Source code for qutip.orbital

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__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