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:
$\large F(\epsilon) = {\Large \frac{1}{e^{(\epsilon-\mu)/kT}+1}}$
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 [8]:
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 [9]:
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 [44]:
def plot_fermidist(mu, kT):
energy = np.linspace(0,10.0,21)
plt.plot(energy, fermidist(energy, mu, kT))
plt.tick_params(direction='out')
plt.xlabel('$Energy$')
plt.ylabel('$F(Energy)$')
plt.title('Fermi Distribution')
In [46]:
plot_fermidist(4.0, 1.0)
In [47]:
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 [48]:
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 slope at $Energy = \mu$ becomes much steeper, while when it's high, the line is much flatter. Changing $\mu$ shifts where the line flips concavity. The higher $\mu$ is, the more particles there are in the system.