QHO-catstate-even2-animation-color.gif (300 × 200 pixels, file size: 371 KB, MIME type: image/gif, looped, 100 frames, 5.0 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. |
DescriptionQHO-catstate-even2-animation-color.gif |
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 |
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
# /info/en/?search=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:
# /info/en/?search=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=bounds0], bottom=bounds1],
right=bounds2], top=bounds3], 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')
|
![]() |
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 |
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 13:01, 4 October 2015 |
![]() | 300 × 200 (371 KB) | Geek3 | legend added |
21:28, 20 September 2015 |
![]() | 300 × 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... |
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
If the file has been modified from its original state, some details may not fully reflect the modified file.
GIF file comment | https://commons.wikimedia.org/wiki/File:QHO-catstate-even2-animation-color.gif |
---|
QHO-catstate-even2-animation-color.gif (300 × 200 pixels, file size: 371 KB, MIME type: image/gif, looped, 100 frames, 5.0 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. |
DescriptionQHO-catstate-even2-animation-color.gif |
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 |
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
# /info/en/?search=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:
# /info/en/?search=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=bounds0], bottom=bounds1],
right=bounds2], top=bounds3], 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')
|
![]() |
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 |
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 13:01, 4 October 2015 |
![]() | 300 × 200 (371 KB) | Geek3 | legend added |
21:28, 20 September 2015 |
![]() | 300 × 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... |
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
If the file has been modified from its original state, some details may not fully reflect the modified file.
GIF file comment | https://commons.wikimedia.org/wiki/File:QHO-catstate-even2-animation-color.gif |
---|