Continuous Spectrums and Line Spectra

The purpose of this lab was to explore continuous specturms and line spectra In three different scenarios. The first case will be a white light bulb with a continuous spectrum; in this case, the continuous spectrum will be measured and analyzed against known values. The second case, an unknown glass tube will be used to create spectral lines. From measuring those spectral lines, it will be possible to determine what the unknown gas is. The third case will be spectral lines created from a hydrogen tube. The lines from the hydrogen tube will be analyzed and compared to known values for hydrogen spectral lines.


The set up consisted of a light source (bulbs or gas tubes), two rulers and a diffraction grating. The diffraction grating was used to split the continuous spectrum or spectral lines. The split colors then showed up on the ruler that is perpendicular to the other ruler. 

The lab set up.

Mysterious green glowing light of doom...... It draws you in......You must obey!

Data for continuous spectrum.

	Start (cm)	End (cm)	Center (cm)
purple	37.1	41	39.05
blue	41	51.9	46.45
green	51.9	56.5	54.2
yellow	56.5	60.2	58.35
orange	60.2	64.5	62.35
red	64.5	76.1	70.3

	Theta (degrees)	Wavelength (nm)
purple	12	365
blue	14.2	431
green	16.2	491
yellow	17.3	523
orange	18.4	555
red	20.4	613

The length from the diffraction grating to the light source was 2.0 m. The angles were obtained using the tangent inverse with the distances from the center of each color to the light source and the two meter distance from the diffraction grating to the light source.The uncertainty in the measurements is on the order of +/- 2.0 cm; however, the uncertainty will be addressed with a linear adjustment to the experimental values. There is also error associated with the math used to obtained wavelength. The equation used to obtain the wavelength is the diffraction equation. 

The diffraction equation uses a small angle approximation. As can be seen in the data, the angle obtained from this setup is not relatively small; hence, the values will be significantly off and require a linear adjustment.The linear adjustment is a "best guess" rather than an actual linear fit to data points. The equation used for the first set of data is:

 Data with adjustments and comparison to known values.

	Wavelength Measured (nm)	Wavelength Adjusted (nm)	Wavelength Actual (nm)	% Difference
purple	365	418	400	4.5
blue	431	497.2	475	4.67
green	491	569.2	510	11.61
yellow	523	607.6	570	6.6
orange	555	646	590	9.5
red	613	715.16	650	10.1

After using the linear adjustment, all the experimental values fall within an acceptable range of known values.The percent difference for some wavelengths border on unacceptable; however, this can be attributed to an estimated linear correction.



 Data for unknown gas tube.

Spectral Line	Center (cm)	Theta (degrees)	Wavelength (nm)
indigo	46.3	13.6	413
green	58.2	16.8	507
yellow	64	18.3	551

Based off of the most visible spectral lines, the unknown gas is determined to be mercury. The wavelength values for the mercury spectra will be used in the data analysis.  A plot of of actual values vs measured values for wavelength was made to determine a linear correction factor. The equation for the line of the following graph is used to adjust the measured values. 





 The linear correction obtained from the actual wavelength vs measured wavelength plot is:

Spectral Line	Wavelength Measured (nm)	Wavelength Adjusted (nm)	Wavelength Actual (nm)	% Difference
indigo	413	433.565	430	0.829069767
green	507	536.495	548	2.099452555
yellow	551	584.675	577	1.330155979

After the correction, the data falls within an acceptable range of difference; furthermore, with these values matching so well the claim of the unknown gas being mercury is reasonably valid.






Data for hydrogen spectra.

Spectral Line	Center (cm)	Theta (degrees)	Wavelength (nm)
violet	46.5	13.6	415
blue	50.6	14.7	448
red	70.8	20	602

Plot of known vs unknown wavelength for linear correction.





Linear adjustment equation:

Spectral Line	Wavelength Measured (nm)	Wavelength Adjusted (nm)	Wavelength Actual (nm)	% Difference
violet	415	440.095	434	1.40437788
blue	448	478.408	486	1.562139918
red	602	657.202	656	0.183231707

The slope is a bit larger than the previous linear equations used for adjustment. This is due to a larger error associated with the measured wavelength. However, the data still falls within an acceptable range of difference for this set up.

Computational Programing: Visualizing Wave Packets

This is a computational project to help understand probability distributions and wave natures of particles. By adding a bunch of sinusoid together, it's possible to arrive at a probability distribution for a particle. The proceeding text and pictures are code (python) being used in the mathematical model and images of the program in work. Following the final stage of the model and pictures, a series of questions will be addressed.




Code (in python)

from pylab import *

#Define constantsConstants

w=1 #sets the frequancey coefficient

Fourier_Series=[] #initialize the list of sine functions

sigma = 10

coeff = 1

numberofharmonics=50

center = numberofharmonics/2

#Calculate the harmonics of the sine functions

for i in range(1,50):#every time the loop repeats this will change the harmonic

    x = [] # plots from -pi to +pi

    gauss = coeff*exp(-(i-center)**2/(2.*sigma**2))#Define the amplitude of the sin function

    sin_list = [] #this includes the sine functions

    for t in arange (-3.14, 3.14, 0.01): # create tsh range of the graph and incriments

        sine= gauss*sin(i*w*t) #the function to be plotted

        sin_list.append(sine) # Adds the calculated vale from the sine function to the list appendex

        x.append(t)

    #plot(x,sin_list)# plots values

#show()

    Fourier_Series.append(sin_list)

superposition = zeros(len(sin_list))

for function in Fourier_Series:

    for i in range(len(function)):

        superposition[i]+=function[i]

plot(x,superposition)

show()

An early step in the program showing waves of different types.

Further along in the program with many waves displayed.

The envelope for the waves.

The program with a limiting factor included to make the wave superposition approach zero.

The final result: a super position of many waves limited to a given region, this can model wave packets. 

Questions:

Click to Enlarge.

A)

Taken from Mastering Physics.

 B)

Taken from Mastering Physics

C) 1.00 L

D) 2.00 L

E) Duplicate Question.

F) h, plank's constant is the answer.

G) h, plank's constant is the answer.

H)The questions in part F and G tie in strongly to the Uncertainty Principle. Both answers for part F and part G come out to be h. This shows there is a constraint on the accuracy to position and momentum which shows a strong parallel to the Uncertainty Principle.

Electron Trap: Particle in a Box

Modules were completed at Pearson's physics module website for section 20.2. The following are screenshots of corresponding answers and models.

20.2

Answers to questions 1-4.

Answer to questions 5-10.

Simulation used for 20.2