Visualizing Wave Packets

Purpose:
The purpose of this experiment is to teach students to understand and visualize wave packets. Student will practice the Python program to plot some functions such as Gaussian function, sinusoidal function and superposition of fourier components. Students will use the integral in   to determine the Shrodinger equation depending on function B(k). The function B(k) is given   .  Moreover, student will use the program called Python to plot the graphs of the Shrodinger equation versus x(L).

Procedure:

Gaussian function:
Sample code:

from pylab import *
center = 2
sigma = .5
coeff = 1/(sqrt(2*pi)*sigma)
gauss_list = []
for x in arange (0,10,.1):
    gauss = coeff * exp(-(x-center)**2/(2.*sigma**2))
    gauss_list.append(gauss)
plot(gauss_list)
show()

Sinusoidal function:
sample code:
from pylab import *
A=1
w=1
for i in range (1,4):
    x=[]
    sine_function=[]
    for t in arange(-3.14,3.14,0.01):
        sine_f=A*sin(i*w*t)
        sine_function.append(sine_f)
        x.append(t)
    plot(x,sine_function)
show()

Superpostion:

from pylab import *
A=1
w=1
Fourier_Series = []
for i in range (1,3):
    x = []
    sine_function = []
    for t in arange (-3.14,3.14,0.01):
        sine_f = A*sin(i*w*t)
        sine_function.append(sine_f)
        x.append(t)
    #plot(x,sine_function)
    Fourier_Series.append(sine_function)

superposition = zeros(len(sine_function))
                     
for sine_function in Fourier_Series :
    for i in range(len(sine_function)):
        superposition[i]+= sine_function[i]
                    
plot(x,superposition)
show()

The Gaussian Wave Packet
from pylab import *
center = 3
sigma = 1
coeff = 1/sqrt(2*pi)*sigma

#define constant
A=1
w=1
Fourier_Series =[]

for i in range (1,6):
    x=[]
    sine_function=[]
    A=coeff*exp(-(i-center)**2/(2.*sigma**2))
    print A
    for t in arange(-3.14,3.14,0.01):
        sine_f=A*sin(i*w*t)
        sine_function.append(sine_f)
        x.append(t)
       # plot(x,sine_function)
    Fourier_Series.append(sine_function)
    superposition = zeros(len(sine_function))
for function in Fourier_Series:
    for i in range(len(function)):
        superposition[i]+=function[i]
plot(x,superposition)
show()

Derive the Shrodinger equation:

Plotting the graph B(x) vs. k for ko = 2pi/L

Sample code:
from pylab import*
A=1
l=1
k=2*pi/l
B_list=[]
for l in arange(-3.14,3.14,0.1):
        B=1/k
        B_list.append(B)

plot(B_list)
show()

B(k) vs. k

Plotting the Graph versus for the case , where is a length.
Sample code:
from pylab import *
A=1
l=1
k=2*pi/l
y_function=[]
for i in range (1,3):
    y=[]
    y_function=[]
    for x in arange(-3.14,3.14,0.01):
        y_f=sin(k*x)/(k*x)
        y_function.append(y_f)
        y.append(x)
plot(y,y_function)
show()

  vs.  x(L)

 Graph versus for the case , where is a length.


 Sample code:
from pylab import *
A=1
l=1
k=pi/l
y_function=[]
for i in range (1,3):
    y=[]
    y_function=[]
    for x in arange(-6.28,6.28,0.01):
        y_f=sin(k*x)/(k*x)
        y_function.append(y_f)
        y.append(x)
plot(y,y_function)
show()

  vs.  x(L)

Summary:
The experiment successfully taught students how to use a simple programming to graph some functions which are hard to do by hands. Moreover, students can understand the wave packets. Students can plot a large number of sine functions and add them together.

Relativity of Time and Length

Purpose:
The purpose of this activity is to understand how the time interval between the two events and the length of the objects in spaces depending on the inertial frame of reference. Based on the relativity, the clock of an object moving with the constant velocity runs slower compared with the rest one. This is demonstrated by the relationship between the time interval of the moving and stationary light clock:

∆t = γ∆t₀,     

where ∆t is the time interval of the moving light clock, γ is the Lorentz factor, and ∆t₀ is the time interval of the stationary light clock.

Not only the time interval, but also the length differs depending in the inertial frame of reference. An object moving with constant velocity sees the length of the object to be contracted compared to the object in the rest frame. The relationship is shown by the equation:

l= l₀/γ 

Where l is the length in the moving frame, and l₀ is the length in the rest frame

Questions and images:

Part 1: relativity of time

Question 1: The distance traveled by the light pulse on the moving light clock is longer than the distance traveled by the pulse on the stationary light clock

Question 2: The time interval for the light pulse to travel on the moving light clock is longer compare to on the stationary light clock.

