69 lines
2.7 KiB
Python
69 lines
2.7 KiB
Python
# P372-PS14-FFT-shift.py
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"""
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A wavepacket is made to move. Based on P372-PS14-FFT.py
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"""
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import numpy as np
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import pylab as pl
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import numpy.fft as fft
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# Some basic numbers, first. These are all design decisions you can change!
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numPoints = 1000 # number of points in x-waveform, how much resolution
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numFreqs = numPoints / 2 + 1 # (numPoints/2 -1) complex & 2 real k-amplitudes
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# This is still same numPoints of bits of information!
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# The wavenumber k's are also called "spatial frequencies"
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# We are going to look at the region between 0 and 2 in x-space,
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# but suppose we don't want anything to repeat within twice this, so
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windowLength = 4.0 # effective window length
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dx = windowLength / numPoints # Step size in x-space
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x = np.arange(0.0, windowLength, dx) # All the x-positions
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# Now prepare the spectrum and k wavenumbers
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dk = 2 * np.pi / windowLength # Step size in k-space
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ks = np.arange(numFreqs) * dk # All the k's, highest k found at ks[-1]
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# HERE IS WHERE THE REAL WORK GETS DONE, the above is all just setting up
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# Choose the <k>
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k0 = 0.1 * ks[-1] # 25% of highest k value <-- PICK THIS!
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alpha = 0.1 # smaller alpha means wider phi_k, narrower psi_x <-- AND THIS!
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# Now define the k-spectrum amplitudes
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phi_k = np.exp(-0.5 * alpha * (ks - k0) ** 2) # the spectral amplitudes (complex)
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# And generate the wavefunction from phi(k) using Inverse Real FFT
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psi_x = fft.irfft(phi_k) # inverse real FFT gets the real wavefunction
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# Now we want the wavepacket to move! Let's define a velocity:
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v = 0.5
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# First let's have it move just 1/4 cycle to the right
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# Thus k0 v dt = pi/2
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dt = 0.5 * np.pi / v / k0
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# Calculate the phase shifts
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phaseShift = np.exp(-1j * ks * v * dt)
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# And apply them before doing the F.T.
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psi_x_shift = fft.irfft(phi_k * phaseShift)
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# And now let's get them to shift all the way to x=1.0, which means vt = 1.0
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t1 = 1.0 / v
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# Calculate the phase shifts
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phaseShift1 = np.exp(-1j * ks * v * t1)
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# And apply them before doing the F.T.
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psi_x_shift1 = fft.irfft(phi_k * phaseShift1)
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pl.figure(1)
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pl.plot(ks, phi_k) # phi_k could be complex, which would throw a warning here
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# if so, could use: pl.plot(ks, np.abs(phi_k) )
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pl.title(r"Spectrum $\phi(k)$")
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pl.xlabel("wavenumber k")
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pl.ylabel("spectral amplitude")
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pl.figure(2)
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pl.plot(x[:numPoints // 2], psi_x[:numPoints // 2]) # only plot half of the points
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pl.plot(x[:numPoints // 2], psi_x_shift[:numPoints // 2]) # These are the shifted wavepacket
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pl.plot(x[:numPoints // 2], psi_x_shift1[:numPoints // 2]) # These are the shifted wavepacket
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pl.title(r"Wavefunction $\psi(x)$ shifted")
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pl.xlabel("position x")
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pl.ylabel(r"wavefunction $\psi(x)$")
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pl.show() # show the graph windows and loop here until they are closed
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print("Done.")
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