Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.
In [4]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
In [5]:
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 [6]:
Image('fermidist.png')
Out[6]:
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 [47]:
def fermidist(energy, mu, kT):
exp = (energy-mu)/kT
F = 1/((np.exp(exp))+1)
if type(energy) or type (mu) or typle(kT) == np.array:
return np.array(F)
else:
return F
In [48]:
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 [49]:
def plot_fermidist(mu, kT):
#plt.figure(figsize = (15,5))
plt.plot(energy,fermidist)
#plt.ylabel('Fermidist Distribution')
#plt.xlabel('Energy')
#plt.title('Distribution vs. Energy')
#plt.grid(True)
#plt.box(True)
#plt.xlim(0,10.0,100);
#plt.ylim(0,10)
#axis = plt.gca()
#axis.spines['top'].set_visible(False)
#axis.spines['right'].set_visible(False)
#axis.get_xaxis().tick_bottom()
#axis.get_yaxis().tick_left();
In [50]:
plot_fermidist(4.0, 1.0)
In [35]:
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 [51]:
#interact(plot_fermidist , mu = (0.0,5.0))
#interact(plot_fermidist , 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.
YOUR ANSWER HERE