PHYS201 PHYSICS IIA Python Lab 7 Exercises
Mike Steel, 2017, modification by James Tocknell 2018
Your Name:Nicholas Arico
(Double-click cell above and fill out your name)
N.B. Generally it's a really good idea to make each of your answers self contained so that all the code necessary for the answer is in cells under the exercise heading. Use comments or text cells to document your work as necessary.
Exercise 1
A narrow wave packet in space must be composed of a broad range of momenta.
How does this explain the spreading of a wave packet in empty space?
Write your answer here
Using the heisengberg uncertainty principle, having a small uncertainty in relation to the position of the particle, implies a large uncertainty in relation the momentum of the particle. In this instance, having a narrow wave packet in space implies a larger spread of momentum of the wave. Since momentum is mv, therefore, the larger the spread of momenta, the larger the velocity of the spread
Exercise 2
By slight alteration of the code above, solve and plot the evolution of a wave packet with a smaller initial width of .
Comment on your result: is the wave packet spreading faster or slower for the smaller width initial wave packet? Explain this physically.
Exercise 3
Now perform the same evolution but with a supergaussian
input wave function. (Here 'super-gaussian') means the form . Choose a tmax
of at least 5.
How is the result different? Can you explain this behaviour at all?
Exercise 4
Now excite a Gaussian wave packet of width 2 but initial momentum p0=2
.
You may wish to shift the starting point on the axis to keep the beam in the simulation domain.
Do the results make sense? Explain.
Exercise 5
Return to a simulation of a wave packet with initial width 0.25, zero momentum and a domain width of 20, but this time let the evolution run until .
Display the result with an animation.
Explain what is going on here? Is the behaviour physical? How can it be fixed?
Exercise 6
Generate an animation of the previous example in the notes to show the 'breathing' in time.
In general, a superposition of energy eigenstates will not show periodic evolution but a continually evolving appearance. Explain why we see perfect periodicity in the harmonic oscillator case?
Exercise 7
- Excite the ground state solution with width of the harmonic potential, but displaced in starting position a little way from the origin. What behaviour do you observe?
- Confirm this by viewing the evolution as an animation.
Exercise 8
We saw similar rapid oscillations earlier when a pulse encountered the edge of the simulation domain.
Do you think the oscillations in the present case should be considered as physical or not? Explain your answer.
Write your answer here
Since we intentionally placed the particle in a box with real edges, we should consider this system to be physical
Exercise 9 (this is a harder problem worth double marks)
- We still see considerable reflection. Why is that? Think about the spread of energies contained in a single pulse.
- How can you estimate the energy spread?
- How wide a pulse do you think is needed to have small energy dispersion and high transmission?
- Test your ideas by running some calculations with different pulse widths. Can you increase the transmission?
Submitting the lab file
Make sure you've entered your name at the top of this notebook.
Once you're happy with your responses, simply ensure this notebook is saved.
Scripts will take a copy of this folder for marking after the deadline.