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]:
def fermidist(energy, mu, kT):
"""Compute the Fermi distribution at energy, mu and kT."""
K=kT
F=1/(np.exp((energy-mu)/K)+1)
return F
In [13]:
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 [68]:
def plot_fermidist(mu, kT):
energy=np.linspace(0.0,10,50)
distr=fermidist(energy,mu,kT)
f=plt.figure(figsize=(8,6))
ax=plt.gca()
plt.plot(energy,distr)
ax=plt.gca()
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.get_xaxis().tick_bottom()
ax.get_yaxis().tick_left()
plt.title('Fermi Distribution')
plt.xlabel('Energy')
plt.ylabel('Distribution') # can't figure why these won't work
In [69]:
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 [58]:
interact(plot_fermidist, mu=(0.0,5.0,0.1),kT=(0.1,10,0.1))
Provide complete sentence answers to the following questions in the cell below:
Use LaTeX to typeset any mathematical symbols in your answer.
When the temperature of $kT$ is low the distribution vs energy becomes exponential. When the temperature of $kT$ is high the distribution vs energy becomes linear. Changing $\mu$ shifts the graph vertically. The higher the chemical potential ($\mu$) the more particles there are.
In [ ]: