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| from sympy.core import S, pi, Rational | |
| from sympy.functions import assoc_laguerre, sqrt, exp, factorial, factorial2 | |
| def R_nl(n, l, nu, r): | |
| """ | |
| 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 | |
| """ | |
| n, l, nu, r = map(S, [n, l, nu, r]) | |
| # formula uses n >= 1 (instead of nodal n >= 0) | |
| n = n + 1 | |
| C = sqrt( | |
| ((2*nu)**(l + Rational(3, 2))*2**(n + l + 1)*factorial(n - 1))/ | |
| (sqrt(pi)*(factorial2(2*n + 2*l - 1))) | |
| ) | |
| return C*r**(l)*exp(-nu*r**2)*assoc_laguerre(n - 1, l + S.Half, 2*nu*r**2) | |
| def E_nl(n, l, hw): | |
| """ | |
| 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) | |
| """ | |
| return (2*n + l + Rational(3, 2))*hw | |