Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.
In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
In [2]:
from IPython.display import Image
from IPython.html.widgets import interact, interactive, fixed
In quantum statistics, the Fermi-Dirac distribution is related to the probability that a particle will be in a quantum state with energy $\epsilon$. The equation for the distribution $F(\epsilon)$ is:
In [3]:
Image('fermidist.png')
Out[3]:
In this equation:
In the cell below, typeset this equation using LaTeX:
Define a function fermidist(energy, mu, kT)
that computes the distribution function for a given value of energy
, chemical potential mu
and temperature kT
. Note here, kT
is a single variable with units of energy. Make sure your function works with an array and don't use any for
or while
loops in your code.
In [12]:
print (np.e)
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def fermidist(energy, mu, kT):
"""Compute the Fermi distribution at energy, mu and kT."""
# YOUR CODE HERE
#raise NotImplementedError()
F = 1/(np.e**((energy-mu)/kT)+1)
return F
In [16]:
assert np.allclose(fermidist(0.5, 1.0, 10.0), 0.51249739648421033)
assert np.allclose(fermidist(np.linspace(0.0,1.0,10), 1.0, 10.0),
np.array([ 0.52497919, 0.5222076 , 0.51943465, 0.5166605 , 0.51388532,
0.51110928, 0.50833256, 0.50555533, 0.50277775, 0.5 ]))
Write a function plot_fermidist(mu, kT)
that plots the Fermi distribution $F(\epsilon)$ as a function of $\epsilon$ as a line plot for the parameters mu
and kT
.
In [66]:
def plot_fermidist(mu, kT):
# YOUR CODE HERE
#raise NotImplementedError()
plt.plot(np.linspace(0.0,10.0), fermidist(np.linspace(0.0,10.0),mu,kT),color="c")
plt.box(False)
plt.tick_params(axis='x', top="off")
plt.tick_params(axis='y',right ="off")
plt.xlabel('Energy')
plt.ylabel('F($\epsilon$)')
plt.title('Fermi Distibution')
plt.xlim(-.1,11)
plt.ylim(-.1,1.01)
In [67]:
plot_fermidist(4.0, 1.0)
In [ ]:
assert True # leave this for grading the plot_fermidist function
Use interact
with plot_fermidist
to explore the distribution:
mu
use a floating point slider over the range $[0.0,5.0]$.kT
use a floating point slider over the range $[0.1,10.0]$.
In [68]:
# YOUR CODE HERE
#raise NotImplementedError()
interact(plot_fermidist, mu=[0.0,5.0], kT=[0.1,10.0]);
Provide complete sentence answers to the following questions in the cell below:
Use LaTeX to typeset any mathematical symbols in your answer.
When $kT$ is low the graph has a steep drop. When $kT$ is high the graph is basically linear. The area under the graph is larger for high $\mu$ than low $\mu$. When the chemical potential ($\mu$) is high, the area under the curve is greater so there is a larger number of particles in the system.