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- from sympy.core import S, pi, Rational
- from sympy.functions import assoc_laguerre, sqrt, exp, factorial, factorial2
- def R_nl(n, l, nu, r):
- """
- Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic
- oscillator.
- Parameters
- ==========
- ``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.abc import 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)*(3/2 - 2*r**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.Half, 2*nu*r**2)
- def E_nl(n, l, hw):
- """
- Returns the Energy of an isotropic harmonic oscillator.
- Parameters
- ==========
- ``n`` :
- The "nodal" quantum number.
- ``l`` :
- The orbital angular momentum.
- ``hw`` :
- The harmonic oscillator parameter.
- Notes
- =====
- 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
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