File:QHO-coherentstate3-animation-color.gif
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QHO-coherentstate3-animation-color.gif (300 × 200 pixels, file size: 449 KB, MIME type: image/gif, looped, 150 frames, 7.5 s)
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Summary
[edit]DescriptionQHO-coherentstate3-animation-color.gif |
English: Animation of the quantum wave function of a coherent state of initial α=3 in a Quantum harmonic oscillator. The probability distribution is drawn along the ordinate, while the phase is encoded by color. The gaussian wave packet oscillates sinusoidally in the harmonic potential. |
Date | |
Source |
Own work This plot was created with Matplotlib. |
Author | Geek3 |
Other versions | QHO-coherentstate3-animation.gif |
Source Code
[edit]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-coherentstate3-animation-color'
width, height = 300, 200
ml, mr, mt, mb, mh, mc = 35, 19, 22, 45, 12, 6
x0, x1 = -7, 7
y0, y1 = 0.0, 0.7
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(alpha0, x, omega_t):
# Definition of coherent states
# https://en.wikipedia.org/wiki/Coherent_states
alpha = alpha0 * e**(-1j * omega_t)
psi = pi**-0.25 * np.exp(-0.5j * omega_t
- 0.5 * ((x - sqrt(2) * alpha.real))**2
- 1j * alpha.imag * (alpha.real - sqrt(2) * x))
return psi
def animate(nframe):
print str(nframe) + ' ',; sys.stdout.flush()
t = float(nframe) / nframes # animation repeats after t=1.0
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.
# Let's cheat a bit: add a phase phi(t)*const(x)
# This will reduce the period from T=2*(2pi/omega) to T=1.0*(2pi/omega)
# and allow fewer frames and less file size for repetition.
# For big alpha the change is hardly visible
psi = coherent(alpha0, x, omega * t) * np.exp(-0.5j * omega * t)
psi2 = coherent(alpha0, x2, omega * t) * np.exp(-0.5j * omega * t)
y = np.abs(psi)**2
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_yticklabels([l for l in ax.get_yticks() if l < y0+0.9*(y1-y0)])
# 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
[edit]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:
- 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.
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 21:42, 10 October 2015 | 300 × 200 (449 KB) | Geek3 (talk | contribs) | better compression | |
13:26, 4 October 2015 | 300 × 200 (603 KB) | Geek3 (talk | contribs) | legend added | ||
22:04, 20 September 2015 | 300 × 200 (608 KB) | Geek3 (talk | contribs) | {{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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