The Laser Activity

Question 1: Absorption
At any given time, the number of photons inputted into the cavity must be equal to the number that have passed through the cavity without exciting an atom plus the number still in the cavity plus the number of excited atoms. Verify this conservation law by stopping the simulation and counting photons.


N(in) = N(out) + # of photons still on the frame + excited atoms

15 = 2 + 1 + 12

Question 2: Direction of Spontaneous Emission
During spontaneous emission, does there appear to be a preferred direction in which the photons are emitted?

No. Photons are emitted in random directions.

 

Question 3: Lifetime of Excited State
Does there appear to be a constant amount of time in which an atom remains in its excited state?

There is no constant amount of time for a single atom. However, there can be a prediction if a bunch of identical atoms are considered instead.

 

Question 4: Stimulated Emission
Carefully describe what happens when a photon interacts with an excited atom. Pay careful attention to the phase and direction of the subsequent photons. (Can you see why this is called stimulated emission?)

When a photon (which come from the left) strikes an excited atom, the photons appear to multiply by a factor of 3. The proton is then driven to their ground state until all of the excited atoms have been hit.

 

Question 5: Pumping
Approximately what pumping level is required to achieve a population inversion? Remember, a population inversion is when the number of atoms in the excited state is at least as great as the number of atoms in the ground state.

 A pumping level of P = 70 is the one that most of the time keeps the number of excited and ground level atoms even.

Question 6: Photon Emission
Although most photons are emitted toward the right in the simulation, occasionally one is emitted in another direction. Are the photons emitted at odd directions the result of stimulated or spontaneous emission?

In a spontaneous emission, the photons are emitted in random directions.

Visualizing Wave Packets Lab

Gaussian Function Plot:

from pylab import *
center = 2
sigma = 2
coeff = 1 / sqrt(2*pi)*sigma
gauss_list = []

for x in arange(-10,10,0.1):
    gauss = coeff * exp(-(x - center)**2/(2. * sigma**2))
    gauss_list.append(gauss)
    print gauss
plot(gauss_list)
show()

Gaussian Function centered at 2 with Standard Deviation of 2

Sinusoidal Function Plot:

from pylab import *
A = 2
k = 1
sine_list = []
#coeff = 1 / sqrt(2*pi)*sigma

for x in arange(-3.14,3.14,0.1):
    f = A*sin(k*x)
    sine_list.append(f)
    print f
plot(sine_list)
show()

Single Sine function with Amplitude = 2 and k = 1

Multiple Sine Functions:

from pylab import *
A = 2
w = 1

for i in range (1,5):
    x = []
    sine_function = []
    for t in arange(-3.14,3.14,0.01):
        sine_f = A*sin(i*w*t)
        sine_function.append(sine_f)
        x.append(t)
    plot(x,sine_function)
show()

First 4 harmonics

Sinusoidal Superposition - Fourier Synthesis:


from pylab import *
A = 2
w = 1
Fourier_Series = []

for i in range (1,5):
    x = []
    sine_function = []
    for t in arange(-3.14,3.14,0.01):
        sine_f = A*sin(i*w*t)
        sine_function.append(sine_f)
        x.append(t)
    #plot(x,sine_function)
    Fourier_Series.append(sine_function)

superposition = zeros(len(sine_function))

for function in Fourier_Series:
    for i in range(len(function)):
        superposition[i]+= function[i]
      
plot(x,superposition)
show()

Superposition of first 4 harmonics

Now, for the first 19 harmonics;

from pylab import *
A = 2
w = 1
Fourier_Series = []

for i in range (1,20):
    x = []
    sine_function = []
    for t in arange(-3.14,3.14,0.01):
        sine_f = A*sin(i*w*t)
        sine_function.append(sine_f)
        x.append(t)
    #plot(x,sine_function)
    Fourier_Series.append(sine_function)

superposition = zeros(len(sine_function))

for function in Fourier_Series:
    for i in range(len(function)):
        superposition[i]+= function[i]
       
plot(x,superposition)
show()

Superposition of the first 19 harmonics

We can see that the more harmonics we add up, we can begin to see that the wave function starts to localize. If we had an infinite number of harmonics, the ripples would 'disappear'.


Gaussian Wave Packet:

from pylab import*
center = 3
sigma = 1
coeff = 1 / sqrt(2*pi)*sigma

A = 1
w = 1
Fourier_Series = []

for i in range (1,6):
    x = []
    sine_function = []
    A = coeff*exp(-(i-center)**2/(2.*sigma**2))

for t in arange(-3.14,3.14,0.01):
    sine_f = A*sin(i*w*t)
    sine_function.append(sine_f)
    x.append(t)
    Fourier_Series.append(sine_function)

superposition = zeros(len(sine_function))

for function in Fourier_Series:
    for i in range(len(function)):
        superposition[i]+= function [i]

plot(x,superposition)
show()

Gaussian centered at harmonic 3

Relativity of Length Activity

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?

The light pulse takes longer if the light clocks are in motion. However, the path is shortened on the way back in the situation where both clocks are moving. With the stationary light clock, the displacement of the pulse is the same. 

 

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?

It takes longer for the pulse to make a roundtrip when it is measured on the Earth.

 

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?

For the times to be the same, the moving light clock should be adjusted so there will be a smaller distance for the light clock to travel. 

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?

Δx = γΔx_proper;                                    1000 = 1.3 Δx_proper;                           Δx_proper = 769.23 m