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)
This is a file from the Wikimedia Commons. Information from its description page there is shown below. Commons is a freely licensed media file repository. You can help. |
Summary
DescriptionQHO-catstate-even3-animation-color.gif |
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 |
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#!/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:
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.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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:
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Items portrayed in this file
depicts
20 September 2015
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 21:46, 10 October 2015 | 300 × 200 (411 KB) | Geek3 | better compression | |
13:06, 4 October 2015 | 300 × 200 (577 KB) | Geek3 | legend added | ||
23:41, 20 September 2015 | 300 × 200 (572 KB) | Geek3 | phase correction | ||
21:34, 20 September 2015 | 300 × 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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