For this experiment we will use VPython in order to show first create a Gaussian distribution, then create a sine function, a superpositioons of various sine functions
To simulate a Gaussian function the following was code used:
from pylab import *
center=5
sigma=1
coeff=1/sqrt(2*pi)*sigma
gauss_list=[]
for x in arange (0,10,0.1):
gauss=coeff*exp(-(x-center)**2/(2.*sigma**2))
gauss_list.append(gauss)
plot(gauss_list)
show()
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To create a sine function the following was used:
from pylab import *
center=5
sigma=1
coeff=1/sqrt(2*pi)*sigma
omega=1
sine_list=[]
for x in arange (0,10,0.1):
sine=coeff*sin(omega*x)
sine_list.append(sine)
plot(sine_list)
show()
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To create a superposition of the sine functions:
from pylab import *
center=5
sigma=1
coeff=1/sqrt(2*pi)*sigma
sine_list=[]
A=5
omega=0.5
supaa = []
for i in range (1,4):
x=[]
sine_list=[]
for t in arange (-2*pi,2*pi,0.01):
sine=A*sin(i*omega*t)
sine_list.append(sine)
x.append(t)
plot(x,sine_list)
supaa.append(sine_list)
superposition= zeros(len(sine_list))
for function in supaa:
#print function
for i in range(len(function)):
superposition[i] += function[i]
#print superposition
plot(x,superposition)
show()
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For a Gaussian we have:from pylab import * #Need this for plotting functions
center = 3 #Define the center of the guassian
sigma = 1.0 #Set the standard deviation to 1
coeff = 1 / ((sqrt(2* pi))*sigma) #This the normalization coefficient#Define Constants
w = 1 #Set the frequency coefficient
gauss_list=[]
A=gauss_list
Fourier_Series = [] #Initialize the list of sine functions
#Calculate the harmonics of the sine functions
for x in arange(1,50):
gauss=coeff*exp(-(x-center)**2/(2.*sigma**2))
gauss_list.append(gauss)
for i in range(1,50):
x = [] #This will let us plot the value from -pi to pi
sine_function = [] #This contains the sine function
for t in arange(-3.14,3.14,0.01): #Loop from -3.14 to 3.14 by 0.1
sine_f=gauss_list[i-1]*sin(i*w*t)
sine_function.append(sine_f) #Add the calculated value to the list of values
x.append(t)
Fourier_Series.append(sine_function)
superposition = zeros(len(sine_function)) #set as zeros of length equal to the sine
for function in Fourier_Series:
for i in range(len(function)):
superposition[i]+= function[i]
plot(x,superposition)
show()
Questions:Using the integral in
, determine the wave function 
for a function 
given by 
. This represents an equal combination of all wave numbers between 0 and 
. Thus 
represents a particle with average wave number 
, with a total spread or uncertainty in wave number of . We will call this spread the width 
of , so 
.a. Graph
versus 
for the case 
, where 
is a length.
versus for the case , where is a length.
, and define the distance along the 
-axis between these two points as 
, the width of . What is the value of if ?This is 1.00 L
d.Repeat part C for the case 
.
.
e. Repeat part D for the case .
.
f. The momentum 
is equal to 
, so the width of 
in momentum is 
. Calculate the product 
for the case 
is equal to , so the width of in momentum is . Calculate the product for the case
This is h
g. Calculate the product for the case 
for the case
This is also h.
h. Discuss your results in light of the Heisenberg uncertainty principle.
The packets follow the uncertainty principle. As we have greater values of k(a wider range of k) which relate the uncertainty in the momentun increases. therefore the positions should be more precise.
This is seen from using various harmonics in the function and you see a large amplitude in the center that quickyly dies off as you leave the center. This corresponds to a morelocalized particle (just as stated by the uncertainty principle. They all also have the same uncertainty which agrees with the principle.
