Source code for qutip.orbital

# This file is part of QuTiP: Quantum Toolbox in Python.
#
#    Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
#
#    Redistribution and use in source and binary forms, with or without
#    modification, are permitted provided that the following conditions are
#    met:
#
#    1. Redistributions of source code must retain the above copyright notice,
#       this list of conditions and the following disclaimer.
#
#    2. Redistributions in binary form must reproduce the above copyright
#       notice, this list of conditions and the following disclaimer in the
#       documentation and/or other materials provided with the distribution.
#
#    3. Neither the name of the QuTiP: Quantum Toolbox in Python nor the names
#       of its contributors may be used to endorse or promote products derived
#       from this software without specific prior written permission.
#
#    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
#    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
#    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
#    PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
#    HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
#    SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
#    LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
#    DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
#    THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
#    (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
#    OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
###############################################################################

__all__ = ['orbital']

import numpy as np
from scipy.special import factorial

[docs]def orbital(theta, phi, *args):
"""Calculates an angular wave function on a sphere.
psi = orbital(theta,phi,ket1,ket2,...) calculates
the angular wave function on a sphere at the mesh of points
defined by theta and phi which is
:math:\sum_{lm} c_{lm} Y_{lm}(theta,phi) where :math:C_{lm} are the
coefficients specified by the list of kets. Each ket has 2l+1 components
for some integer l.

Parameters
----------
theta : list/array
Polar angles

phi : list/array
Azimuthal angles

args : list/array
list of ket vectors.

Returns
-------
array for angular wave function

"""
psi = 0.0
if isinstance(args[0], list):
# use the list in args[0]
args = args[0]

for k in range(len(args)):
ket = args[k]
if not ket.type == 'ket':
raise TypeError('Invalid input ket in orbital')
sk = ket.shape
nchk = (sk[0] - 1) / 2.0
if nchk != np.floor(nchk):
raise ValueError(
'Kets must have odd number of components in orbital')
l = int((sk[0] - 1) / 2)
if l == 0:
SPlm = np.sqrt(2) * np.ones((np.size(theta), 1), dtype=complex)
else:
SPlm = _sch_lpmv(l, np.cos(theta))
fac = np.sqrt((2.0 * l + 1) / (8 * np.pi))
kf = ket.full()
psi += np.sqrt(2) * fac * kf[l, 0] * np.ones((np.size(phi),
np.size(theta)),
dtype=complex) * SPlm[0]
for m in range(1, l + 1):
psi += ((-1.0) ** m * fac * kf[l - m, 0]) * \
np.array([np.exp(1.0j * 1 * phi)]).T * \
np.ones((np.size(phi), np.size(theta)),
dtype=complex) * SPlm[1]
for m in range(-l, 0):
psi = psi + (fac * kf[l - m, 0]) * \
np.array([np.exp(1.0j * 1 * phi)]).T * \
np.ones((np.size(phi), np.size(theta)), dtype=complex) * \
SPlm[abs(m)]
return psi

# Schmidt Semi-normalized Associated Legendre Functions
def _sch_lpmv(n, x):
'''
Outputs array of Schmidt Seminormalized Associated Legendre Functions
S_{n}^{m} for m<=n.

Parameters
----------
n : int
Degree of polynomial.

x : float
Point at which to evaluate

Returns
-------
array of values for Legendre functions.

'''
from scipy.special import lpmv
n = int(n)
sch = np.array([1.0])
sch2 = np.array([(-1.0) ** m * np.sqrt(
(2.0 * factorial(n - m)) / factorial(n + m)) for m in range(1, n + 1)])
sch = np.append(sch, sch2)
if isinstance(x, float) or len(x) == 1:
leg = lpmv(np.arange(0, n + 1), n, x)
return np.array([sch * leg]).T
else:
for j in range(0, len(x)):
leg = lpmv(range(0, n + 1), n, x[j])
if j == 0:
out = np.array([sch * leg]).T
else:
out = np.append(out, np.array([sch * leg]).T, axis=1)
return out