Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.
In [34]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
In [35]:
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 [36]:
Image('fermidist.png')
Out[36]:
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.
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def fermidist(energy, mu, kT):
"""Compute the Fermi distribution at energy, mu and kT."""
f = 1/(np.exp((energy-mu)/kT)+1)
return f
In [38]:
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
.
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def plot_fermidist(mu, kT):
plt.figure(figsize=(9,6))
plt.plot(np.linspace(0.0,10.0,100),fermidist(np.linspace(0.0,10.0,100), mu, kT),'r')
plt.xlabel('Energy')
plt.ylabel('Fermi Distribution')
plt.title('Fermi Distribution as a function of Energy')
plt.xlim(0.0, 10.0)
plt.ylim(0.0, 1.0)
plt.grid(True)
plt.box(False)
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plot_fermidist(4.0, 1.0)
In [41]:
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 [42]:
interact(plot_fermidist, mu=(0.0,10.0),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 $kT$ was low, the curve of $F$ became very steep. When $kT$ was high, the curve became much shallower. Changing the chemical potential $\mu$ shifts the graph to the right with an increase, and to the left with a decrease. A decrease in $\mu$ decreases the number of particles in the system and vice versa.