Physics 4C - CABazua: November 2012

LED's

Objective

To determine a value for h, Plancks constant, using light emitting diodes and a diffraction grading

Procedure

Equipment

Voltage Source

Aligator Clips

Variable Resistance

LED's

Diffraction Grading

Meter Sticks

Diffraction Gratimg

Circuit

Spectra

LED

First each different color LED is hooked up in the circuit and the resistance is adjusted until the light can barely be seen, at this point the voltage across the Light Emitting Diode is recorded. This is the minimum voltage to excite an the electrons in the diode.Two meter sticks are set up at a right angle with a diffraction grading is placed on one meter stick. An Led is then shined towards the diffraction grading and one student looks through it while another writes down the position of the spectral lines based on what the other see.

Figure 1

using the formula and calculations shown in figure 1. Plancks Constant is measured.

The error with each is as follows:

Green 12.6%

Blue 18.4%

Yellow 22.2%

Red 20.6%

These values are all of the right magnitude however there is a large systematic error for this procedure, mostly likely due to the accuracy of the measurements taken in performing this experiment.

However when considering the equipment used and the time spent on the procedure it is quite remarkable to come so close to the value of a constant whose value controls much of what we see everyday.

Active Physics

There are 20 particles in the unexcited states. Photons are coming in from the left with enough energy to excite the particles to an excited state (shown in red)

Observations:

Conservation of energy is observed by the fact that every photon that has entered the cavity is accounted for either by remaining in the cavity, exiting the cavity or being absorbed by a particle. (excited to red)

This spontaneous emission mode shows initially all the excited particles going to a lower energy ground state by emitting a photon. The direction on the photon is random.

The time a particle remains excited is random

Stimulated emission mode sends in photons which remove a particle out of its excited state and it shows 2 photons leaving, one that was incident and the other from lowering the energy. The emitted photon is in the same direction as the incident photon. They are also in phase.

In the laser mode it includes the excitation, spontaneous and stimulated emissions and you can control the amount of photons being put into the cavity. At low pumping levels the amount of photons exiting roughly equal the amount entering and a majority are in the unexcited state.

When the pumping level reaches at least 40 out of 100 a population inversion is seen

It is seen that there is an amplification of the number of photons once this population inversion is seen. Occasionally a photon no in the direction of incidence is observed. This occurs due to spontaneous emission that takes place before a photon can shove it to a lower state.

Color and Spectra

Objective

To visualize the quantization of energy levels by viewing emitted photon wavelengths and using spectroscopy to identify an unknown gas.

Equipment

Apparatus

2 meter sticks (one 2m and the other 1m)
Stands
A Diffraction Grading
Incandescent lamp
Hydrogen Tube
florescent tube
Misc Tube

Procedure

Place the meter sticks at a 90 degree angle as close as possible to the to the light source, taking high voltage precautions (i.e. only handle lamp with one arm)
The lights in the room were then shut off. The diffraction grating was then placed at a known distance from the lamp. One future engineer observes through the grating while another moves his finger over the other meter stick and notes the distances at which the former sees the spectra.
We observed a violet to red spectrum with violet being closest to the bulb. The wavelengths recorded were:

violet edge	36.7	±	0.5	m
blue	45.5	±	0.5	m
green	53.7	±	0.5	m
red	70.3	±	0.5	m
red edge	83.5	±	0.5	m

we derove the following:

Values for our experiment:
L=1.96 0.003m
d=0.0020 mm

Using the true values for the colors viewed in the experiment along with the measured values a correcting linear function was obtained as shown below

Having this equation allowed us to correct any errors in future wavelengths.

For the next part, to test our calculation and the theoretical values of hydrogen spectrum, we examined the spectrum lines for hydrogen.
the procedure was repeated to view the spectra of the Hydrogen lamp yielding the values noted below. as well as the percent error in our measurements.

Diffraction Grating

Finally we received a tube with an unknown gas from our professor and repeated the experiment once again obtaining these values.

Using the internet to look up spectrum lines and values, we discovered that our unknown was Xenon.

Wave Packets

Objective

To simulate wave packets and gain insight about them

VPython was used 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()

----------------------------------------------------

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()

------------------------------------------

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()

-------------------------------------------------------

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 $\psi(x)=\int_{0}^{\infty}B(k) \:{\rm cos}\: kx \: dk$ , determine the wave function $\psi \left( {x} \right)$ for a function $B\left( k \right)$ given by
$B\left( k \right) = \left\{ {\begin{array}{*{20}c} 0 & {k < 0} \\ {1/k_0 {\rm{,}}} & {0 \le k \le k_0 } \\ {0,} & {k > k_0 } \\ \end{array}} \right.$ . This represents an equal combination of all wave numbers between 0 and k_0

. Thus $\psi \left( x \right)$ represents a particle with average wave number k_0 /2

, with a total spread or uncertainty in wave number of k_0

. We will call this spread the width $w_{\rm k}$ of $B\left( k \right)$ , so $w_{\rm k} = k_0$ .

a. Graph $B\left( k \right)$ versus for the case $k_0 = 2\pi /L$ , where is a length.

This is a straight line with a height of 2pi/L

b. Graph $\psi \left( {x} \right)$ versus for the case $k_0 = 2\pi /L$ , where is a length.

c. Locate the two points closest to this maximum (one on each side of it) where $\psi \left( x \right) = 0$ , and define the distance along the -axis between these two points as $w_{\rm x}$ , the width of $\psi \left( x \right)$ . What is the value of $w_{\rm x}$ if $k_0 = 2\pi /L$ ?
This is 1.00 L

d.Repeat part C for the case $k_0 = \pi /L$ .

e. Repeat part D for the case $k_0 = \pi /L$ .

f. The momentum is equal to $hk/2\pi$ , so the width of in momentum is $w_{\rm p} = hw_k /2\pi$ . Calculate the product $w_{\rm p} w_{\rm x}$ for the case $k_0 = 2\pi /L.$

This is h

g. Calculate the product $w_{\rm p} w_{\rm x}$ for the case $k_0 = \pi /L.$

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.

Thursday, November 29, 2012

LED's

Objective

Procedure

Equipment

Active Physics

Color and Spectra

Wave Packets