/

# Source code for sympy.physics.sho

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)/(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
`