+ i B^RIHtHtHt^RIHtHtHtHtH t Rt Rt R#))SpiRational)assoc_laguerresqrtexp factorial factorial2c X\\WW#.4wrr#V^,p\^V,V\^^4,,^W,^,,,\ V^, 4,\\ 4\ ^V,^V,,^, 4,, 4pWCV,,\V)V^,,4,\V^, V\P,^V,V^,,4,#)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&&&& q/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/sho.pyR_nlrs`a!'KA" AA da(1a.( )!aeai. 81q59I I "Xz!A#!)a-0 1 3 A V8CAqDM !.QAFF AbDAI"N NNcR^V,V,\^^4,V,#)a 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)rrhws&&&rE_nlr@s"> aC!Ghq!n $b ((rN) sympy.corerrrsympy.functionsrrrr r rrrrrs&&LL8Ov)r