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QHO-catstate-even2-animation-color.gif (300 × 200 pixels, file size: 371 KB, MIME type: image/gif, looped, 100 frames, 5.0 s)

Summary

Description
English: Animation of the quantum wave function of a Schrödinger cat state of α=2 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-even2-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

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

# image settings
fname = 'QHO-catstate-even2-animation-color'
width, height = 300, 200
ml, mr, mt, mb, mh, mc = 35, 19, 22, 45, 12, 6
x0, x1 = -5, 5
y0, y1 = 0.0, 1.2
nframes = 100
fps = 20

# physics settings
alpha0 = 2.0
omega = 2*pi

def color(phase):
    phase1 = ((phase / (2*pi)) % 1 + 1) % 1
    hue = (interp1d([0, 1./3, 1.2/3, 0.5, 1], # spread yellow a bit
                    [0, 1./3, 1.3/3, 0.5, 1])(phase1) + 2./3.) % 1
    light = interp1d([0, 1, 2, 3, 4, 5, 6], # adjust lightness
                     [0.64, 0.5, 0.56, 0.48, 0.75, 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) + ' ',
    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
anim = animation.FuncAnimation(fig, animate, frames=nframes)
anim.save(fname + '_.gif', writer='imagemagick', fps=fps)

import os
# compress with gifsicle
commons = 'https://commons.wikimedia.org/wiki/File:'
cmd = 'gifsicle -O3 -k256 --careful --comment="' + commons + fname + '.gif"'
cmd += ' < ' + fname + '_.gif > ' + fname + '.gif'
if os.system(cmd) == 0:
    os.remove(fname + '_.gif')
else:
    print 'warning: gifsicle not found!'
    os.remove(fname + '.gif')
    os.rename(fname + '_.gif', fname + '.gif')

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.
You may select the license of your choice.

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20 September 2015

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Date/TimeThumbnailDimensionsUserComment
current13:01, 4 October 2015Thumbnail for version as of 13:01, 4 October 2015300 × 200 (371 KB)Geek3legend added
21:28, 20 September 2015Thumbnail for version as of 21:28, 20 September 2015300 × 200 (391 KB)Geek3{{Information |Description ={{en|1=Animation of the quantum wave function of a Schrödinger cat state of α=2 in a Quantum harmonic oscillator. The [[:en:Probability distrib...

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