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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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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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