# P372-PS14-FFT-shift.py
"""
A wavepacket is made to move.  Based on P372-PS14-FFT.py
"""

import numpy as np
import pylab as pl
import numpy.fft as fft

# Some basic numbers, first.  These are all design decisions you can change!
numPoints = 1000  # number of points in x-waveform, how much resolution
numFreqs = numPoints / 2 + 1  # (numPoints/2 -1) complex & 2 real k-amplitudes
# This is still same numPoints of bits of information!
# The wavenumber k's are also called "spatial frequencies"
# We are going to look at the region between 0 and 2 in x-space,
# but suppose we don't want anything to repeat within twice this, so
windowLength = 4.0  # effective window length
dx = windowLength / numPoints  # Step size in x-space
x = np.arange(0.0, windowLength, dx)  # All the x-positions

# Now prepare the spectrum and k wavenumbers
dk = 2 * np.pi / windowLength  # Step size in k-space
ks = np.arange(numFreqs) * dk  # All the k's, highest k found at ks[-1]

# HERE IS WHERE THE REAL WORK GETS DONE, the above is all just setting up
# Choose the <k>
k0 = 0.1 * ks[-1]  # 25% of highest k value <-- PICK THIS!
alpha = 0.1  # smaller alpha means wider phi_k, narrower psi_x <-- AND THIS!
# Now define the k-spectrum amplitudes
phi_k = np.exp(-0.5 * alpha * (ks - k0) ** 2)  # the spectral amplitudes (complex)
# And generate the wavefunction from phi(k) using Inverse Real FFT
psi_x = fft.irfft(phi_k)  # inverse real FFT gets the real wavefunction

# Now we want the wavepacket to move!  Let's define a velocity:
v = 0.5
# First let's have it move just 1/4 cycle to the right
# Thus k0 v dt = pi/2
dt = 0.5 * np.pi / v / k0
# Calculate the phase shifts
phaseShift = np.exp(-1j * ks * v * dt)
# And apply them before doing the F.T.
psi_x_shift = fft.irfft(phi_k * phaseShift)

# And now let's get them to shift all the way to x=1.0, which means vt = 1.0
t1 = 1.0 / v
# Calculate the phase shifts
phaseShift1 = np.exp(-1j * ks * v * t1)
# And apply them before doing the F.T.
psi_x_shift1 = fft.irfft(phi_k * phaseShift1)

pl.figure(1)
pl.plot(ks, phi_k)  # phi_k could be complex, which would throw a warning here
# if so, could use: pl.plot(ks, np.abs(phi_k) )
pl.title(r"Spectrum $\phi(k)$")
pl.xlabel("wavenumber k")
pl.ylabel("spectral amplitude")

pl.figure(2)
pl.plot(x[:numPoints // 2], psi_x[:numPoints // 2])  # only plot half of the points
pl.plot(x[:numPoints // 2], psi_x_shift[:numPoints // 2])  # These are the shifted wavepacket
pl.plot(x[:numPoints // 2], psi_x_shift1[:numPoints // 2])  # These are the shifted wavepacket
pl.title(r"Wavefunction $\psi(x)$ shifted")
pl.xlabel("position x")
pl.ylabel(r"wavefunction $\psi(x)$")

pl.show()  # show the graph windows and loop here until they are closed

print("Done.")