up to 4.6 x 105
m/s when light with a wavelength of = 625 nm is used. (a) What is the work
function of the surface? (b) What is the cutoff frequency for this surface?
27.3 X-Rays
X-rays are produced in most settings by accelerating electrons to hit a target using a strong voltage. The emitted spectrum consists of lines and a continuum. The lines are due to electron transitions in the inner parts of the atom, while the continuum is due to bremsstrahlung.
|
|
|
|
The maximum energy that can be liberated as light is the KE of the electron when it strikes the cathode:
Example: Problem #20
Calculate the minimum wavelength x-ray that can be produced when a target is struck by an electron that has been accelerated through a potential difference of (a) 15.0 kV, (b) 100 kV.
27.4 Diffraction of X-Rays by Crystals
We have seen how a 2-dimensional diffraction grating can produce interference patterns in light waves. X-rays can penetrate into many materials, so that a regular 3-dimensional structure can produce similar interference patterns.
|
|
|
|
X-rays scattering through a crystal can constructively interfere, producing bright spots. |
Constructive interference (here shown in the case of reflection) occurs when
|
|
X-ray diffraction of the DNA molecule was the crucial piece of evidence for the determination of the helical structure of DNA by F.H.C. Crick and J.D. Watson in 1953, for which they received the Nobel Prize. The diffraction work was done by Rosalind Franklin in 1952, whose precise and careful work was barely acknowledged by Crick & Watson (and their competitor Wilkinson).
|
|
Example: Problem #22
A monochromatic x-ray beam is incident on a NaCl crystal surface where d = 0.353 nm. The second-order maximum in the reflected beam is found when the angle between the incident beam and the surface is 20.5°. Determine the wavelength of the x-rays.
27.6 Photons & EM waves
Light behaves both as a wave and as a particle. Which aspect you see depends on the situation.
High Energy a single photon produces an easily
measurable effect
particle aspect. Example: X-rays
Low Energy a single photon has too little energy to
produce an easily measurable effect
only detect large numbers
wave aspect. Example: Radio waves.
Physicists think of photons as a wave packets or wavicles, which carry both wave and particle aspects.
27.7 Wave Properties of Particles
1924 Louis de Broglie suggests that the
wave-particle duality of photons might extend to all matter particles. Suppose
electrons, protons, and other particles with mass also behaved as photons did,
where
Then we could, in
principle, define a wavelength of a matter particle as and its frequency as
.
At the time de Broglie suggested the idea, there wasn’t a shred of evidence to
support it. However, in 1927 Davisson & Germer (quite by accident) measured
Bragg diffraction of electrons scattering off of crystalline nickel. The effect
has since been observed for other particles as well.
Example: Problem #36
Through
what potential difference would an electron have to be accelerated from rest to
give it a de Broglie wavelength of 1.0 x 1010
m?
Example: Problem #41
The
resolving power of a microscope is proportional to the wavelength used. A
resolution of approximately 1.0 x 1011
m (0.010 nm) would be required in order to “see” an atom. (a) If electrons were
used (electron microscope), what minimum kinetic energy would be required for
the electrons? (b) If photons were used, what minimum photon energy would be
needed to obtain 1.0 x 10
11
m resolution?
27.8 The Wave Function
1926 Erwin Schrödinger develops the wave equation that govern how
matter waves work. This is the fundamental development for modern quantum
mechanics (QM), also called wave
mechanics. It is QMs equivalent of
Newton’s Laws of Mechanics.
The basic “animal” of QM is
the wave function .
It’s essentially the amplitude of the matter wave. Here, a particle is not
located at a specific place
it is “fuzzy” in the sense that we
cannot predict its location exactly ahead of time. The likelihood
probability
of finding the particle at a specific
place is related to the square of the wave function
.
(For those of you familiar with complex numbers, it is actually then product of
the wave function with its complex conjugate
).
QM is all about solving for
what is in different situations. If you go here
you can see a simple case of a “particle in a box”. Look at the wave function,
and the probability distribution.
27.9 The Uncertainty Principle Heisenberg (1927)
If a measurement of
position of a particle is made with precision and a simultaneous measurement of linear
momentum is made with precision
,
then the product of the two uncertainties can never be smaller than
.
One can get a “feel” for this by considering the following experiment:
|
When you try to measure
the position of an electron by “seeing it” using a photon, when the photon
scatters off the electron, it alters the momentum of the electron (and its
position)! It can change by as much as the photon’s momentum |
|
A really careful
calculations gets the equation with the in it.
Is it the act of making the measurement or the intrinsic fuzziness of the wave functions of matter that really causes this limitation? This is the sort of thought that physicists have argued about for years!
Example: Problem #46
(a)
Show that the kinetic energy of a nonrelativistic particle can be written in terms
of its momentum as KE = p2/2m. (b) Use the results of (a) to find
the minimum kinetic energy of a proton confined within a nucleus having a
diameter of 1.0 x 1015
m.
27.10 Scanning Tunneling Microscope (reading exercise for the student) Short version:
|
|
Electrons can cross an
insulating (vacuum) barrier by a QM process called “tunneling”. Essentially,
the
The exponential dependence makes the current between sample and probe very sensitive to the heights of the irregularities on the surface of the sample. |
