*Mike Steel*, 2017, modification by *James Tocknell* 2018

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(Double-click cell above and fill out your name)

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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.

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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?

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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

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By slight alteration of the code above, solve and plot the evolution of a wave packet with a *smaller* initial width of $w=0.25$.

Comment on your result: is the wave packet spreading faster or slower for the smaller width initial wave packet? Explain this physically.

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In [58]:

# write your code here from math import * import numpy as np import matplotlib.pyplot as plt import phys201quantum as phys201q %matplotlib inline psi0 = phys201q.make_psi0([-8, 8], 250, 'gaussian', .25, 0, 0) V = phys201q.make_piecewise_potential([[-8, 0], [8, 0]]) psi_evol = phys201q.schro_evolve(psi0, V, tmax=1.5) phys201q.plot_psi_at(psi_evol)

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Doing 363 steps of size 0.004129 with sigma=dt/(4dx^2)=0.250000
Outputting 100 time slices separated by 3 dt=0.012387

(<Figure size 432x288 with 2 Axes>,
<matplotlib.axes._subplots.AxesSubplot at 0x7f46b104dc50>)

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Now perform the same evolution but with a `supergaussian`

input wave function. (Here 'super-gaussian') means the form $\exp(-x^4)$. Choose a `tmax`

of at least 5.

How is the result different? Can you explain this behaviour at all?

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In [59]:

# write your code here psi0 = phys201q.make_psi0([-8, 8], 250, 'supergaussian', .25, 0, 0) V = phys201q.make_piecewise_potential([[-8, 0], [8, 0]]) psi_evol = phys201q.schro_evolve(psi0, V, tmax=7) phys201q.plot_psi_at(psi_evol) fig, axs = plt.subplots(1, 2) phys201q.plot_psi_pcolor(psi_evol, ax=axs[0]) phys201q.plot_psi_pcolor(psi_evol, show_phase=1, ax=axs[1]) phys201q.animate_psi(psi_evol) #This is because the domain of the simulation acts as an infinite well/barrier. So the waveform is perfectly reflected off the boundaries, and interferes with itself, creating an interference pattern, which is what can be seen different

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Doing 1695 steps of size 0.004129 with sigma=dt/(4dx^2)=0.250000
Outputting 100 time slices separated by 16 dt=0.066063

(<Figure size 432x288 with 2 Axes>,
<matplotlib.axes._subplots.AxesSubplot at 0x7f46b03017f0>)

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Now excite a Gaussian wave packet of width 2 but initial momentum `p0=2`

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You may wish to shift the starting point on the $x$ axis to keep the beam in the simulation domain.

Do the results make sense? Explain.

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In [60]:

# write your code here psi0 = phys201q.make_psi0([-8, 8], 250, 'gaussian', 2, 0, 2) V = phys201q.make_piecewise_potential([[-8, 0], [8, 0]]) psi_evol = phys201q.schro_evolve(psi0, V, tmax=1.5) phys201q.plot_psi_at(psi_evol) phys201q.animate_psi(psi_evol) #Since we increased the momentum from 0 to 2, the particle now has a veloicty, so the wave packet also has a velocity, and hence moves over time.

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Doing 363 steps of size 0.004129 with sigma=dt/(4dx^2)=0.250000
Outputting 100 time slices separated by 3 dt=0.012387

(<Figure size 432x288 with 2 Axes>,
<matplotlib.axes._subplots.AxesSubplot at 0x7f46abd5b828>)

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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 $t=8$.

Display the result with an animation.

Explain what is going on here? Is the behaviour physical? How can it be fixed?

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In [61]:

# write your code here psi0 = phys201q.make_psi0([-10,10], 250, 'gaussian', .5, 0, 0) V = phys201q.make_piecewise_potential([[-8, 0], [8, 0]]) psi_evol = phys201q.schro_evolve(psi0, V, tmax=8) phys201q.animate_psi(psi_evol) # print('Once again, the wave function spreads until reaching the boundaries,then hitting the boundaries and being reflected back and causing an interference pattern. This is not a physical phenomenon, and can be fixed by extending the boundaries"domain" to larger numbers')

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Doing 1240 steps of size 0.006452 with sigma=dt/(4dx^2)=0.250000
Outputting 100 time slices separated by 12 dt=0.077418