from __future__ import print_function, division
from sympy.core import S, pi, Rational
from sympy.functions import assoc_laguerre, sqrt, exp, factorial, factorial2
[docs]def R_nl(n, l, nu, r):
"""
Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic
oscillator.
``n``
the "nodal" quantum number. Corresponds to the number of nodes in
the wavefunction. n >= 0
``l``
the quantum number for orbital angular momentum
``nu``
mass-scaled frequency: nu = m*omega/(2*hbar) where `m` is the mass
and `omega` the frequency of the oscillator.
(in atomic units nu == omega/2)
``r``
Radial coordinate
Examples
========
>>> from sympy.physics.sho import R_nl
>>> from sympy import var
>>> var("r nu l")
(r, nu, l)
>>> R_nl(0, 0, 1, r)
2*2**(3/4)*exp(-r**2)/pi**(1/4)
>>> R_nl(1, 0, 1, r)
4*2**(1/4)*sqrt(3)*(-2*r**2 + 3/2)*exp(-r**2)/(3*pi**(1/4))
l, nu and r may be symbolic:
>>> R_nl(0, 0, nu, r)
2*2**(3/4)*sqrt(nu**(3/2))*exp(-nu*r**2)/pi**(1/4)
>>> R_nl(0, l, 1, r)
r**l*sqrt(2**(l + 3/2)*2**(l + 2)/factorial2(2*l + 1))*exp(-r**2)/pi**(1/4)
The normalization of the radial wavefunction is:
>>> from sympy import Integral, oo
>>> Integral(R_nl(0, 0, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 0, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 1, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
"""
n, l, nu, r = map(S, [n, l, nu, r])
# formula uses n >= 1 (instead of nodal n >= 0)
n = n + 1
C = sqrt(
((2*nu)**(l + Rational(3, 2))*2**(n + l + 1)*factorial(n - 1))/
(sqrt(pi)*(factorial2(2*n + 2*l - 1)))
)
return C*r**(l)*exp(-nu*r**2)*assoc_laguerre(n - 1, l + S(1)/2, 2*nu*r**2)
[docs]def E_nl(n, l, hw):
"""
Returns the Energy of an isotropic harmonic oscillator
``n``
the "nodal" quantum number
``l``
the orbital angular momentum
``hw``
the harmonic oscillator parameter.
The unit of the returned value matches the unit of hw, since the energy is
calculated as:
E_nl = (2*n + l + 3/2)*hw
Examples
========
>>> from sympy.physics.sho import E_nl
>>> from sympy import symbols
>>> x, y, z = symbols('x, y, z')
>>> E_nl(x, y, z)
z*(2*x + y + 3/2)
"""
return (2*n + l + Rational(3, 2))*hw
