19. WAVES OF MATTER
Trivia: Louis was a history student, but his brother Maurice was a physicist.
Intrigued by the philosophical problems of the new physics (in discussions
with Maurice), Louis decided to change careers.
Louis de Broglie proposed the existence of matter waves in 1924 in his
doctoral thesis, but there was little experimental evidence at the time.
His reasoning was basically one of symmetry: if light waves act in some
ways like particles, perhaps particles behave like waves.
For matter and waves, we have
and p=mv (classically, with no relativistic effects)
He got his PhD, but the idea of matter waves wasn’t taken seriously
until Einstein took an interest. Then the experimentalists came along and
proved de Broglie right. He got the Nobel in 1929.
Those experimentalists were Davisson & Germer in the U.S. and Thomson
in Scotland. In 1927, they basically did the Thompson double-slit experiment
with electrons instead of light: still get an interference pattern. They
used a crystal lattice as the ‘slits’. (Need very narrow slits
to get diffraction, because de Broglie wavelengths are very small.)
Davisson and Thomson shared the Nobel in 1937. [Thomson’s father
J.J.Thomson discovered the electron (assumed a particle) in 1897, and won
the Nobel for that in 1906. G.P.Thomson won the Nobel for showing the electron
as a wave. High power family!]
Note that the wavelike behavior of electrons is what enables an electron
microscope to work.
It is the smallness of h that makes these matter waves unnoticeable in
the macroscopic world.
Neils Bohr, principle of complementarity: if a measurement/experiment shows
the wave property of matter or radiation, it can’t also reveal the
particle nature, and vice versa. Matter and radiation are both particles
and waves, but which aspect is manifest depends on the nature of the experiment.
The role of the observer in the Universe...
We can do another variation of Young’s double-slit experiment. Do
it with light, but turn down the brightness so there is only one photon
in the system at a time. What happens is...
Again, the role of the observer...
20. SCHRÖDINGER’S EQUATION
An example of a wave equation would be
Such a wave equation can be used to describe the matter wave associated
with a particle. (Actually, sums of such waves are used, alá Fourier
Series, and they’re complex also.)
Draw a wave packet.
Schrödinger’s Equation
Motivation to learn calculus and differential equations.
21. HEISENBERG, DETERMINISM, AND WAVES OF PROBABILITY
Heisenburg’s Uncertainty Principle
The role of observer/participant...
These can be seen as a consequence of the wave-particle duality of matter and
radiation, and the principle of complementarity.
Classical determinism is killed by the Uncertainty Principle, we can never know
the initial conditions that accurately, and the role of the experiment/observer
tips the balance of the uncertainty trade-offs. All of time is not pre-determined;
there is slop in fate. There is also room for soul (free will).
Again, Young’s double slit experiment with low light intensity. The fallacy
comes from thinking that each photon must pass through one slit or the other.
If we measure its path (position) closely enough that we can determine which
slit it goes through, we are forcing the particle aspect to be manifest, and
the interference pattern disappears.
That’s what many said, including Einstein “God does not play dice.” But
the ‘sense’ is ‘common sense,’ and our everyday experience
does not include the subatomic realm. Nonsense: violates our existing cognitive
structures, formed by the cumulative experience of our senses. Einstein himself
encountered such resistance, and sometimes organized, to his own ideas. When
talking about macroscopic objects, those probabilities go to very close to 1
or 0.
Heisenburg also explains why diffraction occurs...
Copenhagen Interpretation: de Broglie’s matter waves are probability waves.
pauli exclusion and the periodic table
pair production
BH evaporation (Hawking)
PROBLEMS
1. Graph the equation twice — once with t=0, and once with t=1.
2. Show that if the uncertainty in the location of a particle is about equal
to its de Broglie wavelength, then the uncertainty in its velocity is about equal
to one-tenth its velocity.
3. The highest achievable resolving power of a microscope is limited only by
the wavelength used; that is, the smallest detail that can be separated is about
equal to the wavelength. Suppose we wish to “see” inside an atom.
Assuming the atom to have a diameter of 1.0Å (typical) this means we wish
to resolve detail of separation about 0.1Å.
(a) If an electron microscope is used, what minimum energy of electrons is needed?
(b) If a photon microscope is used, what energy of photons is needed? In what
region of the EM spectrum are these photons?
(c) Which microscope seems more practical for this purpose? Explain.
4. Briefly describe the Principle of Complementarity.
5. Does a blackbody always appear black? Explain.
6. The energy required to remove an electron (i.e. the work function B) from
sodium is 2.3 eV.
(a) What is the cutoff wavelength for photoelectric emission from sodium?
(b) Does sodium show a photoelectric effect for yellow light, with ?=589nm?
7. What we call temperature is the motion of atoms within matter. As the temperature
of an object is decreased, the motion of its constituent atoms becomes less vigorous.
Using the Heisenburg Uncertainty Principle, answer this query: Is it possible
for atoms to come to an absolute standstill if they are cooled enough, even to
Absolute Zero? Explain briefly.
8. How does the Heisenburg Uncertainty Principle help invalidate the Newtonian
notion of determinism?
9. Can the de Broglie wavelength of a particle be smaller than the physical size
of the particle? Yes or no, and explain very briefly.
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