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File:QHO-catstate-even3-animation-color.gif

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QHO-catstate-even3-animation-color.gif (300 × 200 pixels, file size: 411 KB, MIME type: image/gif, looped, 150 frames, 7.5 s)

Summary

Description
English: Animation of the quantum wave function of a Schrödinger cat state of α=3 in a Quantum harmonic oscillator. The probability distribution is drawn along the ordinate, while the phase is encoded by color. The two coherent contributions interfere in the center which is characteristic for a cat-state.
Date
Source Own work
 
This plot was created with Matplotlib.
Author Geek3
Other versions QHO-catstate-even3-animation.gif

Source Code

The plot was generated with Matplotlib.


Python Matplotlib source code
#!/usr/bin/python
# -*- coding: utf8 -*-

from math import *
import matplotlib.pyplot as plt
from matplotlib import animation, colors, colorbar
import numpy as np
import colorsys
from scipy.interpolate import interp1d
import os, sys

plt.rc('path', snap=False)
plt.rc('mathtext', default='regular')

# image settings
fname = 'QHO-catstate-even3-animation-color'
width, height = 300, 200
ml, mr, mt, mb, mh, mc = 35, 19, 22, 45, 12, 6
x0, x1 = -6.5, 6.5
y0, y1 = 0.0, 1.2
nframes = 150
fps = 20

# physics settings
alpha0 = 3.0
omega = 2*pi

def color(phase):
    hue = (phase / (2*pi) + 2./3.) % 1
    light = interp1d([0, 1, 2, 3, 4, 5, 6], # adjust lightness
                     [0.64, 0.5, 0.55, 0.48, 0.70, 0.57, 0.64])(6 * hue)
    hls = (hue, light, 1.0) # maximum saturation
    rgb = colorsys.hls_to_rgb(*hls)
    return rgb

def coherent(alpha, x, omega, t, l=1.0):
    # Definition of coherent states
    # https://en.wikipedia.org/wiki/Coherent_states
    psi = (pi*l**2)**-0.25 * np.exp(
                -0.5/l**2 * (x - sqrt(2)*l * alpha.real)**2
                + 1j*sqrt(2)/l * alpha.imag * x
                + 0.5j * (alpha0**2*sin(2*omega*t) - omega*t))
    return psi

def animate(nframe):
    print str(nframe) + ' ',; sys.stdout.flush()
    t = float(nframe) / nframes * 0.5 # animation repeats after t=0.5
    alpha = e ** (-1j * omega * t) * alpha0
    
    ax.cla()
    ax.grid(True)
    ax.axis((x0, x1, y0, y1))
    
    x = np.linspace(x0, x1, int(ceil(1+w_px)))
    x2 = x - px_w/2.
    
    # Definition of cat states in terms of coherent states:
    # https://en.wikipedia.org/wiki/Cat_state
    psi = coherent(alpha, x, omega, t) + coherent(-alpha, x, omega, t)
    psi /= sqrt(2 * (1 + exp(-2*alpha0**2)))
    
    # Let's cheat a bit: discard the constant phase from the zero-point energy!
    # This will reduce the period from T=2*(2pi/omega) to T=0.5*(2pi/omega)
    # and allow fewer frames and less file size for repetition.
    # For big alpha the change is hardly visible
    psi *= np.exp(0.5j * omega * t)
    y = np.abs(psi)**2
    
    psi2 = coherent(alpha, x2, omega, t) + coherent(-alpha, x2, omega, t)
    psi2 *= np.exp(0.5j * omega * t)
    phi = np.angle(psi2)
    
    # plot color filling
    for x_, phi_, y_ in zip(x, phi, y):
        ax.plot([x_, x_], [0, y_], color=color(phi_), lw=2*0.72)
    
    ax.plot(x, y, lw=2, color='black')
    ax.set_yticks(ax.get_yticks()[:-1])
    

