Download Physics 120 Homework Set #1 (due Sunday

Physics 120 Reading Assignment #4
Due: Sunday 10 PM, February 12
Name:_________________________
Please insert your name where indicated in the upper right-hand corner of this page.
Expected length of answers is one paragraph for each question. Please explain your
answers so that they could be understood by another student. Please expand this word-file
and insert your answers in-place below.
Please feel free to discuss questions and concepts with other students from the class. This
is encouraged. However, when you sit down to answer the questions in the reading
assignment, you should submit your own answers.
Your completed homework assignments must be uploaded in assignments for this class
on Canvas by the specified due date and time in order to receive credit.
1) Blackbody radiation always has a characteristic distribution or shape when
plotted as the amount of radiation as a function of wavelength or frequency (a
spectrum with a peak determined by the average temperature like that on slide 22 of
Lesson 2 for the CMB).
a) What is a “black-body cavity”?
b) What happens to the spectrum of its radiation, when you heat it up?
c) What did classical theory predict about the amount of radiation at the shortest
wavelengths? [note – wavelength and frequency are inversely proportional.]
d) Why was this a catastrophe and what was its role in the formation of quantum
mechanics?
e) How did Planck and Einstein eventually resolve this “ultraviolet catastrophe”?
a) A cavity made with walls opaque to radiation with a small hole is a very good
approximation of a black-body. If any electromagnetic radiation is made to enter
through the hole, it will be trapped inside and bounced around the inside walls of
the container (the hole is very small so the probability that by chance it escapes back
out is negligible).
b) The spectrum is modified in such a way that its peak will be shifted towards
shorter wavelengths.
c) Classical theory predicted that the amount of radiation should increase
indefinitely at the shortest wavelengths, proportional to the increasing number of
modes of vibrations.
d) This behavior was called the “ultraviolet catastrophe” because it leads to an
infinite amount of energy stored in the short-wavelength mode of vibrations (high
frequency). This odd behavior of the classical theory of radiation sparked the need
for a new understanding of microscopic phenomena, which was then provided by
quantum mechanics.
e) The “ultraviolet catastrophe” was resolved via the quantization of the
electromagnetic energy in the black-body cavity. The energy is emitted by each of
the electromagnetic oscillators in quanta (photons) which correspond to an
indivisible amount of energy; each photon carries an energy E proportional to the
frequency  of the oscillator via Planck’s constant: E = h. This solves the
ultraviolet catastrophe: at high frequency (short wavelength) the energy associated
with each quanta is very large and only a few oscillators will have this much energy
at any finite temperature.
2) a) What is particle-wave duality?
b) Explain how Einstein’s equations for the energy and momentum of light quanta
were used by De Broglie to encapsulate this duality.
c) Identify the particle and wave aspects of each of these equations.
a) Particle-wave duality corresponds to the observation that different experiments
or observations can present a particle-like behavior as well as a wave-like behavior
for the same “object”. For example, light behaves like a wave in the double-slit
experiment, but it behaves like a particle in the photoelectric experiment. The same
applies to small matter particles, such as electrons.
b+c) Einstein’s equations for energy and momentum of light quanta are: E = h and
p = h / c. In both equations particle properties (energy and momentum) appear on
the left, whereas frequency (which is a wave property) appears on the right. For de
Broglie, these equations, rather than just being a mathematical “trick”, meant that
the two kinds of behavior are inextricably coupled.
3)
a) What is the Exclusion Principle? To which type of particles does it apply?
The Exclusion Principle states that no two particles can have the exact same set of
quantum numbers. This Principle only applies to fermions, i.e. particles with halfinteger spin. It applies to both elementary particles (such as electrons) and
compound particles (such as protons and nuclei).
The Exclusion Principle is a consequence of the Fermi-Dirac statistics that is
characteristic of or rather describes the behavior of fermions.
b) What is the other general type of particles that exists and which general rule
applies to it?
Particles with integer spin are bosons. They obey the Bose-Einstein statistics.
4)
a) What is the Uncertainty Principle? Explain what it means.
The Uncertainty Principle states that it is impossible to have at the same time an
exact measure of the momentum and the position of a particle.
More precisely the uncertainty on the position multiplied by the uncertainty on the
momentum must always be greater than the value of the Planck constant divided by
2: p  x  h / 2
b) Does the principle describe a property of a quantum object (e.g. electron) or the
limitations of an action (i.e. observation or measurement)? Explain.
The limit posed by the Uncertainty Principle is not a consequence of some deficiency
in the experimental techniques. Rather, it signifies that an electron which is located
at a precise position in space, does not posses any particular value of momentum;
conversely an electron with a precise value of momentum is not located in any
specific position, but can be found anywhere with equal probability.
c) How does the principle connect the particle and wave descriptions?
An electron for which we know its exact position behaves as a particle; conversely
an electron for which we know its momentum but is located anywhere in space
behaves as a wave does.
5) a) Which topics did you find particularly complicated and had difficulty
understanding? What specific questions do you have about this (these) topic(s)?
b) Which topics did you find particularly interesting and would like to discuss
further in class? Any specifics or questions that you wish to add on each topic?
Any answer to this question gives credit.