B ˜‘[² ã@sTddlmZmZddlmZmZmZddlmZm Z m Z m Z m Z dd„Z dd„ZdS) é)Úprint_functionÚdivision)ÚSÚpiÚRational)Úassoc_laguerreÚsqrtÚexpÚ factorialÚ factorial2cCs¾tt||||gƒ\}}}}|d}td||tddƒd||dt|dƒttƒtd|d|dƒƒ}|||t| |dƒt|d|tdƒdd||dƒS)a? 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 ééé) Úmaprrrr rr r r)ÚnÚlZnuÚrÚC©rú0lib/python3.7/site-packages/sympy/physics/sho.pyÚR_nls /0$rcCsd||tddƒ|S)a 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) r r)r)rrZhwrrrÚE_nlAsrN)Z __future__rrZ sympy.corerrrZsympy.functionsrrr r r rrrrrrÚs: