Wavelength of a gas lamp

Quantization of energy levels in atomic nuclei results in different colors of emission spectra for different elements. For example, this picture features a mecury gas lamp viewed through a diffraction grating.

It is clear that there are three spectral images formed to the right of the actual lamp. These three spectral images have colors corresponding to the wavelengths of photons emitted from mercury when one of its electrons is excited and then relaxes to a lower energy state.

Our experimental apparatus, as visible in the picture, consists of a 2 meter stick, and a meter stick arranged orthogonally on a table. The horizontal displacement of the spectral images can thus be converted into an angular relationship, or it can be left in units of length to solve for the wavelength of the light directly.

Similar to the CD diffraction experiment, the position of the spectral images and the spacing of the diffraction grating can be used to determine wavelength. The governing equation is:

λ = (D*d)/sqrt(L^2+D^2)
Where:
D = the horizontal distance of the image
d = the diffraction spacing
L = the distance between the diffraction grating and the gas lamp

This experimental setup yielded the following data for a hydrogen lamp placed in the same location:

position of image in cm:
1=48
2=54
3=74.5
uncertainty: .5cm


L = 2 meters
d = .000002 meters

This setup yields the following prediction for the wavelengths

1 = 466nm

2 = 523nm
3 = 698nm

Calibration of our apparatus against a white light, yielded a correction factor for our setup which included a systematic error of +37.7nm

When factored into the previous data, our final results are

1 = 428.3nm
2 = 485.3nm
3 = 660.3nm

when compared to the actual spectra for hydrogen:

1 = 434 nm
2 = 486 nm
3 = 656 nm

It appears our predictions are correct to within a single percent. This falls approximately in the same range as our uncertainty in horizontal position.

Particle in a box

Welcome to another edition of SOLVE THAT ACTIVPHYSICS PROBLEM. Our contestant today must answer 10 questions related to the particle in a box simulation at:
http://wps.aw.com/aw_young_physics_11/13/3510/898597.cw/index.html

Lets get this contest started.
Questions in order:
Question 1: Standing Waves
From your study of mechanical waves, what is the longest wavelength standing wave on a string of length L?

The longest wavelength for a standing wave is twice the length of the string, or twice the distance between the two hard boundaries.

Question 2: The de Broglie Relation
What is the momentum of the longest wavelength standing wave in a box of length L?

 de Broglie hypothesized that the momentum of a particle must be equal to plank's constant divided by the wavelength, which in this case is 2L.

Question 3: Ground State Energy
Assuming the particle is not traveling at relativistic speeds, determine an expression for the ground state energy. 

Ground state energy here refers to the kinetic energy stored in the particle, if we are operating at non-relativistic speeds then:
p=mv
K=.5mv^2
p^2 = (mv)^2
p^2/(2m) = K

then, applying p = h/2L
K = (h/2L)^2/(2m)
K = h^2/(8mL)
Question 4: Increasing L
If the size of the box is increased, will the ground state energy increase or decrease?
It will decrease, as the relationship places L as the divisor of the energy equation.


Question 5: The Correspondence Principle: Large Size
In the limit of a very large box, what will happen to the ground state energy and the spacing between allowed energy levels? Can this result explain why quantum effects are not noticable in everyday, macroscopic situations?

As L approaches the scale of human objects, the ground state energy takes on a magnitude of 10^-37 Joules, which is negligible on macroscopic scale.

Question 6: The Correspondence Principle: Large Mass
In the limit of a very massive particle, what will happen to the ground state energy and the spacing between allowed energy levels?

Massive particles also act as divisors in the energy equation, thus large particles have low levels of ground stage energy.


Question 7: Ground State Probability
If a measurement is made of the particle's position while in the ground state, at what position is it most likely to be detected?

The particle is most likely to be found in the middle of the trap.

Question 8: Probability: Dependence on Mass and Size
The most likely position to detect the particle, when it is in the ground state, is in the center of the box. Does this observation depend on either the mass of the particle or the size of the box?

No, the wave function of the particle is independent of mass, and always has a maximum at L/2.


Question 9: Probability: Dependence on Energy Level
The most likely position to detect the particle, when it is in the ground state, is in the center of the box. Does this observation hold true at higher energy levels?