 

Question 3: If the time is measured in the moving moving light clock, it takes the same amount of time interval to complete a round trip.

Question 4: When the velocity of the light clock is decreased, the difference in the light pulse travel time between the earth's timer and the light clock's timer will be decreased.

Question 5: when the light clock has a Lorentz factor (γ) of 1.2, it will take Δt = 8.00 µs for the light pulse to travel back and forth as measured on the earth. 

Question 6: Calculate the Lorentz factor (γ) if Δt = 7.45 µs. Lorentz factor (γ) = 1.116

 Part 2: Relativity of length

Question 1: As a pulse of light travels back and forth the light clock, the time interval of this round-trip does not depend on whether the light clock is moving or stationary relative to the earth.

Question 2: The round-trip time interval for the light pulse as measured on earth is longer than the time interval measured on the light clock.

Question 3: Yes, the length of the moving light clock is contracted by the Lorentz factor (γ) in order to obey the time-dilation relation.

Question 4: The light clock is 1000m long when measured at rest, it would be 769 m when the light clock had the Lorentz factor of 1.3.

Summary:
      The first part explains the concept of relativity of time. As shown in the pictures, the distance of the light pulse traveled on the moving light clock is lager the stationary light clock. Because the moving light clock moving with constant velocity, the light pulse leaved the light clock and returned at the different position in space. The path of the light pulse traveled is the hypotenuses in the moving light clock. Because its path is hypotenuses, it takes longer time for the moving light clock to complete a round-trip than the stationary light clock. If the observer is moving with the moving light clock, the time interval for a round-trip of the light pulse does not change, because the observer is moving the same speed with the light clock. In that case, the light clock is always stationary in the observer's frame of reference.
      The second part explains the concept of length contraction in space. If the observer is riding on the left end of the light clock, the time interval for the light pulse travels a round-trip is the same whether the light clock is moving or stationary relative to the earth. This is because the observer is moving with the light clock, in the observer's frame, the light clock is stationary. However, if the light clocked is measured on the earth, the light clock is moving with the a constant velocity relative to the earth frame. The light pulse travels a longer distance than the object. Thus, it takes longer time interval for the light pulse travel back to the left end. As shown in the pictures, the length of the object is smaller in the moving light clock than in the stationary light clock. If the length is the same, the time interval as measured on the earth is not equal to γ∆t₀

. This is because the light clock is moving with a certain speed, so the time interval will become even greater since it has to travel a greater distance. The length of the moving object is equal to l₀/γ 

Experiment 12: CD Diffraction

Purpose:
The purpose of this experiment is to use the concept of light interference and diffraction when a light beam goes through a grating to measure the distance between the grooves for a CD and a DVD. When the grooves spacing for a CD and a DVD are experimentally found, the results will be compared with the actual values are computed by the factory. The standard grooves spacing for a  CD is 1600nm and DVD is 740nm.

Procedure:
1. Obtained a meter stick, a laser beam, a CD, a DVD, and a screen with a small center hole.
2. Set a laser beam pointing through the hole of a screen to the CD or DVD.

3. Adjusted the disks so that the zero order maximum reflected light strikes through the hole of the screen and the first order maxima appeared on the screen.

Data Analysis for the CD:

	λ (nm)	Distance (l)  between the screen and the disk (cm)	Distance (x) between the  hole and the bright fringe (cm)
Trial 1	633	32.4 ± 0.5	14.8 ± 0.5
Trial 2	633	34.8 ± 0.5	16.0 ± 0.5
Average	633	33.6 ± 0.5	15.4 ± 0.5

Results:

Experimental grooves spacing (nm)	Standard spacing  (nm)	% error	% uncertainty error
1519 ± 112	1600nm	5 %	7 %

Data Analysis for the  DVD:

	λ (nm)	Distance (l)  between the screen and the disk (cm)	Distance (x) between the  hole and the bright fringe (cm)
Trial 1	633	5.5 ± 0.3	9.1 ± 0.3

Results:

Experimental grooves spacing (nm)	Standard spacing  (nm)	% error	% uncertainty error
739 ± 89	740nm	0.14 %	12 %

Sample calculations:

how to calculate the grooves spacing

 uncertainty calculation

Summary:
According to the results, the % error between the experimental grooves spacing of the CD and DVD comparing to the standard values are 5% and 0.14% respectively. These % error are within experimental uncertainties. The error could be from the scratch on the surface of the disks and some uncertainties of the measurement.
The grooves spacing depends on the wavelength of the laser light they use to shine the disk. Therefore, the shortest wavelength they can use now is the the gamma ray which should give the spacing in order of

10^-12 m. One way to improve the capacity of a DVD or CD is to add more layers. However, these layers have to have different pitch so the incoming light will not be destructive interference.