Physics 4C dlbruns: 2012

LED: Energy Bands of a Semiconductor

Purpose: To use an understanding of p-n junctions in an LED light to measure plank's constant.

Introduction: A light-emitting-diode is a semiconducting diode which is used for the emission of light. A voltage difference excited electrons across a band gap and when they return to the lower energy band a photon is emitted. We can use the emitted wavelength and voltage to calculate plank's constant.

Equipment:

Voltage source
Alligator Clips
Variable Resistance
Volt Meter
Various colored LED lights
A Diffraction Grading
2 Meter Sticks

Procedure: Set up the circuit shown in the picture to the side where the source is attached to your variable resistance then to the LED. Make sure to keep some resistance in the circuit at all times, and to make sure your LED is connected correctly as voltage will only flow in one direction.

Next increase the resistance until the light is barely visible (we want the minimum voltage required to light the bulb (excite the electrons). Record the voltage across the LED. Do this for various colors.

The second part of this experiment requires the determination of wavelengths. To do this we set up a spectrometer (as is described in my Color and Spectra Lab). Using this set up sine the bulb at the diffraction grading from the end on the meter stick and with one person at the diffraction grading direct another person the spot of most intensity on that color as seen on the second meter stick. Use this to determine the wavelengths as shown below.

The LEDs will give a mixture of colors for the spectra unlike the isolated gasses.

Data and Analysis:

The error with each is as follows:

Green 12.6%

Blue 18.4%

Yellow 22.2%

Red 20.6%

The values of Plank's constant are fairly close. At our best we are 12.6% low of the accepted value. There is a systematically low error for this lab which agreed with the rest of the class but we are very close.

The green was the most accurate and this is because the human eye is most sensitive to the wavelengths corresponding to green.

ActivPhysics Lasers

Purpose: To understand how lasers function and how they are practical applications of quantum mechanics.
We will use ActivPhysics simulations from the following website:
http://media.pearsoncmg.com/bc/aw_young_physics_11/pt2a/Media/AtomicPhysics/1803Laser/Main.html

Introduction: 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 seems to be random as well.

Stimulated emission mode sends in photons which "knock" a particle out of its excited state and (since it is already excited) 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 (more particles in the excited state).

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 "knock" it to a lower state.

Color and Spectra

Purpose: To visualize the quatization of emery levels by viewing emitted photon wavelengths and to use spectroscopy to identify an unknown gas.

Equipment:

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

Procedure: Place the 2-meter meters stick on a flat surface and place a diffraction grading at one end and an Incandescent lamp at the other. Perpendicular to this stick and close to the lamp place the other meter stick. Since we are using high voltage we must take the appropriate precautions. We place the florescent tube in the lamp being careful to only use one hand.
We turn on the lamp and dim the other lights. By looking through the diffraction grating at the tube you should notice a rainbow of colors. With a partners help to identify the location on the second meter stick locate the edges of this spectrum and identify the middle of red, blue and the green color regions.

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

Using the ideas of constructive interference through a double slit we derive the following:

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

Then using the edge of the spectra we are examining and the accepted span of the visible spectra we can create an equation to correct any errors with our set up. To do this we calculate the wavelengths of the edge of the spectra (the lowest violet and furthest red) and then set up the following system of equations and solve.

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.
We set up a lamp that is high voltage and place a hydrogen tube in it with the same set up and practice as before (using two partners to determine where the spectrum lines are and plugging them into the wavelength equation and the correction equation) we calculate the wavelengths. We also include the percent error.

For the final part of this project we take a look at the application of the process of spectroscopy: Identifying an unknown. We were given an unknown gas and were asked to identify it. The gas in plasma state is shown below and the same procedures were used as in the hydrogen spectra to identify the wavelengths of the spectra.

We observed very faint lines in the red violet and cyan color regions. The black shows the wavelengths and their uncertainties.

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

Conclusion: Spectrometery is a very simple but very useful tool and we were able to use it to not only identify an unknown but verify very important aspects of quantum mechanics.

Gaussian Wave Packets

Purpose: It is to visualize and understand wave packets and see how the uncertainty principle relates to their construction. This will also prove as a visualization of the Schrodinger equation and probability distribution functions by superimposing various waves that will cause a localization of a particle and display its wave property.

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()

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

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()

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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 $\psi(x)=\int_{0}^{\infty}B(k) \:{\rm cos}\: kx \: dk$ , 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 2*pi/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. This corresponds to a more
localized particle (just as stated by the uncertainty principle. They all also have the same uncertainty which agrees with the principle.

Tuesday, November 27, 2012

LED: Energy Bands of a Semiconductor

ActivPhysics Lasers

Color and Spectra

Gaussian Wave Packets