No, for instance the first excited state is more likely to be found at L/4 and 3L/4.

Question 10: The Correspondence Principle: Large n
In the limit of large n, what will happen to the spacing between regions of high and low probability of detection? Does this agree with what is observed in everyday, macroscopic situations?

As n approaches large values, the probability density function becomes almost continuously flat, or equally likely in every location. This is consistent with a free particle, which has no definite location.

Experiment 12: CD diffraction

Using the properties of diffraction, it is possible to measure the space between adjacent grooves on a CD. If you shine a laser on a CD and observe the spacing between the central maxima and the resulting accessory maxima it is possible to calculate the groove spacing by using the following geometric relationship.

Photo credits to sciwebhop

the distance between points A and B is notated as d. Θ is measured in degrees from the normal to the diffraction grating, and λ is the wavelength of the light incident on the diffraction grating. The following equation relates these variables.

d*sin(Θ)= m * λ

 Where m is the number of the maxima counting away from the central maxima.

When a laser is shone upon a CD the reflected pattern may appear something like this. 

It is clear from this image that some form of diffraction is taking place. By measuring the spacing between these maxima, and measuring the distance from the wall to our CD we can determine the angular separation of these maxima, additionally we know the wavelength of the laser to be 630-680 nanometers.

This table holds all of the relevant data.

 table

It is clear from the calculations that there is some sort of systematic error in our data collection, and it most likely stems from the fact that we were not measuring the appropriate phenomena. 

The predicted groove spacing based on our data is 2.6 * 10^-5 meters which is 16 times larger than the specifications listed for a CD(1.6 micrometers).

Unfortunately, we were unable to collect data after determining the flaw in our experiment and are unable to verify the CD groove spacing with the current data.

Additionally, here is an interesting video on another method for determining the groove spacing.

http://www.youtube.com/watch?v=IKKFPtcaZpQ

Time dialation

    Einstein based the special theory of relativity on two postulates.
The first being that the laws of physics are the same in any inertial frame.
The second being that the speed of light is measured to be the same in any inertial frame. Take these two light clocks for example...

The guy standing next to the stationary clock measures the speed of light to be c.

A guy riding on the moving clock measures the speed of light to be c.

On the surface, these simple postulates seem both logical and convenient, but they lead to a plethora of paradoxes. Take for example a person in a spaceship travelling at half the speed of light, if they turn on the headlights for their spaceship, the light travels away from them at the speed of light. 

But how does this appear to a stationary observer watching the same ship?

Classical Newtonian physics predicts that if light has a velocity of c compared to the spaceship, and the spaceship has velocity .5c compared to the observer, then the light must be travelling at  1.5c compared to the stationary observer.This however violates the second postulate, that the speed of light is measured to be the same by the people in the spaceship AND the stationary observer. How is this possible?

In a deft side step, Albert invents this half baked theory of time dilation.

What is time dilation you say?

In the simple form it means that time is travelling slower for the people on the spaceship than it is for the stationary observer. (A more through approach reveals that this too produces a paradox, but lets just wave our hands and say 'OKAY Albert, what ever you say must be true.')

Well, if time is travelling at different rates for different people, how do we compare them to one another? 

BAM! We have this convenient formula.

The t' represents the magnitude of a unit of time, lets call it a second, compared to the stationary observer's time, t. The denominator has terms v, for the velocity of the travelling thing, and c for the speed of light. The denominator can also be represented by multiplying t by a term we refer to as gamma. Interesting to note is that in the limit where v approaches c, t' approaches infinity. As such, it appears all things happen simultaneously to things travelling at the speed of light.

One might ask, how do we use this? Well, the nice people over at activphysics

provided us with an example using the light clocks from above. If told that the round trip time for the moving clock is 7.45 us, measured from the stationary frame, and that the stationary clock takes 6.67 us, we can determine the value that the coefficient of t must take in the above equation. The value that expression takes is also known as the Lorentz Factor. 

A simple division results that the Lorentz factor for the moving clock is 1.117, meaning that a single second in the stationary frame lasts 1.117 seconds in the moving frame. Solving this expression for velocity results in a v of, .445c.