Fig. 1.Example of program session
Before the exercise, you should carefully study the following
sections in Reif's book:
Section 9.2 (Formulation of the statistical problem)
Section 9.3 (The quantum distribution functions)
Section 9.4 (Maxwell-Boltzmann statistics)
Section 9.8 (Quantum statistics in the classical limit)
During the exercise, you will use a computer program,
specially designed for this exercise. For the user, the program is a dialogue
session. By answering the questions put by the program, the user specifies
the physical parameters, such as temperature, number of particles, type of
single-particle energy spectrum and type of statistics to be used. An example
of a program dialogue is shown below. The questions and answers should be
self-explaining. If something is unclear, ask about it before starting your
calculations.
Fig. 1.Example of program session
ei = c in
where ei is the energy of level i and c is a constant, such that the highest level in the spectrum gets the energy of your choice. The lowest level gets the energy c. Once you have made your choice, use the same spectrum in all your calculations. Avoid using exactly the same spectrum as other participating students.
What to investigate?
The following investigations should be made:
1. Distribution functions at "normal" temperatures.
2. Distribution functions at low temperatures.
3. Distribution functions at high temperatures.
4. How does the Fermi energy depend on the number of particles?
5. How does the Fermi energy depend on the temperature?
Not to forget!
Do not forget to save and print out the results you need for your
written report.
The written report.
You are required to write a report on your results from the
computer exercise. You are allowed to write either in English
or in Swedish. The report must at least contain the following:
The purpose of your investigation.
When writing your report, you must make sure that you
give the answers to the questions asked in the five points above.
How to look at your results?
For looking at the results of your calculations, you will use a
a graphics program nusma. In the setting for this exercise,
the active nusma menus will appear as in the example below.
You use nusma for saving (printing) your results.
"Normal" temperatures should here be understood as such temperatures for
which kT is comparable to or a few times larger than the energy difference
between single-particle levels near the Fermi energy. Start by finding
a temperature, which gives a Fermi-distribution similar to the one shown
in fig. 3. Then calculate the distributions for the other two kinds
of statistics, using the same parameters. Plot the three distributions in
the same figure. For the Fermi-Dirac and Bose-Einstein distributions,
the Fermi energy should be shown. A suggestion: Choose the number of particles
to be about half as big as the number of single particle levels. How do you
interpret the results?
In this part the properties of the distribution functions at low temperatures
shall be studied. Putting T=0 in the computer program will not work, since
the computer will refuse to divide by zero (compare with the analytical
expressions for the distribution functions). Instead, you have to use some
very low, but non-zero, temperature. All other parameters should be the
same as in part 1. You should pay special attention to the different
behavior of the Fermi-Dirac and Bose-Einstein distributions. What is
the underlying physical explanation? What happens to the Fermi energy
in the two cases? How do you explain it? What is there to say about the
Maxwell-Boltzmann distribution?
In the high temperature limit, it is expected that the difference between
the Fermi-Dirac and Bose-Einstein distributions will disappear. It is
also expected that the Maxwell-Boltzmann distribution function will
become a good approximation to the quantum mechanical distribution
functions. Investigate this by letting the temperature increase to
much higher values than you used in part 1. For this investigation,
it could be an advantage to use a small particle number (why?).
Will there be a temperature for which the expected high temperature
properties show up? If so, what is this temperature? Can you explain
why such temperatures are required?
It may be obvious that the Fermi energy must increase when the number
of particles gets larger. However, it is probably not so obvious to
realize in which energy range, the Fermi energy is allowed to vary.
A comparison between the Fermi-Dirac and Bose-Einstein distributions
will also give results, which in the first place may seem surprising.
You should make the calculations needed for determining in which
energy interval the Fermi energy will vary as the particle number
changes. You should also give an explanation to the different
results obtained for the two quantum distribution functions.
You must keep the temperature constant in this investigation.
In this part, you keep the number of particles fixed while letting
the temperature vary. It may be particularly interesting to
investigate what happens when you have a small number of
particles (for example the same as in part 3) and let the temperature
increase from low to very high. How do you explain the results?
A presentation of the results you have obtained.
A physical interpretation of your results.
Fig. 2. Example of nusma session
Fig. 3. Example of nusma output