Planck's Constant from an LED

Statement of Purpose

With the ideas from diffraction, this experiment will show evidence supporting planck's constant.

Procedure 

Setup an L-shape for simple Pythagorean math use.

 Measure the distance of spectra.

Data

The white light is actually a combination of the 4 wavelengths from the colored LEDs. Which explained the gaps in the spectrum.

Conclusion
The small angle approximation makes calculation and measuring much simpler.

Color and Spectra

Statement of Purpose
To see individual wavelengths of white light through a diffraction grating.

Procedure

 The L-shape allows us to use pythagorean theorem.

Picture of the spectrum taken by the phone.

Data

With hydrogen light,

 

Conclusion
The hydrogen light showed discrete wavelengths whereas the white light showed a continuous spectrum of wavelengths.

Quantum Mechanics: Potential Well

A particle is trapped in a one-dimensional region of space by a potential energy function which is zero between positions zero and L, and equal to U₀ at all other positions. This is referred to as a potential well of depth U₀.
Examine a proton in a potential well of depth 50 MeV and width 10 x 10^-15 m.

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Question 1: Infinite Well
If the potential well was infinitely deep, determine the ground state energy. Is this also the ground state energy in the finite well?

E_1(infinite well) = (1)^2(h)^2/(8*(mass of proton)*(10*10^-15) = 2.05MeV
E_1(finite well) = 1.8MeV

The ground state energy of an infinite well is more than the ground state energy of a finite well.

Question 2: First Excited State
If the potential well was infinitely deep, determine the energy of the first excited state (n = 2). Is this also the energy of the first excited state in the finite well?

E_2(infinite) = 4E_1 = 8.20MeV.
E_2(finite) = 6.8 MeV
The energy of the first excited state in the finite well is not the same as the one in the infinite well.

Question 3: "Forbidden" Regions
Since the wavefunction can penetrate into the "forbidden" regions, will the energy of the first excited state in the finite well to be greater than or less than the energy of the first excited state in the infinite well? Why?

The energy in the first excited state in the finite well is less than the first excited state in the infinite well due to the greater probability of tunneling.

Question 4: More Shallow Well
Will the energy of the n = 3 state increase or decrease if the depth of the potential well is decreased from 50 MeV to 25 MeV? Why?

The energy of the n=3 state decreases if the potential well is decreased from 50MeV to 25MeV due to less tunneling.

Question 5: Penetration Depth
What will happen to the penetration depth as the mass of the particle is increased?
As the mass of the particle increases the penetration depth decreases

Quantum Mechanics: Potential Energy Diagrams

A particle of energy 12 x 10^-7 J moves in a region of space in which the potential energy is 10 x 10^-7 J between the points -5 cm and 0 cm, zero between the points 0 cm and +5 cm, and 20 x 10^-7 J everywhere else.

Question 1: Range of Motion
What will be the range of motion of the particle when subject to this potential energy function?


The particle is between +-5 cm.


Question 2: Turning Points
Clearly state why the particle can not travel more than 5 cm from the origin.

The energy that the particle has is less than the energy at the top of the well.

Question 3: Probability of Detection
Assume we measure the position of the particle at several random times. Is there a higher probability of detecting the particle between -5 cm and 0 cm or between 0 cm and +5 cm?

The particle is most likely to be found between -5cm and 0cm because the particle has less kinetic energy at U_1 it moves slower, thus spending more time there.

Question 4: Range of Motion
What will happen to the range of motion of the particle if its energy is doubled?


The range of motion increases

Question 5: Kinetic Energy
Clearly describe the shape of the graph of the particle's kinetic energy vs. position.

The shape is an concave down parabola.

Question 6: Most Likely Location(s)
Assume we measure the position of the particle at several random times. Where will the particle most likely be detected?

The edges.

Measuring Plank's Constant

Relativity of Time and Length

Relativity of Time
 
Question 1: Distance traveled by the light pulse

How does the distance traveled by the light pulse on the moving light clock compare to the distance traveled by the light pulse on the stationary light clock?

Answer 1: The distance travelled by the stationary clock is longer by a factor of gamma=1.41

Question 2: Time interval required for light pulse travel, as measured on the earth
Given that the speed of the light pulse is independent of the speed of the light clock, how does the time interval for the light pulse to travel to the top mirror and back on the moving light clock compare to on the stationary light clock?

Answer 2: The time of the stationary clock is longer by (9.4 – 6.67)*10^-6 s = 2.73 *10^-6 s .

Question 3: Time interval required for light pulse travel, as measured on the light clock
Imagine yourself riding on the light clock. In your frame of reference, does the light pulse travel a larger distance when the clock is moving, and hence require a larger time interval to complete a single round trip?

Answer 3: You observe a shorter distance and a longer time interval than the stationary observer.

Question 4: The effect of velocity on time dilation
Will the difference in light pulse travel time between the earth's timers and the light clock's timers increase, decrease, or stay the same as the velocity of the light clock is decreased?

Answer 4: The difference in light pulse travel time will decrease.

Question 5: The time dilation formula
Using the time dilation formula, predict how long it will take for the light pulse to travel back and forth between mirrors, as measured by an earth-bound observer, when the light clock has a Lorentz factor (γ) of 1.2.

Answer 5: (6.67*10^-6)(1.2) = 8.004*10^-6s

Question 6: The time dilation formula, one more time
If the time interval between departure and return of the light pulse is measured to be 7.45 µs by an earth-bound observer, what is the Lorentz factor of the light clock as it moves relative to the earth?

Answer 6:

7.45*10^-6 = (6.67*10^-6) γ

γ = 7.45/6.67

γ = 1.12

Relativity of Length

Question 1: Round-trip time interval, as measured on the light clock
Imagine riding on the left end of the light clock. A pulse of light departs the left end, travels to the right end, reflects, and returns to the left end of the light clock. Does your measurement of this round-trip time interval depend on whether the light clock is moving or stationary relative to the earth?