# create figure and axes
plt.close('all')
fig, ax = plt.subplots(1, figsize=(width/100., height/100.))
bounds = [float(ml)/width, float(mb)/height,
          1.0 - float(mr+mc+mh)/width, 1.0 - float(mt)/height]
fig.subplots_adjust(left=bounds[0], bottom=bounds[1],
                    right=bounds[2], top=bounds[3], hspace=0)
w_px = width - (ml+mr+mh+mc) # plot width in pixels
px_w = float(x1 - x0) / w_px # width of one pixel in plot units

# axes labels
fig.text(0.5 + 0.5 * float(ml-mh-mc-mr)/width, 4./height,
         r'$x\ \ [(\hbar/(m\omega))^{1/2}]$', ha='center')
fig.text(5./width, 1.0, '$|\psi|^2$', va='top')

# colorbar for phase
cax = fig.add_axes([1.0 - float(mr+mc)/width, float(mb)/height,
                    float(mc)/width, 1.0 - float(mb+mt)/height])
cax.yaxis.set_tick_params(length=2)
cmap = colors.ListedColormap([color(phase) for phase in
                              np.linspace(0, 2*pi, 384, endpoint=False)])
norm = colors.Normalize(0, 2*pi)
cbar = colorbar.ColorbarBase(cax, cmap=cmap, norm=norm,
                    orientation='vertical', ticks=np.linspace(0, 2*pi, 3))
cax.set_yticklabels(['$0$', r'$\pi$', r'$2\pi$'], rotation=90)
fig.text(1.0 - 10./width, 1.0, '$arg(\psi)$', ha='right', va='top')
plt.sca(ax)

# start animation
if 0 != os.system('convert -version > ' +  os.devnull):
    print 'imagemagick not installed!'
    # warning: imagemagick produces somewhat jagged and therefore large gifs
    anim = animation.FuncAnimation(fig, animate, frames=nframes)
    anim.save(fname + '.gif', writer='imagemagick', fps=fps)
else:
    # unfortunately the matplotlib imagemagick backend does not support
    # options which are necessary to generate high quality output without
    # framewise color palettes. Therefore save all frames and convert then.
    if not os.path.isdir(fname):
        os.mkdir(fname)
    fnames = []
    
    for frame in range(nframes):
        animate(frame)
        imgname = os.path.join(fname, fname + '{:03d}'.format(frame) + '.png')
        fig.savefig(imgname)
        fnames.append(imgname)
    
    # compile optimized animation with ImageMagick
    cmd = 'convert -loop 0 -delay ' + str(100 / fps) + ' '
    cmd += ' '.join(fnames) # now create optimized palette from all frames
    cmd += r' \( -clone 0--1 \( -clone 0--1 -fill black -colorize 100% \) '
    cmd += '-append +dither -colors 255 -unique-colors '
    cmd += '-write mpr:colormap +delete \) +dither -map mpr:colormap '
    cmd += '-alpha activate -layers OptimizeTransparency '
    cmd += fname + '.gif'
    os.system(cmd)
    
    for fnamei in fnames:
        os.remove(fnamei)
    os.rmdir(fname)

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution
This file is licensed under the Creative Commons Attribution 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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20 September 2015

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Date/TimeThumbnailDimensionsUserComment
current21:46, 10 October 2015Thumbnail for version as of 21:46, 10 October 2015300 × 200 (411 KB)Geek3better compression
13:06, 4 October 2015Thumbnail for version as of 13:06, 4 October 2015300 × 200 (577 KB)Geek3legend added
23:41, 20 September 2015Thumbnail for version as of 23:41, 20 September 2015300 × 200 (572 KB)Geek3phase correction
21:34, 20 September 2015Thumbnail for version as of 21:34, 20 September 2015300 × 200 (577 KB)Geek3{{Information |Description ={{en|1=Animation of the quantum wave function of a Schrödinger cat state of α=3 in a Quantum harmonic oscillator. The [[:en:Probability distrib...

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