7^(h @ddlmZmZmZddlmZmZmZmZm Z dZ dZ y))SpiRational)assoc_laguerresqrtexp factorial factorial2c tt||||g\}}}}|dz}td|z|tddzzd||zdzzzt |dz ztt t d|zd|zzdz zz }|||zzt| |dzzzt|dz |tjzd|z|dzzzS)a= 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 ) maprrrr rr rrHalf)nlnurCs O/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/physics/sho.pyR_nlrs`a!QA'KAq"a AA da(1a.( )!a!eai. 81q59I I "Xz!A#!)a-0 1 3 A QV8CAqDM !.QAFF AbDAI"N NNc2d|z|ztddz|zS)aD 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) r r)r)rrhws rE_nlr@s"> aC!Ghq!n $b ((rN) sympy.corerrrsympy.functionsrrrr r rrrrrs&&LL8Ov)r