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