Download Physics 120 Homework Set #1 (due Sunday

Solutions Physics 120 Reading Assignment #5
Due: Sunday 10 PM, February 19
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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
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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) 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.
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) 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.
2) Give two examples (each for a and b below) of
a) how a photon (or electron) exhibits particle behavior and
In the photo-electric effect the energy appears to be transferred in indivisible pieces
rather than emerging as a continuous energy flow (as we would expect for a
propagating wave). In fact, if the single indivisible pieces of energy carried by each
“quanta” (photons) are not large enough to knock out an electron out of a metallic
surface, shooting more and more energy flow does not trigger the photo-electric
effect.
The photo-electric effect occurs for wavelengths of the electromagnetic radiation
which correspond to the optical range or slightly above (ultraviolet or soft X-rays).
If the wavelength is much shorter (hard X-rays) the Compton effect can occur. In
this case the energy of the single photon is much higher than the binding energy of
the electron (namely the energy that keeps the electrons bound to the nucleus). As a
consequence, the electron can be treated as if it were free. The result of the
Compton effect is similar to the photo-electric effect in that electrons are usually
knocked out of the material struck by the electromagnetic radiation; however, in the
Compton effect photons are not completely absorbed but are scattered off with a
longer wavelength. A similar effect would occur also for waves, however the effect
should vanish for low-intensity radiation. Instead Compton observed that the effect
did not depend on the intensity of the radiation but rather on the wavelength,
confirming that energy is propagated in indivisible quanta the amount of which
depends solely on the wavelength. Compton also calculated the wavelength shift as a
function of the scattering angle of the photon.
Note: you do not need to include all these details in your answer.
Electrons behave like particles in all daily phenomena involving electricity, where
electrons are responsible for resistance in wires.
Single electrons are also observed in radioactive decay of heavy nuclei (the so-called
 radiation).
b) how it exhibits wave behavior.
The double-slit experiment is an example of light behaving as a wave. If light was
just a stream of particles we would expect a simple shadow of the two slits to appear
in the screen. However, if the slit separation is comparable to the wavelength of the
electromagnetic radiation, an interference pattern appears. This can be interpreted
as the result of constructive and destructive interference between spherical waves
produced in each of the two slits.
Electrons behave like waves in the electron microscope. In this device electrons are
used in lieu of optical light to resolve small structures such as molecules and atoms.
Another example of electrons behaving like waves is the scattering of electrons
through a crystal, which shows also an interference pattern similar to that observed
in the double-slit experiment. In this case the two “slits” are provided by the
crystalline structure. This experiment was proposed by de Broglie and carried out
for the first time by Davisson and Germer at Bell Labs.
3-4) a) Describe the Copenhagen Interpretation of quantum mechanics?
b) What is the Many Worlds Interpretation of quantum mechanics?
c) Which non-trivial issue of quantum mechanics do both attempt to resolve?
d) Make an argument why you would subscribe to one rather than the other.
a) In the Copenhagen Interpretation of quantum mechanics a separation is devised
between the quantum object being observed and the observer (together with their
experimental devices), which is treated as a classical (i.e. non-quantum) object. The
act of the observation causes the wave function of the quantum object to “collapse”
from the superposition of many possible states (such as different positions or
different momenta) into a single state which can be interpreted classically (a definite
position or momentum of an object).
b) In the Many World Interpretation the quantum system being observed and the
observer are thought as being part of a universal wave function (i.e. everything is
treated quantum-mechanically). Before the act of measurement, the object and the
observer are two isolated quantum systems. The act of measurement consists of
putting them in close contact. In so doing, the quantum system being observed
“decoheres”. For example, in the double-slit experiment the individual photons are
initially part of a coherent quantum state; when they go through the double-slit and
to the screen they “decohere” from each other in that the subsequent evolution of
each photon is not anymore linked with that of the others. The universal wave
function contains the superposition of all possible outcomes in the interaction
between the object and the observer (i.e. all possible “decohered” states). Each of
these possible outcomes is one of the many worlds.
c) The two interpretations try to address the problem of the act of measurement.
d)
Some possible arguments for the Copenhagen Interpretation:
- It can be thought as a “minimal” interpretation, since it uses quantum mechanics
only when it is really needed, relying on the more intuitive concept of classical
mechanics to describe the observer and the outcome of the observation
- It does not rely on the concept of the “universal wave function” which cannot be
measured by definition (it would require an observer to be “outside” of it, but by
definition all the universe is part of the universal wave function)
Some possible arguments for the Many World Interpretation:
- It is a more “natural” interpretation because it does not devise a separation
between the object and the observer, asserting that everything is described
quantum-mechanically by the universal wave function (contrary to the Copenhagen
Interpretation that describes the observer classically)
- It does not need additional postulates about how the act of observation induces a
sudden wave function collapse
5) a) What is the Principle of Complementarity?
b) How does it preclude a deterministic approach to the future?
a) The Principle of Complementarity asserts that certain complementary properties
cannot be observed or measured at the same time. Some example are: wave and
particle, the Heisenberg Uncertainty Principle (momentum and position).
b) It precludes a deterministic approach in the classical sense, which consists in
following the trajectory of a particle. In fact, in classical mechanics following a
trajectory of a particle means knowing exactly the position and momentum of the
particle at any given time. This is however forbidden by the Principle of
Complementarity (or the Heisenberg Uncertainty Principle for momentum and
position).
6) 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.