Answer 1: Yes, the moving light clock will experience a time interval longer than the stationary clock by a Lorentz factor.

 

Question 2: Round-trip time interval, as measured on the earth
Will the round-trip time interval for the light pulse as measured on the earth be longer, shorter, or the same as the time interval measured on the light clock?

Answer 2: The round-trip time interval measured on the earth is longer by a Lorentz factor.

 

Question 3: Why does the moving light clock shrink?
You have probably noticed that the length of the moving light clock is smaller than the length of the stationary light clock. Could the round-trip time interval as measured on the earth be equal to the product of the Lorentz factor and the proper time interval if the moving light clock were the same size as the stationary light clock?

Answer 3:  Yes, since there is no length contraction the Lorentz factor is 1.

Question 4: The length contraction formula
A light clock is 1000 m long when measured at rest. How long would earth-bound observer's measure the clock to be if it had a Lorentz factor of 1.3 relative to the earth?

Answer 4: 1000m/1.3 = 769.23m


Conclusion
When objects move at speeds approaching the speed of light, length contracts and time dilates from the moving observer's perspective relative to a stationary observer's perspective.  

Polarization

Statement of Purpose
To observe the change of light intensity through a polarizer as a function of the polarizer's angle.

Procedure

Setup the polarizer to 0 degrees so that maximum intensity passes.

LoggerPro sensor is setup at the other end

Data

The curve is roughly the same as a cos^2 function.

This time an additional polarizer is placed

Interestingly, when two polarizers are 90 degrees with respect to each other which means that no light can pass through two such polarizers, but when a third mediating polarizer is placed in the middle the light is polarized by 45 degrees. When it passes through the last polarizer it changes by another 45 degrees, summing to a total of 90 degrees. This means that the the outgoing light is 0 degrees out of phase with respect to  

Conclusion
Does the light from the fluorescent bulb have any polarization to it? If so, in what plane is the light polarized? How can you tell?

No, since the light that passes through the polarizer at all angles, the light can not be polarized

Does the reflected light have a polarization to it? If so, in what plane is the light polarized? How can you tell?

Yes, the polarizer is polarizing the light and decreasing the intensity of the light.

CD Diffraction

Statement of Purpose
Observe the diffraction of parallel light rays through multiple slits.

Procedure

Part 1
Set up the following

With given distance between slits in the diffraction grating, the target variable is the wavelength of the light. The distance from the Huygen sources to the screen is separated by one meter stick and the distance between maximas is measured by another meter stick.

The following calculation determined that the wavelength of the laser is (660 +- 27.6) nm which affirms the wavelength specified on the laser.

Part 2

With the given wavelength, this setup attempts to find the separation of the gratings on a compact disk (CD). The following measurements were made: the distance from the CD to the screen and the average distance between maximas.

Conclusion
The actual separation of the gratings in the CD is 1600 micrometers which compared to our experimental data makes an error of about 40.75%. This error comes from the angle at which we adjusted for the CD. A more accurate experiment would have the CD perfectly parallel to the screen.

Light Interference

Statement of Purpose
To measure the width of a hair using the two-slit interference principle.

Procedure

Set up the following.

The distance from the source to the screen was separated by one meter stick.

 

The average distance between two maximas were measured by a caliper. 

The control measurement of the hair was measured by a micrometer.

Data

Conclusion
According to CAQTI Cosmetics, Inc. the diameter of a typical black hair is between 56-181 micrometers which our experiment affirms.

Thin Lenses

Statement of Purpose
To observe the effects on light through a thin lense.

Procedure
Measure the focal length of the lense with two parallel rays.

The distance at which the two parallel rays converge is the focal length of the lense.

f = 5 +- 1cm

We focus light at a particular distance for several distances and measure the height of the object for which we used the tip of the arrow.

Data
d0 (± .1 cm) di (± .1 cm) h0 (± .1 cm) hi (± .1cm) M
5f = 25.000 7.000 1.800 0.400 0.23 ± 0.07
4f = 20.000 7.300 1.800 0.600 0.34 ± 0.07
3f = 15.000 8.200 1.800 0.700 0.39 ± 0.08
2f = 10.000 10.500 1.800 1.800 1.0 ± 0.1
1.5f =7.500 18.700 1.800 4.100 2.3 ± 0.2

 Object distance vs Image distance

Inverse Image Distance vs Negative Inverse Object Distance

Slope = .9475
y-intercept = .184

We see that at distances behind the focal length the virtual image reflects off of the back wall of the lense.

Conclusion
The y-intercept of the graph represents the 1/f in the lense maker's equation and since the slope is close to one we can write the equation of the graph.

y = x + (1/.184)

This equation relates the object distance vs the image distance.

Concave and Convex Mirrors

Statement of Purpose
To explore the image formed by both concave and convex mirrors

Procedure

Convex Mirror

 

Nearby image

The image appears at the center of the mirror about the same size, maybe smaller and upright.

Distant Image

At a further distance, the image decreased in size.

 The observations that were made agree with phase diagrams listed here.

Sample calculation

 

 

Convex mirror

Distant image

 The image here is focused at the center of the mirror which appears smaller and inverted.

Nearby Image
 

At a closer distance the image increases in size and is still inverted.

The phase diagram shows that if the object distance decreases and the image distance increases the height of the object will increase, perfectly describing our observation.

sample calculation

Conclusions 
The purpose of the mirrors has been observed. That is why convex mirrors are used in small markets for a lower magnification, hence wider view of the market, whereas concave mirrors are handy for putting on make-up because the higher magnification.