Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.
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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
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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:
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Image('fermidist.png')
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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):
F=1/(np.exp((energy-mu)/kT)+1)
return F
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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):
energy = np.linspace(0,10.0,100) # Energy ranges from 0-10 with 100 points
F=1/(np.exp((energy-mu)/kT)+1) # We want this to be our Y axis.
plt.plot(energy,F,'g') # We want to plot the fermidist versus energy
plt.xlabel('Energy')
plt.ylabel('Fermidist Function')
plt.title(' My Beautiful Graph')
plt.box(False)
plt.grid(True)
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plot_fermidist(4.0, 1.0)
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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 $[0.1,10.0]$.
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interact(plot_fermidist, mu=(0.0,5.0,.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 kT is low, there is a high probability of locating the particle at low energies
-When kT is high, probability decreases for low energies but increases for high energies
-changing the chemical potential increases the probability of finding the particle at higher energies
-increasing chemical potential increases number of particles ( because area is increasing)
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